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CM& Morgan Claypool Publishers&SYNTHESIS LECTURES ONCOMPUTER MATHEMATICS AND STATISTICS

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9 781627 052375

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SYNTHESIS LECTURES ONMATHEMATICS AND STATISTICSMorgan Claypool Publishers&

w w w . m o r g a n c l a y p o o l . c o m

Steven G. Krantz, Series Editor

Series ISSN: 1938-1743

An Easy Path to Convex Analysis andApplicationsBoris S. Mordukhovich, Wayne State University

Nguyen Mau Nam, Portland State University

Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical

applications. It is now being taught at many universities and being used by researchers of different fields. As

convex analysis is the mathematical foundation for convex optimization, having deep knowledge of convex

analysis helps students and researchers apply its tools more effectively. The main goal of this book is to provide

an easy access to the most fundamental parts of convex analysis and its applications to optimization. Modern

techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis

and build the theory of generalized differentiation for convex functions and sets in finite dimensions. We also

present new applications of convex analysis to location problems in connection with many interesting geometric

problems such as the Fermat-Torricelli problem, the Heron problem, the Sylvester problem, and their

generalizations. Of course, we do not expect to touch every aspect of convex analysis, but the book consists

of sufficient material for a first course on this subject. It can also serve as supplemental reading material for a

course on convex optimization and applications.

An Easy Path to ConvexAnalysis and Applications

Boris S. MordukhovichNguyen Mau Nam

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AnEasy Path toConvex Analysis and Applications

Synthesis Lectures onMathematics and Statistics

EditorStevenG. Krantz,Washington University, St. Louis

An Easy Path to Convex Analysis and ApplicationsBoris S. Mordukhovich and Nguyen Mau Nam2014

Applications of Affine and Weyl GeometryEduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo2013

Essentials of Applied Mathematics for Engineers and Scientists, Second EditionRobert G. Watts2012

Chaotic Maps: Dynamics, Fractals, and Rapid FluctuationsGoong Chen and Yu Huang2011

Matrices in Engineering ProblemsMarvin J. Tobias2011

e Integral: A Crux for AnalysisSteven G. Krantz2011

Statistics is Easy! Second EditionDennis Shasha and Manda Wilson2010

Lectures on Financial Mathematics: Discrete Asset PricingGreg Anderson and Alec N. Kercheval2010

iii

Jordan Canonical Form: eory and PracticeSteven H. Weintraub2009

e Geometry of Walker ManifoldsMiguel Brozos-Vázquez, Eduardo García-Río, Peter Gilkey, Stana Nikčević, and RamónVázquez-Lorenzo2009

An Introduction to Multivariable MathematicsLeon Simon2008

Jordan Canonical Form: Application to Differential EquationsSteven H. Weintraub2008

Statistics is Easy!Dennis Shasha and Manda Wilson2008

A Gyrovector Space Approach to Hyperbolic GeometryAbraham Albert Ungar2008

Copyright © 2014 by Morgan & Claypool

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted inany form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotationsin printed reviews, without the prior permission of the publisher.

An Easy Path to Convex Analysis and Applications

Boris S. Mordukhovich and Nguyen Mau Nam

www.morganclaypool.com

ISBN: 9781627052375 paperbackISBN: 9781627052382 ebook

DOI 10.2200/S00554ED1V01Y201312MAS014

A Publication in the Morgan & Claypool Publishers seriesSYNTHESIS LECTURES ONMATHEMATICS AND STATISTICS

Lecture #14Series Editor: Steven G. Krantz, Washington University, St. LouisSeries ISSNSynthesis Lectures on Mathematics and StatisticsPrint 1938-1743 Electronic 1938-1751

www.morganclaypool.com

AnEasy Path toConvex Analysis and Applications

Boris S. MordukhovichWayne State University

Nguyen Mau NamPortland State University

SYNTHESIS LECTURES ONMATHEMATICS AND STATISTICS #14

CM&

cLaypoolMorgan publishers&

ABSTRACTConvex optimization has an increasing impact on many areas of mathematics, applied sciences,and practical applications. It is now being taught at many universities and being used by re-searchers of different fields. As convex analysis is the mathematical foundation for convex opti-mization, having deep knowledge of convex analysis helps students and researchers apply its toolsmore effectively. e main goal of this book is to provide an easy access to the most fundamentalparts of convex analysis and its applications to optimization. Modern techniques of variationalanalysis are employed to clarify and simplify some basic proofs in convex analysis and build thetheory of generalized differentiation for convex functions and sets in finite dimensions. We alsopresent new applications of convex analysis to location problems in connection with many in-teresting geometric problems such as the Fermat-Torricelli problem, the Heron problem, theSylvester problem, and their generalizations. Of course, we do not expect to touch every aspect ofconvex analysis, but the book consists of sufficient material for a first course on this subject. It canalso serve as supplemental reading material for a course on convex optimization and applications.

KEYWORDSAffine set, Carathéodory theorem, convex function, convex set, directional deriva-tive, distance function, Fenchel conjugate, Fermat-Torricelli problem, generalizeddifferentiation, Helly theorem, minimal time function, Nash equilibrium, normalcone, Radon theorem, optimal value function, optimization, smallest enclosing ballproblem, set-valued mapping, subdifferential, subgradient, subgradient algorithm,support function, Weiszfeld algorithm

vii

e first author dedicates this book to the loving memory of his fatherS M (1924–1993),

a kind man and a brave warrior.

e second author dedicates this book to the memory of his parents.

ix

ContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Convex Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Relative Interiors of Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5 e Distance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.6 Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Subdifferential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 Convex Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Normals to Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.3 Lipschitz Continuity of Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4 Subgradients of Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.5 Basic Calculus Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.6 Subgradients of Optimal Value Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.7 Subgradients of Support Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.8 Fenchel Conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.9 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822.10 Subgradients of Supremum Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.11 Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3 Remarkable Consequences of Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.1 Characterizations of Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.2 Carathéodory eorem and Farkas Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.3 Radon eorem and Helly eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

x

3.4 Tangents to Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.5 Mean Value eorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.6 Horizon Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.7 Minimal Time Functions and Minkowski Gauge . . . . . . . . . . . . . . . . . . . . . . . 1113.8 Subgradients of Minimal Time Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.9 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.10 Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4 Applications to Optimization and Location Problems . . . . . . . . . . . . . . . . . . . 1294.1 Lower Semicontinuity and Existence of Minimizers . . . . . . . . . . . . . . . . . . . . 1294.2 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.3 Subgradient Methods in Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . 1404.4 e Fermat-Torricelli Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.5 A Generalized Fermat-Torricelli Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.6 A Generalized Sylvester Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.7 Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Solutions andHints for Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Authors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

xi

PrefaceSome geometric properties of convex sets and, to a lesser extent, of convex functions hadbeen studied before the 1960s by many outstanding mathematicians, first of all by HermannMinkowski and Werner Fenchel. At the beginning of the 1960s convex analysis was greatly de-veloped in the works of R. Tyrrell Rockafellar and Jean-JacquesMoreau who initiated a systematicstudy of this new field. As a fundamental part of variational analysis, convex analysis contains ageneralized differentiation theory that can be used to study a large class of mathematical mod-els with no differentiability assumptions imposed on their initial data. e importance of convexanalysis for many applications in which convex optimization is the first to name has been wellrecognized. e presence of convexity makes it possible not only to comprehensively investigatequalitative properties of optimal solutions and derive necessary and sufficient conditions for opti-mality but also to develop effective numerical algorithms to solve convex optimization problems,even with nondifferentiable data. Convex analysis and optimization have an increasing impacton many areas of mathematics and applications including control systems, estimation and signalprocessing, communications and networks, electronic circuit design, data analysis and modeling,statistics, economics and finance, etc.

ere are several fundamental books devoted to different aspects of convex analysis andoptimization. Among themwemention “ConvexAnalysis” by Rockafellar [26], “ConvexAnalysisand Minimization Algorithms” (in two volumes) by Hiriart-Urruty and Lemaréchal [8] and itsabridge version [9], “Convex Analysis and Nonlinear Optimization” by Borwein and Lewis [4],“Introductory Lectures on Convex Optimization” by Nesterov [21], and “Convex Optimization”by Boyd and Vandenberghe [3] as well as other books listed in the bibliography below.

In this big picture of convex analysis and optimization, our book serves as a bridge forstudents and researchers who have just started using convex analysis to reach deeper topics in thefield. We give detailed proofs for most of the results presented in the book and also include manyfigures and exercises for better understanding the material. e powerful geometric approachdeveloped in modern variational analysis is adopted and simplified in the convex case in orderto provide the reader with an easy path to access generalized differentiation of convex objects infinite dimensions. In this way, the book also serves as a starting point for the interested reader tocontinue the study of nonconvex variational analysis and applications. It can be of interest fromthis viewpoint to experts in convex and variational analysis. Finally, the application part of thisbook not only concerns the classical topics of convex optimization related to optimality conditionsand subgradient algorithms but also presents some recent while easily understandable qualitativeand numerical results for important location problems.

xii PREFACE

e book consists of four chapters and is organized as follows. In Chapter 1 we studyfundamental properties of convex sets and functions while paying particular attention to classesof convex functions important in optimization. Chapter 2 is mainly devoted to developing basiccalculus rules for normals to convex sets and subgradients of convex functions that are in themainstream of convex theory. Chapter 3 concerns some additional topics of convex analysis thatare largely used in applications. Chapter 4 is fully devoted to applications of basic results of convexanalysis to problems of convex optimization and selected location problems considered from bothqualitative and numerical viewpoints. Finally, we present at the end of the book Solutions andHints to selected exercises.

Exercises are given at the end of each chapter while figures and examples are providedthroughout the whole text. e list of references contains books and selected papers, which areclosely related to the topics considered in the book and may be helpful to the reader for advancedstudies of convex analysis, its applications, and further extensions.

Since only elementary knowledge in linear algebra and basic calculus is required, this bookcan be used as a textbook for both undergraduate and graduate level courses in convex analysis andits applications. In fact, the authors have used these lecture notes for teaching such classes in theiruniversities as well as while visiting some other schools. We hope that the book will make convexanalysis more accessible to large groups of undergraduate and graduate students, researchers indifferent disciplines, and practitioners.

Boris S. Mordukhovich and Nguyen Mau NamDecember 2013

xiii

Acknowledgmentse first author acknowledges continued support by grants from the National Science Foundationand the second author acknowledges support by the Simons Foundation.

Boris S. Mordukhovich and Nguyen Mau NamDecember 2013

xv

List of Symbols

R the real numbersR D .�1;1� the extended real lineRC the nonnegative real numbersR> the positive real numbersN the positive integersco˝ convex hull of ˝aff˝ affine hull of ˝int˝ interior of ˝ri˝ relative interior of ˝span˝ linear subspace generated by ˝cone˝ cone generated by ˝K˝ convex cone generated by ˝dim˝ dimension of ˝˝ closure of ˝bd˝ boundary of ˝IB closed unit ballIB. NxI r/ closed ball with center Nx and radius rdomf domain of fepif epigraph of fgphF graph of mapping FLŒa; b� line connecting a and bd.xI˝/ distance from x to ˝˘.xI˝/ projection of x to ˝hx; yi inner product of x and yA� adjoint/transpose of linear mapping/matrix AN. NxI˝/ normal cone to ˝ at Nx

kerA kernel of linear mapping AD�F. Nx; Ny/ coderivative to F at . Nx; Ny/

T . NxI˝/ tangent cone to ˝ at Nx

F1 horizon/asymptotic cone of FT F

˝ minimal time function defined by constant dynamic F and target set ˝L.x; �/ Lagrangian�F Minkowski gauge of F

xvi LISTOF SYMBOLS

˘F .�I˝/ generalized projection defined by the minimal time functionGF . Nx; t/ generalized ball defined by dynamic F with center Nx and radius t

1

C H A P T E R 1

Convex Sets and Functionsis chapter presents definitions, examples, and basic properties of convex sets and functions inthe Euclidean space Rn and also contains some related material.

1.1 PRELIMINARIESWe start with reviewing classical notions and properties of the Euclidean space Rn. e proofsof the results presented in this section can be found in standard books on advanced calculus andlinear algebra.

Let us denote by Rn the set of all n�tuples of real numbers x D .x1; : : : ; xn/. en Rn isa linear space with the following operations:

.x1; : : : ; xn/C .y1; : : : ; yn/ D .x1 C y1; : : : ; xn C yn/;

�.x1; : : : ; xn/ D .�x1; : : : ; �xn/;

where .x1; : : : ; xn/; .y1; : : : ; yn/ 2 Rn and � 2 R. e zero element of Rn and the number zeroof R are often denoted by the same notation 0 if no confusion arises.

For any x D .x1; : : : ; xn/ 2 Rn, we identify it with the column vector x D Œx1; : : : ; xn�T ,

where the symbol “T ” stands for vector transposition. Given x D .x1; : : : ; xn/ 2 Rn and y D

.y1; : : : ; yn/ 2 Rn, the inner product of x and y is defined by

hx; yi WD

nXiD1

xiyi :

e following proposition lists some important properties of the inner product in Rn.

Proposition 1.1 For x; y; z 2 Rn and � 2 R, we have:(i) hx; xi � 0, and hx; xi D 0 if and only if x D 0.(ii) hx; yi D hy; xi.(iii) h�x; yi D �hx; yi.(iv) hx; y C zi D hx; yi C hx; zi.

e Euclidean norm of x D .x1; : : : ; xn/ 2 Rn is defined by

kxk WD

qx2

1 C : : :C x2n:

2 1. CONVEX SETSANDFUNCTIONS

It follows directly from the definition that kxk Dp

hx; xi.

Proposition 1.2 For any x; y 2 Rn and � 2 R, we have:(i) kxk � 0, and kxk D 0 if and only if x D 0.(ii) k�xk D j�j � kxk.(iii) kx C yk � kxk C kyk .the triangle inequality/.(iv) jhx; yij � kxk � kyk .the Cauchy-Schwarz inequality/.

Using the Euclidean norm allows us to introduce the balls in Rn, which can be used todefine other topological notions in Rn.

Definition 1.3 e centered at Nx with radius r � 0 and the of Rn

are defined, respectively, by

IB. NxI r/ WD˚x 2 Rn

ˇkx � Nxk � r

and IB WD

˚x 2 Rn

ˇkxk � 1

:

It is easy to see that IB D IB.0I 1/ and IB. NxI r/ D Nx C rIB .

Definition 1.4 Let˝ � Rn. en Nx is an of˝ if there is ı > 0 such that

IB. NxI ı/ � ˝:

e set of all interior points of˝ is denoted by int˝. Moreover,˝ is said to be if every point of˝ is its interior point.

We get that˝ is open if and only if for every Nx 2 ˝ there is ı > 0 such that IB. NxI ı/ � ˝.It is obvious that the empty set ; and the whole space Rn are open. Furthermore, any open ballB. NxI r/ WD fx 2 Rn

j kx � Nxk < rg centered at Nx with radius r is open.

Definition 1.5 A set˝ � Rn is if its complement˝c D Rnn˝ is open in Rn.

It follows that the empty set, the whole space, and any ball IB. NxI r/ are closed in Rn.

Proposition 1.6 (i)e union of any collection of open sets in Rn is open.(ii)e intersection of any finite collection of open sets in Rn is open.(iii)e intersection of any collection of closed sets in Rn is closed.(iv)e union of any finite collection of closed sets in Rn is closed.

Definition 1.7 Let fxkg be a sequence in Rn. We say that fxkg to Nx ifkxk � Nxk ! 0 as k ! 1. In this case we write

limk!1

xk D Nx:

1.1. PRELIMINARIES 3

is notion allows us to define the following important topological concepts for sets.

Definition 1.8 Let˝ be a nonempty subset of Rn. en:(i) e of ˝, denoted by ˝ or cl˝, is the collection of limits of all convergent sequences be-longing to˝.(ii)e of˝, denoted by bd˝, is the set˝ n int ˝.

We can see that the closure of˝ is the intersection of all closed sets containing˝ and thatthe interior of ˝ is the union of all open sets contained in ˝. It follows from the definition thatNx 2 ˝ if and only if for any ı > 0 we have IB. NxI ı/ \˝ ¤ ;. Furthermore, Nx 2 bd ˝ if and onlyif for any ı > 0 the closed ball IB. NxI˝/ intersects both sets ˝ and its complement ˝c .

Definition 1.9 Let fxkg be a sequence in Rn and let fk`g be a strictly increasing sequence of positiveintegers. en the new sequence fxk`

g is called a of fxkg.

We say that a set ˝ is bounded if it is contained in a ball centered at the origin with someradius r > 0, i.e., ˝ � IB.0I r/. us a sequence fxkg is bounded if there is r > 0 with

kxkk � r for all k 2 N :

e following important result is known as the Bolzano-Weierstrass theorem.

eorem 1.10 Any bounded sequence in Rn contains a convergent subsequence.

e next concept plays a very significant role in analysis and optimization.

Definition 1.11 We say that a set˝ is inRn if every sequence in˝ contains a subsequenceconverging to some point in˝.

e following result is a consequence of the Bolzano-Weierstrass theorem.

eorem 1.12 A subset ˝ of Rn is compact if and only if it is closed and bounded.

For subsets ˝, ˝1, and ˝2 of Rn and for � 2 R, we define the operations:

˝1 C˝2 WD˚x C y

ˇx 2 ˝1; y 2 ˝2

; �˝ WD

˚�x

ˇx 2 ˝

:

e next proposition can be proved easily.

Proposition 1.13 Let˝1 and˝2 be two subsets of Rn.(i) If˝1 is open or˝2 is open, then˝1 C˝2 is open.(ii) If˝1 is closed and˝2 is compact, then˝2 C˝2 is closed.


Recall now the notions of bounds for subsets of the real line.

Definition 1.14 LetD be a subset of the real line. A numberm 2 R is a ofD if wehave

x � m for all x 2 D:

If the setD has a lower bound, then it is . Similarly, a numberM 2 R is an ofD if

x � M for all x 2 D;

andD is if it has an upper bound. Furthermore, we say that the setD is if it is simultaneously bounded below and above.

Now we are ready to define the concepts of infimum and supremum of sets.

Definition 1.15 LetD � R be nonempty and bounded below. e infimum ofD, denoted by infD,is the greatest lower bound of D. When D is nonempty and bounded above, its supremum, denoted bysupD, is the least upper bound ofD. IfD is not bounded below .resp. above/, thenwe set infD WD �1

.resp. supD WD 1/. We also use the convention that inf ; WD 1 and sup ; WD �1.

e following fundamental axiom ensures that these notions are well-defined.CompletenessAxiom. For every nonempty subsetD ofR that is bounded above, the least upper boundofD exists as a real number.

Using the Completeness Axiom, it is easy to see that if a nonempty set is bounded below,then its greatest lower bound exists as a real number.

roughout the book we consider for convenience extended-real-valued functions, whichtake values in R WD .�1;1�. e usual conventions of extended arithmetics are that aC 1 D

1 for any a 2 R, 1 C 1 D 1, and t � 1 D 1 for t > 0.

Definition 1.16 Let f W ˝ ! R be an extended-real-valued function and let Nx 2 ˝ with f . Nx/ <

1. en f is at Nx if for any � > 0 there is ı > 0 such that

jf .x/ � f . Nx/j < � whenever kx � Nxk < ı; x 2 ˝:

We say that f is continuous on˝ if it is continuous at every point of˝.

It is obvious from the definition that if f W ˝ ! R is continuous at Nx (with f . Nx/ < 1),then it is finite on the intersection of˝ and a ball centered at Nx with some radius r > 0. Further-more, f W˝ ! R is continuous at Nx (with f . Nx/ < 1) if and only if for every sequence fxkg in˝converging to Nx the sequence ff .xk/g converges to f . Nx/.

1.2. CONVEX SETS 5

Definition 1.17 Let f W ˝ ! R and let Nx 2 ˝ with f . Nx/ < 1. We say that f has a - at Nx relative to˝ if there is ı > 0 such that

f .x/ � f . Nx/ for all x 2 IB. NxI ı/ \˝:

We also say that f has a / at Nx relative to˝ if

f .x/ � f . Nx/ for all x 2 ˝:

e notions of local and global maxima can be defined similarly.Finally in this section, we formulate a fundamental result of mathematical analysis and

optimization known as the Weierstrass existence theorem.

eorem1.18 Let f W˝ ! R be a continuous function, where˝ is a nonempty, compact subsetof Rn. en there exist Nx 2 ˝ and Nu 2 ˝ such that

f . Nx/ D inf˚f .x/

ˇx 2 ˝

and f . Nu/ D sup

˚f .x/

ˇx 2 ˝

:

In Section 4.1 we present some “unilateral” versions of eorem 1.18.

1.2 CONVEX SETSWe start the study of convexity with sets and then proceed to functions. Geometric ideas playan underlying role in convex analysis, its extensions, and applications. us we implement thegeometric approach in this book.

Given two elements a and b in Rn, define the interval/line segment

Œa; b� WD˚�aC .1 � �/b

ˇ� 2 Œ0; 1�

:

Note that if a D b, then this interval reduces to a singleton Œa; b� D fag.

Definition 1.19 A subset˝ of Rn is if Œa; b� � ˝ whenever a; b 2 ˝. Equivalently,˝ isconvex if �aC .1 � �/b 2 ˝ for all a; b 2 ˝ and � 2 Œ0; 1�.

Given !1; : : : ; !m 2 Rn, the element x DPm

iD1 �i!i , wherePm

iD1 �i D 1 and �i � 0 forsome m 2 N, is called a convex combination of !1; : : : ; !m.

Proposition 1.20 A subset˝ of Rn is convex if and only if it contains all convex combinations of itselements.


Figure 1.1: Convex set and nonconvex set.

Proof. e sufficient condition is trivial. To justify the necessity, we show by induction that anyconvex combination x D

PmiD1 �i!i of elements in˝ is an element of˝. is conclusion follows

directly from the definition form D 1; 2. Fix now a positive integerm � 2 and suppose that everyconvex combination of k 2 N elements from ˝, where k � m, belongs to ˝. Form the convexcombination

y WD

mC1XiD1

�i!i ;

mC1XiD1

�i D 1; �i � 0

and observe that if �mC1 D 1, then �1 D �2 D : : : D �m D 0, so y D !mC1 2 ˝. In the casewhere �mC1 < 1 we get the representations

mXiD1

�i D 1 � �mC1 andmX

iD1

�i

1 � �mC1

D 1;

which imply in turn the inclusion

z WD

mXiD1

�i

1 � �mC1

!i 2 ˝:

It yields therefore the relationships

y D .1 � �mC1/

mXiD1

�i

1 � �mC1

!i C �mC1!mC1 D .1 � �mC1/z C �mC1!mC1 2 ˝

and thus completes the proof of the proposition. �

Proposition 1.21 Let ˝1 be a convex subset of Rn and let ˝2 be a convex subset of Rp. en theCartesian product˝1 �˝2 is a convex subset of Rn

� Rp.

1.2. CONVEX SETS 7

Proof. Fix a D .a1; a2/; b D .b1; b2/ 2 ˝1 �˝2, and � 2 .0; 1/. en we have a1; b1 2 ˝1 anda2; b2 2 ˝2. e convexity of ˝1 and ˝2 gives us

�ai C .1 � �/bi 2 ˝i for i D 1; 2;

which implies therefore that

�aC .1 � �/b D��a1 C .1 � �/b1; �a2 C .1 � �/b2

�2 ˝1 �˝2:

us the Cartesian product ˝1 �˝2 is convex. �

Let us continue with the definition of affine mappings.

Definition 1.22 AmappingB W Rn! Rp is if there exist a linear mappingA W Rn

! Rp

and an element b 2 Rp such that B.x/ D A.x/C b for all x 2 Rn.

Every linear mapping is affine. Moreover, B W Rn! Rp is affine if and only if

B.�x C .1 � �/y/ D �B.x/C .1 � �/B.y/ for all x; y 2 Rn and � 2 R :

Now we show that set convexity is preserved under affine operations.

Proposition 1.23 Let B W Rn! Rp be an affine mapping. Suppose that˝ is a convex subset of Rn

and � is a convex subset of Rp. en B.˝/ is a convex subset of Rp and B�1.�/ is a convex subsetof Rn.

Proof. Fix any a; b 2 B.˝/ and � 2 .0; 1/. en a D B.x/ and b D B.y/ for x; y 2 ˝. Since˝is convex, we have �x C .1 � �/y 2 ˝. en

�aC .1 � �/b D �B.x/C .1 � �/B.y/ D B.�x C .1 � �/y/ 2 B.˝/;

which justifies the convexity of the image B.˝/.Taking now any x; y 2 B�1.�/ and � 2 .0; 1/, we get B.x/; B.y/ 2 �. is gives us

�B.x/C .1 � �/B.y/ D B��x C .1 � �/y

�2 �

by the convexity of �. us we have �x C .1 � �/y 2 B�1.�/, which verifies the convexity ofthe inverse image B�1.�/. �

Proposition 1.24 Let ˝1;˝2 � Rn be convex and let � 2 R. en both sets ˝1 C˝2 and �˝1

are also convex in Rn.

Proof. It follows directly from the definitions. �


Next we proceed with intersections of convex sets.

Proposition1.25 Let f˝˛g˛2I be a collection of convex subsets ofRn.enT

˛2I ˝˛ is also a convexsubset of Rn.

Proof. Taking any a; b 2T

˛2I ˝˛, we get that a; b 2 ˝˛ for all ˛ 2 I . e convexity of each˝˛ ensures that �aC .1 � �/b 2 ˝˛ for any � 2 .0; 1/. us �aC .1 � �/b 2

T˛2I ˝˛ and

the intersectionT

˛2I ˝˛ is convex. �

Definition 1.26 Let˝ be a subset of Rn. e of˝ is defined by

co˝ WD\ n

CˇC is convex and ˝ � C

o:

e next proposition follows directly from the definition and Proposition 1.25.

Figure 1.2: Nonconvex set and its convex hull.

Proposition 1.27 e convex hull co˝ is the smallest convex set containing˝.

Proof. e convexity of the set co˝ � ˝ follows from Proposition 1.25. On the other hand, forany convex set C that contains ˝ we clearly have co˝ � C , which verifies the conclusion. �

Proposition 1.28 For any subset˝ of Rn, its convex hull admits the representation

co˝ D

n mXiD1

�iai

ˇ mXiD1

�i D 1; �i � 0; ai 2 ˝; m 2 No:

1.2. CONVEX SETS 9

Proof. Denoting by C the right-hand side of the representation to prove, we obviously have˝ �

C . Let us check that the set C is convex. Take any a; b 2 C and get

a WD

pXiD1

˛iai ; b WD

qXj D1

j bj ;

where ai ; bj 2 ˝, ˛i ; j � 0 withPp

iD1 ˛i DPq

j D1 j D 1, and p; q 2 N. It follows easily thatfor every number � 2 .0; 1/, we have

�aC .1 � �/b D

pXiD1

�˛iai C

qXj D1

.1 � �/ j bj :

en the resulting equalitypX

iD1

�˛i C

qXj D1

.1 � �/ j D �

pXiD1

˛i C .1 � �/

qXj D1

j D 1

ensures that �aC .1 � �/b 2 C , and thus co˝ � C by the definition of co˝. Fix now any a DPmiD1 �iai 2 C with ai 2 ˝ � co˝ for i D 1; : : : ; m. Since the set co˝ is convex, we conclude

by Proposition 1.20 that a 2 co˝ and thus co˝ D C . �

Proposition 1.29 e interior int˝ and closure˝ of a convex set˝ � Rn are also convex.

Proof. Fix a; b 2 int˝ and � 2 .0; 1/. en find an open set V such that

a 2 V � ˝ and so �aC .1 � �/b 2 �V C .1 � �/b � ˝:

Since �V C .1 � �/b is open, we get �aC .1 � �/b 2 int˝, and thus the set int˝ is convex.To verify the convexity of ˝, we fix a; b 2 ˝ and � 2 .0; 1/ and then find sequences fakg

and fbkg in ˝ converging to a and b, respectively. By the convexity of ˝, the sequence f�ak C

.1 � �/bkg lies entirely in ˝ and converges to �aC .1 � �/b. is ensures the inclusion �aC

.1 � �/b 2 ˝ and thus justifies the convexity of the closure ˝. �

To proceed further, for any a; b 2 Rn, define the interval

Œa; b/ WD˚�aC .1 � �/b

ˇ� 2 .0; 1�

:

We can also define the intervals .a; b� and .a; b/ in a similar way.

Lemma 1.30 For a convex set ˝ � Rn with nonempty interior, take any a 2 int˝ and b 2 ˝.en Œa; b/ � int˝.


Proof. Since b 2 ˝, for any � > 0, we have b 2 ˝ C �IB . Take now � 2 .0; 1� and let x� WD

�aC .1 � �/b. Choosing � > 0 such that aC �2 � �

�IB � ˝ gives us

x� C �IB D �aC .1 � �/b C �IB

� �aC .1 � �/�˝ C �IB

�C �IB

D �aC .1 � �/˝ C .1 � �/�IB C �IB

� �haC �

2 � �

�IB

iC .1 � �/˝

� �˝ C .1 � �/˝ � ˝:

is shows that x� 2 int˝ and thus verifies the inclusion Œa; b/ � int˝. �

Now we establish relationships between taking the interior and closure of convex sets.

Proposition 1.31 Let˝ � Rn be a convex set with nonempty interior. en we have:(i) int˝ D ˝ and .ii/ int˝ D int˝.

Proof. (i) Obviously, int˝ � ˝. For any b 2 ˝ and a 2 int˝, define the sequence fxkg by

xk WD1

kaC

�1 �

1

k

�b; k 2 N:

Lemma 1.30 ensures that xk 2 int˝. Since xk ! b, we have b 2 int˝ and thus verify (i).(ii) Since the inclusion int˝ � int˝ is obvious, it remains to prove the opposite inclusionint˝ � int˝. To proceed, fix any b 2 int˝ and a 2 int˝. If � > 0 is sufficiently small, then

c WD b C �.b � a/ 2 ˝, and hence b D�

1C �aC

1

1C �c 2 .a; c/ � int˝;

which verifies that int˝ � int˝ and thus completes the proof. �

1.3 CONVEXFUNCTIONSis section collects basic facts about general (extended-real-valued) convex functions includingtheir analytic and geometric characterizations, important properties as well as their specificationsfor particular subclasses. We also define convex set-valued mappings and use them to study a re-markable class of optimal value functions employed in what follows.

Definition 1.32 Let f W ˝ ! R be an extended-real-valued function define on a convex set ˝ �

Rn. en the function f is on˝ if

f��x C .1 � �/y

�� f .x/C .1 � �/f .y/ for all x; y 2 ˝ and � 2 .0; 1/: (1.1)

1.3. CONVEXFUNCTIONS 11

If the inequality in (1.1) is strict for x ¤ y, then f is on˝.

Given a function f W ˝ ! R, the extension of f to Rn is defined by

ef .x/ WD

(f .x/ if x 2 ˝;

1 otherwise:

Obviously, if f is convex on ˝, where ˝ is a convex set, then ef is convex on Rn. Furthermore,if f W Rn

! R is a convex function, then it is also convex on every convex subset of Rn. isallows to consider without loss of generality extended-real-valued convex functions on the wholespace Rn.

Figure 1.3: Convex function and nonconvex function.

Definition 1.33 e and of f W Rn! R are defined, respectively, by

domf WD˚x 2 Rn

ˇf .x/ < 1

and

epif WD˚.x; t/ 2 Rn

� Rˇx 2 Rn; t � f .x/

D

˚.x; t/ 2 Rn

� Rˇx 2 domf; t � f .x/

:

Let us illustrate the convexity of functions by examples.

Example 1.34 e following functions are convex:(i) f .x/ WD ha; xi C b for x 2 Rn, where a 2 Rn and b 2 R.(ii) g.x/ WD kxk for x 2 Rn.(iii) h.x/ WD x2 for x 2 R.


Indeed, the function f in (i) is convex since

f��x C .1 � �/y

�D ha; �x C .1 � �/yi C b D �ha; xi C .1 � �/ha; yi C b

D �.ha; xi C b/C .1 � �/.ha; yi C b/

D �f .x/C .1 � �/f .y/ for all x; y 2 Rn and � 2 .0; 1/:

e function g in (ii) is convex since for x; y 2 Rn and � 2 .0; 1/, we have

g��x C .1 � �/y

�D k�x C .1 � �/yk � �kxk C .1 � �/kyk D �g.x/C .1 � �/g.y/;

which follows from the triangle inequality and the fact that k˛uk D j˛j � kuk whenever ˛ 2 Rand u 2 Rn.e convexity of the simplest quadratic function h in (iii) follows from amore generalresult for the quadratic function on Rn given in the next example.

Example1.35 LetA be an n � n symmetricmatrix. It is called positive semidefinite if hAu; ui � 0

for all u 2 Rn. Let us check that A is positive semidefinite if and only if the function f W Rn! R

defined byf .x/ WD

1

2hAx; xi; x 2 Rn;

is convex. Indeed, a direct calculation shows that for any x; y 2 Rn and � 2 .0; 1/ we have

�f .x/C .1 � �/f .y/ � f��x C .1 � �/y

�D1

2�.1 � �/hA.x � y/; x � yi: (1.2)

If the matrix A is positive semidefinite, then hA.x � y/; x � yi � 0, so the function f is convexby (1.2). Conversely, assuming the convexity of f and using equality (1.2) for x D u and y D 0

verify that A is positive semidefinite.

e following characterization of convexity is known as the Jensen inequality.

eorem 1.36 A function f W Rn! R is convex if and only if for any numbers �i � 0 as i D

1; : : : ; m withPm

iD1 �i D 1 and for any elements xi 2 Rn, i D 1; : : : ; m, it holds that

f� mX

iD1

�ixi

��

mXiD1

�if .xi /: (1.3)

Proof. Since (1.3) immediately implies the convexity of f , we only need to prove that any convexfunction f satisfies the Jensen inequality (1.3). Arguing by induction and taking into account thatfor m D 1 inequality (1.3) holds trivially and for m D 2 inequality (1.3) holds by the definitionof convexity, we suppose that it holds for an integer m WD k with k � 2. Fix numbers �i � 0,i D 1; : : : ; k C 1, with

PkC1iD1 �i D 1 and elements xi 2 Rn, i D 1; : : : ; k C 1. We obviously have

the relationshipkX

iD1

�i D 1 � �kC1:


If �kC1 D 1, then �i D 0 for all i D 1; : : : ; k and (1.3) obviously holds for m WD k C 1 in thiscase. Supposing now that 0 � �kC1 < 1, we get

kXiD1

�i

1 � �kC1

D 1

and by direct calculations based on convexity arrive at

f� kC1X

iD1

�ixi

�D f

h.1 � �kC1/

PkiD1 �ixi

1 � �kC1

C �kC1xkC1

i� .1 � �kC1/f

�PkiD1 �ixi

1 � �kC1

�C �kC1f .xkC1/

D .1 � �kC1/f� kX

iD1

�i

1 � �kC1

xi

�C �kC1f .xkC1/

� .1 � �kC1/

kXiD1

�i

1 � �kC1

f .xi /C �kC1f .xkC1/

D

kC1XiD1

�if .xi /:

is justifies inequality (1.3) and completes the proof of the theorem. �

e next theorem gives a geometric characterization of the function convexity via the con-vexity of the associated epigraphical set.

eorem 1.37 A function f W Rn! R is convex if and only if its epigraph epi f is a convex

subset of the product space Rn� R.

Proof. Assuming that f is convex, fix pairs .x1; t1/; .x2; t2/ 2 epif and a number � 2 .0; 1/.en we have f .xi / � ti for i D 1; 2. us the convexity of f ensures that

f��x1 C .1 � �/x2

�� f .x1/C .1 � �/f .x2/ � �t1 C .1 � �/t2:

is implies therefore that

�.x1; t1/C .1 � �/.x2; t2/ D .�x1 C .1 � �/x2; �t1 C .1 � �/t2/ 2 epif;

and thus the epigraph epi f is a convex subset of Rn� R.

Conversely, suppose that the set epi f is convex and fix x1; x2 2 domf and a number� 2 .0; 1/. en .x1; f .x1//; .x2; f .x2// 2 epif . is tells us that�

�x1 C .1 � �/x2; �f .x1/C .1 � �/f .x2/�

D ��x1; f .x1/

�C .1 � �/

�x2; f .x2/

�2 epif


and thus implies the inequality

f��x1 C .1 � �/x2

�� f .x1/C .1 � �/f .x2/;

which justifies the convexity of the function f . �

Figure 1.4: Epigraphs of convex function and nonconvex function.

Now we show that convexity is preserved under some important operations.

Proposition 1.38 Let fi W Rn! R be convex functions for all i D 1; : : : ; m. en the following

functions are convex as well:(i)e multiplication by scalars �f for any � > 0.(ii)e sum function

PmiD1 fi .

(iii)e maximum function max1�i�m fi .

Proof. e convexity of �f in (i) follows directly from the definition. It is sufficient to prove (ii)and (iii) for m D 2, since the general cases immediately follow by induction.(ii) Fix any x; y 2 Rn and � 2 .0; 1/. en we have�

f1 C f2

��x C .1 � �/y

�D f1

��x C .1 � �/y

�C f2

��x C .1 � �/y

�� f1.x/C .1 � �/f1.y/C �f2.x/C .1 � �/f2.y/

D �.f1 C f2/.x/C .1 � �/.f1 C f2/.y/;

which verifies the convexity of the sum function f1 C f2.(iii) Denote g WD maxff1; f2g and get for any x; y 2 Rn and � 2 .0; 1/ that

fi

��x C .1 � �/y

�� fi .x/C .1 � �/fi .y/ � �g.x/C .1 � �/g.y/

for i D 1; 2. is directly implies that

g��x C .1 � �/y

�D max

˚f1

��x C .1 � �/y

�; f2

��x C .1 � �/y

�� g.x/C .1 � �/g.y/;

which shows that the maximum function g.x/ D maxff1.x/; f2.x/g is convex. �


e next result concerns the preservation of convexity under function compositions.

Proposition 1.39 Let f W Rn! R be convex and let ' W R ! R be nondecreasing and convex on

a convex set containing the range of the function f . en the composition ' ı f is convex.

Proof. Take any x1; x2 2 Rn and � 2 .0; 1/. en we have by the convexity of f that

f��x1 C .1 � �/x2

�� f .x1/C .1 � �/f .x2/:

Since ' is nondecreasing and it is also convex, it follows that�' ı f

��x1 C .1 � �/x2

�D '

�f

��x1 C .1 � �/x2

�� '

��f .x1/C .1 � �/f .x2/

�� '

�f .x1/

�C .1 � �/'

�f .x2/

�D �.' ı f /.x1/C .1 � �/.' ı f /.x2/;

which verifies the convexity of the composition ' ı f . �

Now we consider the composition of a convex function and an affine mapping.

Proposition 1.40 Let B W Rn! Rp be an affine mapping and let f W Rp

! R be a convex func-tion. en the composition f ı B is convex.

Proof. Taking any x; y 2 Rn and � 2 .0; 1/, we have�f ı B

��x C .1 � �/y

�D f .B.�x C .1 � �/y// D f

��B.x/C .1 � �/B.y/

�� f

�B.x/

�C .1 � �/f

�B.y/

�D �.f ı B/.x/C .1 � �/.f ı B/.y/

and thus justify the convexity of the composition f ı B . �

e following simple consequence of Proposition 1.40 is useful in applications.

Corollary 1.41 Let f W Rn! R be a convex function. For any Nx; d 2 Rn, the function 'W R ! R

defined by '.t/ WD f . Nx C td / is convex as well. Conversely, if for every Nx; d 2 Rn the function 'defined above is convex, then f is also convex.

Proof. Since B.t/ D Nx C td is an affine mapping, the convexity of ' immediately follows fromProposition 1.40. To prove the converse implication, take any x1; x2 2 Rn, � 2 .0; 1/ and letNx WD x2, d WD x1 � x2. Since '.t/ D f . Nx C td / is convex, we have

f��x1 C .1 � �/x2

�D f

�x2 C �.x1 � x2/

�D '.�/ D '

��.1/C .1 � �/.0/

�� '.1/C .1 � �/'.0/ D �f .x1/C .1 � �/f .x2/;

which verifies the convexity of the function f . �


e next proposition is trivial while useful in what follows.

Proposition 1.42 Let f W Rn� Rp

! R be convex. For . Nx; Ny/ 2 Rn� Rp, the functions '.y/ WD

f . Nx; y/ and .x/ WD f .x; Ny/ are also convex.

Now we present an important extension of Proposition 1.38(iii).

Proposition 1.43 Let fi W Rn! R for i 2 I be a collection of convex functions with a nonempty

index set I . en the supremum function f .x/ WD supi2I fi .x/ is convex.

Proof. Fix x1; x2 2 Rn and � 2 .0; 1/. For every i 2 I , we have

fi

��x1 C .1 � �/x2

�� fi .x1/C .1 � �/fi .x2/ � �f .x1/C .1 � �/f .x2/;

which implies in turn that

f��x1 C .1 � �/x2

�D sup

i2I

fi

��x1 C .1 � �/x2

�� f .x1/C .1 � �/f .x2/:

is justifies the convexity of the supremum function. �

Our next intention is to characterize convexity of smooth functions of one variable. To pro-ceed, we begin with the following lemma.

Lemma 1.44 Given a convex function f W R ! R, assume that its domain is an open interval I .For any a; b 2 I and a < x < b, we have the inequalities

f .x/ � f .a/

x � a�f .b/ � f .a/

b � a�f .b/ � f .x/

b � x:

Proof. Fix a; b; x as above and form the numbers t WDx � a

b � a2 .0; 1/. en

f .x/ D f�aC .x � a/

�D f

�aC

x � a

b � a

�b � a

��D f

�aC t.b � a/

�D f

�tb C .1 � t/a

�:

is gives us the inequalities f .x/ � tf .b/C .1 � t/f .a/ and

f .x/ � f .a/ � tf .b/C .1 � t/f .a/ � f .a/ D t�f .b/ � f .a/

�Dx � a

b � a

�f .b/ � f .a/

�;

which can be equivalently written as

f .x/ � f .a/

x � a�f .b/ � f .a/

b � a:


Similarly, we have the estimate

f .x/ � f .b/ � tf .b/C .1 � t /f .a/ � f .b/ D .t � 1/�f .b/ � f .a/

�Dx � b

b � a

�f .b/ � f .a/

�;

which finally implies thatf .b/ � f .a/

b � a�f .b/ � f .x/

b � x

and thus completes the proof of the lemma. �

eorem1.45 Suppose that f W R ! R is differentiable on its domain, which is an open intervalI . en f is convex if and only if the derivative f 0 is nondecreasing on I .

Proof. Suppose that f is convex and fix a < b with a; b 2 I . By Lemma 1.44, we have

f .x/ � f .a/

x � a�f .b/ � f .a/

b � a

for any x 2 .a; b/. is implies by the derivative definition that

f 0.a/ �f .b/ � f .a/

b � a:

Similarly, we arrive at the estimate

f .b/ � f .a/

b � a� f 0.b/

and conclude that f 0.a/ � f 0.b/, i.e., f 0 is a nondecreasing function.To prove the converse, suppose that f 0 is nondecreasing on I and fix x1 < x2 with x1; x2 2

I and t 2 .0; 1/. enx1 < xt < x2 for xt WD tx1 C .1 � t/x2:

By the mean value theorem, we find c1; c2 such that x1 < c1 < xt < c2 < x2 and

f .xt / � f .x1/ D f 0.c1/.xt � x1/ D f 0.c1/.1 � t /.x2 � x1/;

f .xt / � f .x2/ D f 0.c2/.xt � x2/ D f 0.c2/t.x1 � x2/:

is can be equivalently rewritten as

tf .xt / � tf .x1/ D f 0.c1/t.1 � t /.x2 � x1/;

.1 � t /f .xt / � .1 � t/f .x2/ D f 0.c2/t.1 � t/.x1 � x2/:

Since f 0.c1/ � f 0.c2/, adding these equalities yields

f .xt / � tf .x1/C .1 � t/f .x2/;

which justifies the convexity of the function f . �


Corollary 1.46 Let f W R ! R be twice differentiable on its domain, which is an open interval I .en f is convex if and only if f 00.x/ � 0 for all x 2 I .

Proof. Since f 00.x/ � 0 for all x 2 I if and only if the derivative function f 0 is nondecreasing onthis interval. en the conclusion follows directly from eorem 1.45. �

Example 1.47 Consider the function

f .x/ WD

8<:1

xif x > 0;

1 otherwise:

To verify its convexity, we get that f 00.x/ D2

x3> 0 for all x belonging to the domain of f , which

is I D .0;1/. us this function is convex on R by Corollary 1.46.

Next we define the notion of set-valued mappings (or multifunctions), which plays an im-portant role in modern convex analysis, its extensions, and applications.

Definition 1.48 We say that F is a - between Rn and Rp and denote it byF W Rn

!! Rp if F.x/ is a subset of Rp for every x 2 Rn. e and of F are defined,respectively, by

domF WD˚x 2 Rn

ˇF.x/ ¤ ;

and gphF WD

˚.x; y/ 2 Rn

� Rpˇy 2 F.x/

:

Any single-valued mapping F W Rn! Rp is a particular set-valued mapping where the set

F.x/ is a singleton for every x 2 Rn. It is essential in the following definition that the mappingF is set-valued.

Definition 1.49 Let F W Rn!! Rp and let ' W Rn

� Rp! R. e or

associated with F and ' is defined by

�.x/ WD inf˚'.x; y/

ˇy 2 F.x/

; x 2 Rn : (1.4)

roughout this section we assume that �.x/ > �1 for every x 2 Rn.

Proposition 1.50 Assume that 'W Rn� Rp

! R is a convex function and that F W Rn!! Rp is of

convex graph. en the optimal value function � in (1.4) is convex.


Figure 1.5: Graph of set-valued mapping.

Proof. Take x1; x2 2 dom�, � 2 .0; 1/. For any � > 0, find yi 2 F.xi / such that

'.xi ; yi / < �.xi /C � for i D 1; 2:

It directly implies the inequalities

�'.x1; y1/ < ��.x1/C ��; .1 � �/'.x2; y2/ < .1 � �/�.x2/C .1 � �/�:

Summing up these inequalities and employing the convexity of ' yield

'��x1 C .1 � �/x2; �y1 C .1 � �/y2

�� '.x1; y1/C .1 � �/'.x2; y2/

< ��.x1/C .1 � �/�.x2/C �:

Furthermore, the convexity of gphF gives us��x1 C .1 � �/x2; �y1 C .1 � �/y2

�D �.x1; y1/C .1 � �/.x2; y2/ 2 gphF;

and therefore �y1 C .1 � �/y2 2 F.�x1 C .1 � �/x2/. is implies that

��x1 C .1 � �/x2

�� '

��x1 C .1 � �/x2; �y1 C .1 � �/y2

�< ��.x1/C .1 � �/�.x2/C �:

Letting finally � ! 0 ensures the convexity of the optimal value function �. �

Using Proposition 1.50, we can verify convexity in many situations. For instance, giventwo convex functions fi W Rn

! R, i D 1; 2, let '.x; y/ WD f1.x/C y and F.x/ WD Œf2.x/;1/.en the function ' is convex and set gphF D epif2 is convex as well, and hence we justify theconvexity of the sum

�.x/ D infy2F .x/

'.x; y/ D f1.x/C f2.x/:


Another example concerns compositions. Let f W Rp! R be convex and let B W Rn

! Rp beaffine. Define '.x; y/ WD f .y/ and F.x/ WD fB.x/g. Observe that ' is convex while F is of con-vex graph. us we have the convex composition

�.x/ D infy2F .x/

'.x; y/ D f�B.x/

�; x 2 Rn :

e examples presented above recover the results obtained previously by direct proofs. Now weestablish via Proposition 1.50 the convexity of three new classes of functions.

Proposition 1.51 Let ' W Rp! R be convex and let B W Rp

! Rn be affine. Consider the set-valued inverse image mapping B�1W Rn

!! Rp, define

f .x/ WD inf˚'.y/

ˇy 2 B�1.x/

; x 2 Rn;

and suppose that f .x/ > �1 for all x 2 Rn. en f is a convex function.

Proof. Let '.x; y/ � '.y/ and F.x/ WD B�1.x/. en the set

gphF D˚.u; v/ 2 Rn

� RpˇB.v/ D u

is obviously convex. Since we have the representation

f .x/ D infy2F .x/

'.y/; x 2 Rn;

the convexity of f follows directly from Proposition 1.50. �

Proposition 1.52 For convex functions f1; f2 W Rn! R, define the

.f1 ˚ f2/.x/ WD inf˚f1.x1/C f2.x2/

ˇx1 C x2 D x

and suppose that .f1 ˚ f2/.x/ > �1 for all x 2 Rn. en f1 ˚ f2 is also convex.

Proof. Define ' W Rn� Rn

! R by '.x1; x2/ WD f1.x1/C f2.x2/ and B W Rn� Rn

! Rn byB.x1; x2/ WD x1 C x2. We have

inf˚'.x1; x2/

ˇ.x1; x2/ 2 B�1.x/

D .f1 ˚ f2/.x/ for all x 2 Rn;

which implies the convexity of .f1 ˚ f2/ by Proposition 1.51. �

1.4. RELATIVE INTERIORSOFCONVEX SETS 21

Definition 1.53 A function g W Rp! R is called if�

xi � yi for all i D 1; : : : ; p�

H)�g.x1; : : : ; xp/ � g.y1; : : : ; yp/

�:

Now we are ready to present the final consequence of Proposition 1.50 in this section thatinvolves the composition.

Proposition 1.54 Define h W Rn! Rp by h.x/ WD .f1.x/; : : : ; fp.x//, where fi W Rn

! R fori D 1; : : : ; p are convex functions. Suppose that g W Rp

! R is convex and nondecreasing componen-twise. en the composition g ı h W Rn

! R is a convex function.

Proof. Let F W Rn!! Rp be a set-valued mapping defined by

F.x/ WD Œf1.x/;1/ � Œf2.x/;1/ � : : : � Œfp.x/;1/:

en the graph of F is represented by

gphF D˚.x; t1; : : : ; tp/ 2 Rn

� Rpˇti � fi .x/

:

Since all fi are convex, the set gphF is convex as well. Define further ' W Rn� Rp

! R by'.x; y/ WD g.y/ and observe, since g is increasing componentwise, that

inf˚'.x; y/

ˇy 2 F.x/

D g

�f1.x/; : : : ; fp.x/

�D .g ı h/.x/;

which ensures the convexity of the composition g ı h by Proposition 1.50. �

1.4 RELATIVE INTERIORSOFCONVEX SETSWe begin this section with the definition and properties of affine sets. Given two elements a andb in Rn, the line connecting them is

LŒa; b� WD˚�aC .1 � �/b

ˇ� 2 R

:

Note that if a D b, then LŒa; b� D fag.

Definition 1.55 A subset˝ of Rn is if for any a; b 2 ˝ we have LŒa; b� � ˝.

For instance, any point, line, and plane in R3 are affine sets. e empty set and the wholespace are always affine. It follows from the definition that the intersection of any collection ofaffine sets is affine. is leads us to the construction of the affine hull of a set.

Definition 1.56 e of a set˝ � Rn is

aff˝ WD\ ˚

CˇC is affine and ˝ � C

:


An element x in Rn of the form

x D

mXiD1

�i!i withmX

iD1

�i D 1; m 2 N;

is called an affine combination of !1; : : : ; !m. e proof of the next proposition is straightforwardand thus is omitted.

Proposition 1.57 e following assertions hold:(i) A set˝ is affine if and only if˝ contains all affine combinations of its elements.(ii) Let˝,˝1, and˝2 be affine subsets of Rn. en the sum˝1 C˝2 and the scalar product �˝ forany � 2 R are also affine subsets of Rn.(iii) Let B W Rn

! Rp be an affine mapping. If ˝ is an affine subset of Rn and � is an affine subsetof Rp, then the image B.˝/ is an affine subset of Rp and the inverse image B�1.�/ is an affine subsetof Rn.(iv) Given˝ � Rn, its affine hull is the smallest affine set containing˝. We have

aff˝ D

n mXiD1

�i!i

ˇ mXiD1

�i D 1; !i 2 ˝; m 2 No:

(v) A set˝ is a .linear/ subspace if and only if˝ is an affine set containing the origin.

Next we consider relationships between affine sets and (linear) subspaces.

Lemma 1.58 A nonempty subset ˝ of Rn is affine if and only if ˝ � ! is a subspace of Rn for any! 2 ˝.

Proof. Suppose that a nonempty set ˝ � Rn is affine. It follows from Proposition 1.57(v) that˝ � ! is a subspace for any ! 2 ˝. Conversely, fix ! 2 ˝ and suppose that˝ � ! is a subspacedenoted by L. en the set ˝ D ! C L is obviously affine. �

e preceding lemma leads to the following notion.

Definition 1.59 An affine set˝ is to a subspace L if˝ D ! C L for some ! 2 ˝.

e next proposition justifies the form of the parallel subspace.

Proposition 1.60 Let˝ be a nonempty, affine subset of Rn. en it is parallel to the unique subspaceL of Rn given by L D ˝ �˝.


Proof. Given a nonempty, affine set˝, fix! 2 ˝ and come up to the linear subspaceL WD ˝ � !

parallel to ˝. To justify the uniqueness of such L, take any !1; !2 2 ˝ and any subspacesL1; L2 � Rn such that ˝ D !1 C L1 D !2 C L2. en L1 D !2 � !1 C L2. Since 0 2 L1, wehave !1 � !2 2 L2. is implies that !2 � !1 2 L2 and thus L1 D !2 � !1 C L2 � L2. Simi-larly, we get L2 � L1, which justifies that L1 D L2.

It remains to verify the representation L D ˝ �˝. Let ˝ D ! C L with the unique sub-space L and some ! 2 ˝. en L D ˝ � ! � ˝ �˝. Fix any x D u � ! with u; ! 2 ˝ andobserve that˝ � ! is a subspace parallel to˝. Hence˝ � ! D L by the uniqueness of L provedabove. is ensures that x 2 ˝ � ! D L and thus ˝ �˝ � L. �

e uniqueness of the parallel subspace shows that the next notion is well defined.

Definition 1.61 e ; ¤ ˝ � Rn is the dimension of the linearsubspace parallel to˝. Furthermore, the ; ¤ ˝ � Rn is the dimensionof its affine hull aff˝.

To proceed further, we need yet another definition important in what follows.

Definition 1.62 e elements v0; : : : ; vm in Rn,m � 1, are ifh mXiD0

�ivi D 0;

mXiD0

�i D 0i

H)��i D 0 for all i D 0; : : : ; m

�:

It is easy to observe the following relationship with the linear independence.

Proposition 1.63 e elements v0; : : : ; vm in Rn are affinely independent if and only if the elementsv1 � v0; : : : ; vm � v0 are linearly independent.

Proof. Suppose that v0; : : : ; vm are affinely independent and consider the systemmX

iD1

�i .vi � v0/ D 0; i.e., �0v0 C

mXiD1

�ivi D 0;

where �0 WD �Pm

iD1 �i . Since the elements v0; : : : ; vm are affinely independent andPm

iD0 �i D

0, we have that �i D 0 for all i D 1; : : : ; m. us v1 � v0; : : : ; vm � v0 are linearly independent.e proof of the converse statement is straightforward. �

Recall that the span of some set C , spanC , is the linear subspace generated by C .

Lemma 1.64 Let˝ WD afffv0; : : : ; vmg, where vi 2 Rn for all i D 0; : : : ; m. en the span of theset fv1 � v0; : : : ; vm � v0g is the subspace parallel to˝.


Proof. Denote by L the subspace parallel to ˝. en ˝ � v0 D L and therefore vi � v0 2 L forall i D 1; : : : ; m. is gives

span˚vi � v0

ˇi D 1; : : : ; m

� L:

To prove the converse inclusion, fix any v 2 L and get v C v0 2 ˝. us we have

v C v0 D

mXiD0

�ivi ;

mXiD0

�i D 1:

is implies the relationship

v D

mXiD1

�i .vi � v0/ 2 span˚vi � v0

ˇi D 1; : : : ; m

;

which justifies the converse inclusion and hence completes the proof. �

e proof of the next proposition is rather straightforward.

Proposition 1.65 e elements v0; : : : ; vm are affinely independent inRn if and only if its affine hull˝ WD afffv0; : : : ; vmg ism-dimensional.

Proof. Suppose that v0; : : : ; vm are affinely independent. en Lemma 1.64 tells us that thesubspace L WD spanfvi � v0 j i D 1; : : : ; mg is parallel to ˝. e linear independence of v1 �

v0; : : : ; vm � v0 by Proposition 1.63 means that the subspace L is m-dimensional and so is ˝.e proof of the converse statement is also straightforward. �

Affinely independent systems lead us to the construction of simplices.

Definition 1.66 Let v0; : : : ; vm be affinely independent in Rn. en the set

�m WD co˚vi

ˇi D 0; : : : ; m

is called anm- in Rn with the vertices vi , i D 0; : : : ; m.

An important role of simplex vertices is revealed by the following proposition.

Proposition 1.67 Consider anm-simplex�m with vertices vi for i D 0; : : : ; m. For every v 2 �m,there is a unique element .�0; : : : ; �m/ 2 RmC1

C such that

v D

mXiD0

�ivi ;

mXiD0

�i D 1:


Proof. Let .�0; : : : ; �m/ 2 RmC1C and .�0; : : : ; �m/ 2 RmC1

C satisfy

v D

mXiD0

�ivi D

mXiD0

�ivi ;

mXiD0

�i D

mXiD0

�i D 1:

is immediately implies the equalitiesmX

iD0

.�i � �i /vi D 0;

mXiD0

.�i � �i / D 0:

Since v0; : : : ; vm are affinely independent, we have �i D �i for i D 0; : : : ; m. �

Now we are ready to define a major notion of relative interiors of convex sets.

Definition 1.68 Let ˝ be a convex set. We say that an element v 2 ˝ belongs to the ri˝ of˝ if there exists � > 0 such that IB.vI �/ \ aff˝ � ˝.

We begin the study of relative interiors with the following lemma.

Lemma 1.69 Any linear mapping A W Rn! Rp is continuous.

Proof. Let fe1; : : : ; eng be the standard orthonormal basis of Rn and let vi WD A.ei /, i D 1; : : : ; n.For any x 2 Rn with x D .x1; : : : ; xn/, we have

A.x/ D A� nX

iD1

xiei

�D

nXiD1

xiA.ei / D

nXiD1

xivi :

en the triangle inequality and the Cauchy-Schwarz inequality give us

kA.x/k �

nXiD1

jxi jkvik �

vuut nXiD1

jxi j2

vuut nXiD1

kvik2 D Mkxk;

whereM WD

qPniD1 kvik

2. It follows furthermore that

kA.x/ � A.y/k D kA.x � y/k � Mkx � yk for all x; y 2 Rn;

which implies the continuity of the mapping A. �

e next proposition plays an essential role in what follows.

Proposition 1.70 Let �m be anm-simplex in Rn with somem � 1. en ri�m ¤ ;.


Proof. Consider the vertices v0; : : : ; vm of the simplex �m and denote

v WD1

mC 1

mXiD0

vi :

We prove the proposition by showing that v 2 ri�m. Define

L WD span˚vi � v0

ˇi D 1; : : : ; m

and observe that L is the m-dimensional subspace of Rn parallel to aff�m D afffv0; : : : ; vmg. Itis easy to see that for every x 2 L there is a unique collection .�0; : : : ; �m/ 2 RmC1 with

x D

mXiD0

�ivi ;

mXiD0

�i D 0:

Consider the mapping A W L ! RmC1, which maps x to the corresponding coefficients.�0; : : : ; �m/ 2 RmC1 as above. en A is linear, and hence it is continuous by Lemma 1.69.Since A.0/ D 0, we can choose ı > 0 such that

kA.u/k <1

mC 1whenever kuk � ı:

Let us now show that .v C ıIB/ \ aff�m � �m, which means that v 2 ri�m. To proceed, fix anyx 2 .v C ıIB/ \ aff�m and get that x D v C u for some u 2 ıIB . Since v; x 2 aff�m and u D

x � v, we haveu 2 L. DenotingA.u/ WD .˛0; : : : ; ˛m/ gives us the representationu DPm

iD0 ˛ivi

withPm

iD0 ˛i D 0 and the estimate

j˛i j � kA.u/k <1

mC 1for all i D 0; : : : ; m:

en implies in turn that

v C u D

mXiD0

.1

mC 1C ˛i /vi D

mXiD0

�ivi ;

where �i WD1

mC 1C ˛i � 0 for i D 0; : : : ; m. Since

PmiD0 �i D 1, this ensures that x 2 �m.

us .v C ıIB/ \ aff�m � �m and therefore v 2 ri�m. �

Lemma 1.71 Let ˝ be a nonempty, convex set in Rn of dimension m � 1. en there exist mC 1

affinely independent elements v0; : : : ; vm in˝.


Proof. Let �k WD fv0; : : : ; vkg be a k�simplex of maximal dimension contained in ˝. env0; : : : ; vk are affinely independent. To verify now that k D m, form K WD afffv0; : : : ; vkg andobserve that K � aff˝ since fv0; : : : ; vkg � ˝. e opposite inclusion also holds since we have˝ � K. Justifying it, we argue by contradiction and suppose that there exists w 2 ˝ such thatw … K.en a direct application of the definition of affine independence shows that v0; : : : ; vk; w

are affinely independent being a subset of˝, which is a contradiction. usK D aff˝, and hencewe get k D dimK D dim aff ˝ D dim˝ D m. �

e next is one of the most fundamental results of convex finite-dimensional geometry.

eorem 1.72 Let ˝ be a nonempty, convex set in Rn. e following assertions hold:(i) We always have ri˝ ¤ ;.(ii) We have Œa; b/ � ri˝ for any a 2 ri˝ and b 2 ˝.

Proof. (i) Letm be the dimension of˝. Observe first that the case wherem D 0 is trivial since inthis case˝ is a singleton and ri˝ D ˝. Suppose thatm � 1 and findmC 1 affinely independentelements v0; : : : ; vm in ˝ as in Lemma 1.71. Consider further the m-simplex

�m WD co˚v0; : : : ; vm

:

We can show that aff�m D aff˝. To complete the proof, take v 2 ri�m, which exists by Propo-sition 1.70, and get for any small � > 0 that

IB.v; �/ \ aff˝ D IB.v; �/ \ aff�m � �m � ˝:

is verifies that v 2 ri˝ by the definition of relative interior.(ii) Let L be the subspace of Rn parallel to aff˝ and let m WD dimL. en there is a bijectivelinear mapping A W L ! Rm such that both A and A�1 are continuous. Fix x0 2 aff˝ and definethe mapping f W aff˝ ! Rm by f .x/ WD A.x � x0/. It is easy to check that f is a bijectiveaffine mapping and that both f and f �1 are continuous. We also see that a 2 ri˝ if and only iff .a/ 2 intf .˝/, and that b 2 ˝ if and only if f .b/ 2 f .˝/. en Œf .a/; f .b// � intf .˝/ byLemma 1.30. is shows that Œa; b/ � ri˝. �

We conclude this section by the following properties of the relative interior.

Proposition 1.73 Let ˝ be a nonempty, convex subset of Rn. For the convex sets ri˝ and ˝, wehave that (i) ri˝ D ˝ and (ii) ri˝ D ri˝.


Proof. Note that the convexity of ri˝ follows fromeorem 1.72(ii) while the convexity of˝ wasproved in Proposition 1.29. To justify assertion (i) of this proposition, observe that the inclusionri˝ � ˝ is obvious. Fix b 2 ˝, choose a 2 ri˝, and form the sequence

xk WD1

kaC

�1 �

1

k

�b; k 2 N;

which converges to b as k ! 1. Since xk 2 ri˝ by eorem 1.72(ii), we have b 2 ri˝. us˝ � ri˝, which verifies (i). e proof (ii) is similar to that of Proposition 1.31(ii). �

1.5 THEDISTANCEFUNCTIONe last section of this chapter is devoted to the study of the class of distance functions for con-vex sets, which belongs to the most interesting and important subjects of convex analysis andits extensions. Functions of this type are intrinsically nondifferentiable while they naturally andfrequently appear in analysis and applications.

Figure 1.6: Distance function.

Given a set ˝ � Rn, the distance function associated with ˝ is defined by

d.xI˝/ WD inf˚kx � !k

ˇ! 2 ˝

: (1.5)

Recall further that a mapping f W Rn! Rp is Lipschitz continuous with constant ` � 0 on some

set C � Rn if we have the estimate

kf .x/ � f .y/k � `kx � yk for all x; y 2 C: (1.6)

Note that the Lipschitz continuity of f in (1.6) specifies its continuity with a linear rate.

Proposition 1.74 Let˝ be a nonempty subset of Rn. e following hold:

1.5. THEDISTANCEFUNCTION 29

(i) d.xI˝/ D 0 if and only if x 2 ˝.(ii)e function d.xI˝/ is Lipschitz continuous with constant ` D 1 on Rn.

Proof. (i) Suppose that d.xI˝/ D 0. For each k 2 N, find !k 2 ˝ such that

0 D d.xI˝/ � kx � !kk < d.xI˝/C1

kD1

k;

which ensures that the sequence f!kg converges to x, and hence x 2 ˝.Conversely, let x 2 ˝ and find a sequence f!kg � ˝ converging to x. en

0 � d.xI˝/ � kx � !kk for all k 2 N;

which implies that d.xI˝/ D 0 since kx � !kk ! 0 as k ! 1.(ii) For any ! 2 ˝, we have the estimate

d.xI˝/ � kx � !k � kx � yk C ky � !k;


d.xI˝/ � kx � yk C inf˚ky � !k

ˇ! 2 ˝

D kx � yk C d.yI˝/:

Similarly, we get d.yI˝/ � ky � xk C d.xI˝/ and thus jd.xI˝/ � d.yI˝/j � kx � yk, whichjustifies by (1.6) the Lipschitz continuity of d.xI˝/ on Rn with constant ` D 1. �

For each x 2 Rn, the Euclidean projection from x to ˝ is defined by

˘.xI˝/ WD˚! 2 ˝

ˇkx � !k D d.xI˝/

: (1.7)

Proposition 1.75 Let ˝ be a nonempty, closed subset of Rn. en for any x 2 Rn the Euclideanprojection˘.xI˝/ is nonempty.

Proof. By definition (1.7), for each k 2 N there exists !k 2 ˝ such that

d.xI˝/ � kx � !kk < d.xI˝/C1

k:

It is clear that f!kg is a bounded sequence. us it has a subsequence f!k`g that converges to !.

Since ˝ is closed, ! 2 ˝. Letting ` ! 1 in the inequality

d.xI˝/ � kx � !k`k < d.xI˝/C

1

k`

;

we have d.xI˝/ D kx � !k, which ensures that ! 2 ˘.xI˝/. �


An interesting consequence of convexity is the following unique projection property.

Corollary 1.76 If˝ is a nonempty, closed, convex subset of Rn, then for each x 2 Rn the Euclideanprojection˘.xI˝/ is a singleton.

Proof. e nonemptiness of the projection˘.xI˝/ follows from Proposition 1.75. To prove theuniqueness, suppose that !1; !2 2 ˘.xI˝/ with !1 ¤ !2. en

kx � !1k D kx � !2k D d.xI˝/:

By the classical parallelogram equality, we have that

2kx � !1k2

D kx � !1k2

C kx � !2k2

D 2 x �

!1 C !2

2

2

Ck!1 � !2k2

2:

is directly implies that x �!1 C !2

2

2

D kx � !1k2

�k!1 � !2k2

4< kx � !1k

2D

�d.xI˝/

�2;

which is a contradiction due to the inclusion !1 C !2

22 ˝. �

Now we show that the convexity of a nonempty, closed set and its distance function areequivalent. It is an easy exercise to show that the convexity of an arbitrary set ˝ implies theconvexity of its distance function.

Proposition 1.77 Let ˝ be a nonempty, closed subset of Rn. en the function d.�I˝/ is convex ifand only if the set˝ is convex.

Proof. Suppose that ˝ is convex. Taking x1; x2 2 Rn and !i WD ˘.xi I˝/, we have

kxi � !ik D d.xi I˝/ for i D 1; 2:

e convexity of ˝ ensures that �!1 C .1 � �/!2 2 ˝ for any � 2 .0; 1/. It yields

d��x1 C .1 � �/x2I˝

�� k�x1 C .1 � �/x2 � Œ�!1 C .1 � �/!2�k

� �kx1 � !1k C .1 � �/kx2 � !2k

D �d.x1I˝1/C .1 � �/d.x2I˝2/;

which implies therefore the convexity of the distance function d.�I˝/ by

d��x1 C .1 � �/x2I˝

�� d.x1I˝/C .1 � �/d.x2I˝/:

1.5. THEDISTANCEFUNCTION 31

To prove the converse implication, suppose that d.�I˝/ is convex and fix any !i 2 ˝ for i D 1; 2

and � 2 .0; 1/. en we have

d��!1 C .1 � �/!2I˝

�� d.!1I˝/C .1 � �/d.!2I˝/ D 0:

Since ˝ is closed, this yields �!1 C .1 � �/!2 2 ˝ and so justifies the convexity of ˝. �

Next we characterize the Euclidean projection to convex sets in Rn. In the propositionbelow and in what follows we often identify the projection˘.xI˝/ with its unique element if˝is a nonempty, closed, convex set.

Figure 1.7: Euclidean projection.

Proposition 1.78 Let ˝ be a nonempty, convex subset of Rn and let N! 2 ˝. en we have N! 2

˘. NxI˝/ if and only ifh Nx � N!;! � N!i � 0 for all ! 2 ˝: (1.8)

Proof. Take N! 2 ˘. NxI˝/ and get for any ! 2 ˝, � 2 .0; 1/ that N! C �.! � N!/ 2 ˝. us

k Nx � N!k2

D�d. NxI˝/

�2� k Nx �

�N! C �.! � N!/k2

D k Nx � N!k2

� 2�h Nx � N!;! � N!i C �2k! � N!k

2:

is readily implies that2h Nx � N!;! � N!i � �k! � N!k

2:

Letting � ! 0C, we arrive at (1.8).


To verify the converse, suppose that (1.8) holds. en for any ! 2 ˝ we get

k Nx � !k2

D k Nx � N! C N! � !k2

D k Nx � N!k2

C k N! � !k2

C 2h Nx � N!; N! � !i

D k Nx � N!k2

C k N! � !k2

� 2h Nx � N!;! � N!i � k Nx � N!k2:

us k Nx � N!k � k Nx � !k for all ! 2 ˝, which implies N! 2 ˘. NxI˝/ and completes the proof.�

We know from the above that for any nonempty, closed, convex set˝ in Rn the Euclideanprojection˘.xI˝/ is a singleton. Now we show that the projection mapping is in fact nonexpan-sive, i.e., it satisfies the following Lipschitz property.

Proposition 1.79 Let˝ be a nonempty, closed, convex subset of Rn. en for any elements x1; x2 2

Rn we have the estimate ˘.x1I˝/ �˘.x2I˝/ 2

�˝˘.x1I˝/ �˘.x2I˝/; x1 � x2

˛:

In particular, it implies the Lipschitz continuity of the projection with constant ` D 1: ˘.x1I˝/ �˘.x2I˝/ � kx1 � x2k for all x1; x2 2 Rn :

Proof. It follows from the preceding proposition that˝˘.x2I˝/ �˘.x1I˝/; x1 �˘.x1I˝/

˛� 0 for all x1; x2 2 Rn :

Changing the roles of x1; x2 in the inequality above and summing them up give us˝˘.x1I˝/ �˘.x2I˝/; x2 � x1 C˘.x1I˝/ �˘.x2I˝/

˛� 0:

is implies the first estimate in the proposition. Finally, the nonexpansive property of the Eu-clidean projection follows directly from ˘.x1I˝/ �˘.x2I˝/

2�

˝˘.x1I˝/ �˘.x2I˝/; x1 � x2

˛� k˘.x1I˝/ �˘.x2I˝/k � kx1 � x2k

for all x1; x2 2 Rn, which completes the proof of the proposition. �

1.6 EXERCISES FORCHAPTER 1

Exercise 1.1 Let˝1 and˝2 be nonempty, closed, convex subsets of Rn such that˝1 is boundedor˝2 is bounded. Show that˝1 �˝2 is a nonempty, closed, convex set. Give an example of twononempty, closed, convex sets ˝1;˝2 for which ˝1 �˝2 is not closed.

1.6. EXERCISES FORCHAPTER 1 33

Exercise 1.2 Let ˝ be a subset of Rn. We say that ˝ is a cone if �x 2 ˝ whenever � � 0 andx 2 ˝. Show that the following are equivalent:(i) ˝ is a convex cone.(ii) x C y 2 ˝ whenever x; y 2 ˝, and �x 2 ˝ whenever x 2 ˝ and � � 0.

Exercise 1.3 (i) Let ˝ be a nonempty, convex set that contains 0 and let 0 � �1 � �2. Showthat �1˝ � �2˝:

(ii) Let ˝ be a nonempty, convex set and let ˛; ˇ � 0. Show that ˛˝ C ˇ˝ � .˛ C ˇ/˝.

Exercise 1.4 (i) Let ˝i for i D 1; : : : ; m be nonempty, convex sets in Rn. Show thatx 2 co

SmiD1˝i if and only if there exist elements !i 2 ˝i and �i � 0 for i D 1; : : : ; m withPm

iD1 �i D 1 such that x DPm

iD1 �i!i .(ii) Let ˝i for i D 1; : : : ; m be nonempty, convex cones in Rn. Show that

mXiD1

˝i D co f

m[iD1

˝ig:

Exercise 1.5 Let˝ be a nonempty, convex cone in Rn. Show that˝ is a linear subspace of Rn

if and only if ˝ D �˝.

Exercise 1.6 Show that the following functions are convex on Rn:(i) f .x/ D ˛kxk, where ˛ � 0.(ii) f .x/ D kx � ak2, where a 2 Rn.(iii) f .x/ D kAx � bk, where A is an p � n matrix and b 2 Rp.(iv) f .x/ D kxkq , where q � 1.

Exercise 1.7 Show that the following functions are convex on the given domains:(i) f .x/ D eax; x 2 R, where a is a constant.(ii) f .x/ D xq , x 2 Œ0;1/, where q � 1 is a constant.(iii) f .x/ D � ln.x/, x 2 .0;1/.(iv) f .x/ D x ln.x/, x 2 .0;1/.

Exercise 1.8 (i)Give an example of a function f W Rn� Rp

! R, which is convex with respectto each variable but not convex with respect to both.(ii)Let fi W Rn

! R for i D 1; 2 be convex functions. Canwe conclude that theminimum functionminff1; f2g.x/ WD minff1.x/; f2.x/g is convex?

Exercise 1.9 Give an example showing that the product of two real-valued convex functions isnot necessarily convex.


Exercise 1.10 Verify that the set f.x; t/ 2 Rn� R j t � kxkg is closed and convex.

Exercise 1.11 e indicator function associated with a set ˝ is defined by

ı.xI˝/ WD

(0 if x 2 ˝;

1 otherwise:

(i) Calculate ı.�I˝/ for ˝ D ;, ˝ D Rn, and ˝ D Œ�1; 1�.(ii) Show that the set ˝ is convex set if and only if its indicator function ı.�I˝/ is convex.

Exercise 1.12 Show that if f W R ! Œ0;1/ is a convex function, then its q-power f q.x/ WD

.f .x//q is also convex for any q � 1.

Exercise 1.13 Let fi W Rn! R for i D 1; 2 be convex functions. Define

˝1 WD f.x; �1; �2/ 2 Rn� R � R j �1 � f1.x/g;

˝2 WD˚.x; �1; �2/ 2 Rn

� R � R j �2 � f2.x/g:

(i) Show that the sets ˝1;˝2 are convex.(ii)Define the set-valued mapping F W Rn

!! R2 by F.x/ WD Œf1.x/;1/ � Œf2.x/;1/ and verifythat the graph of F is ˝1 \˝2.

Exercise 1.14 We say that f W Rn! R is positively homogeneous if f .˛x/ D f .x/ for all ˛ >

0, and that f is subadditive if f .x C y/ � f .x/C f .y/ for all x; y 2 Rn. Show that a positivelyhomogeneous function is convex if and only if it is subadditive.

Exercise 1.15 Given a nonempty set ˝ � Rn, define

K˝ WD˚�x

ˇ� � 0; x 2 ˝

D

[��0

�˝:

(i) Show that K˝ is a cone.(ii) Show that K˝ is the smallest cone containing ˝.(iii) Show that if ˝ is convex, then the cone K˝ is convex as well.

Exercise 1.16 Let ' W Rn� Rp

! R be a convex function and let K be a nonempty, convexsubset of Rp. Suppose that for each x 2 Rn the function '.x; �/ is bounded below on K. Verifythat the function on Rn defined by f .x/ WD inff'.x; y/ j y 2 Kg is convex.

Exercise 1.17 Let f W Rn! R be a convex function.

(i) Show that for every ˛ 2 R the level set fx 2 Rnj f .x/ � ˛g is convex.

(ii) Let ˝ � R be a convex set. Is it true in general that the inverse image f �1.˝/ is a convexsubset of Rn?


Exercise 1.18 Let C be a convex subset of RnC1.(i) For x 2 Rn, define F.x/ WD f� 2 R j .x; �/ 2 C g. Show that F is a set-valued mapping withconvex graph and give an explicit formula for its graph.(ii) For x 2 Rn, define the function

fC .x/ WD inf˚� 2 R

ˇ.x; �/ 2 C

: (1.9)

Find an explicit formula for fC when C is the closed unit ball of R2.(iii) For the function fC defined in (ii), show that if fC .x/ > �1 for all x 2 Rn, then fC is aconvex function.

Exercise 1.19 Let f W Rn! R be bounded from below. Define its convexification

.cof /.x/ WD infn mX

iD1

�if .xi /ˇ�i � 0;

mXiD1

�i D 1;

mXiD1

�ixi D x; m 2 No: (1.10)

Verify that (1.10) is convex with co f D fC , C WD co .epif /, and fC defined in (1.9).

Exercise 1.20 We say that f W Rn! R is quasiconvex if

f��x C .1 � �/y/ � max

˚f .x/; f .y/

for all x; y 2 Rn; � 2 .0; 1/:

(i) Show that a function f is quasiconvex if and only if for any ˛ 2 R the level set fx 2

Rnj f .x/ � ˛g is a convex set.

(ii) Show that any convex function f W Rn! R is quasiconvex. Give an example demonstrating

that the converse is not true.

Exercise 1.21 Let a be a nonzero element in Rn and let b 2 R. Show that

˝ WD˚x 2 Rn

ˇha; xi D b

is an affine set with dim˝ D 1.

Exercise 1.22 Let the set fv1; : : : ; vmg consist of affinely independent elements and let v …

aff fv1; : : : ; vmg. Show that v1; : : : ; vm; v are affinely independent.

Exercise 1.23 Suppose that ˝ is a convex subset of Rn with dim ˝ D m, m � 1. Let the setfv1; : : : ; vmg � ˝ consist of affinely independent elements and let

�m WD co˚v1; : : : ; vm

:

Show that aff˝ D aff�m D aff˚v1; : : : ; vm

.


Exercise 1.24 Let ˝ be a nonempty, convex subset of Rn.(i) Show that aff˝ D aff˝.(ii) Show that for any Nx 2 ri˝ and x 2 ˝ there exists t > 0 such that Nx C t. Nx � x/ 2 ˝.(iii) Prove Proposition 1.73(ii).

Exercise 1.25 Let ˝1;˝2 � Rn be convex sets with ri˝1 \ ri˝2 ¤ ;. Show that:(i) ˝1 \˝2 D ˝1 \˝2.(ii) ri .˝1 \˝2/ D ri˝1 \ ri˝2.

Exercise 1.26 Let ˝1;˝2 � Rn be convex sets with ˝1 D ˝2. Show that ri˝1 D ri˝2.

Exercise 1.27 (i) Let B W Rn! Rp be an affine mapping and let ˝ be a convex subset of Rn.

Prove the equalityB.ri˝/ D riB.˝/:

(ii) Let ˝1 and ˝2 be convex subsets of Rn. Show that ri .˝1 �˝2/ D ri˝1 � ri˝2.

Exercise 1.28 Let f W Rn! R be a convex function. Show that:

(i) aff .epif / D aff .domf / � R :(ii) ri .epif / D

˚.x; �/

ˇx 2 ri .domf /; � > f .x/

:

Exercise 1.29 Find the explicit formulas for the distance function d.xI˝/ and the Euclideanprojection ˘.xI˝/ in the following cases:(i) ˝ is the closed unit ball of Rn.(ii) ˝ WD

˚.x1; x2/ 2 R2

ˇjx1j C jx2j � 1

.

(iii) ˝ WD Œ�1; 1� � Œ�1; 1�.

Exercise 1.30 Find the formulas for the projection ˘.xI˝/ in the following cases:(i) ˝ WD

˚x 2 Rn

ˇha; xi D b

, where 0 ¤ a 2 Rn and b 2 R :

(ii) ˝ WD˚x 2 Rn

ˇAx D b

, where A is an m � n matrix with rankA D m and b 2 Rm.

(iii) ˝ is the nonnegative orthant ˝ WD RnC.

Exercise 1.31 For ai � bi with i D 1; : : : ; n, define

˝ WD˚x D .x1; : : : ; xn/ 2 Rn

ˇai � x � bi for all i D 1; : : : ; n

:

Show that the projection ˘.xI˝/ has the following representation:

˘.xI˝/ D˚u D .u1; : : : ; un/ 2 Rn

ˇui D maxfai ;minfb; xigg for i 2 f1; : : : ; ng

:

37

C H A P T E R 2

Subdifferential CalculusIn this chapter we present basic results of generalized differentiation theory for convex functionsand sets. A geometric approach of variational analysis based on convex separation for extremalsystems of sets allows us to derive general calculus rules for normals to sets and subgradients offunctions. We also include into this chapter some related results on Lipschitz continuity of convexfunctions and convex duality. Note that some extended versions of the presented calculus resultscan be found in exercises for this chapter, with hints to their proofs given at the end of the book.

2.1 CONVEX SEPARATIONWe start this section with convex separation theorems, which play a fundamental role in convexanalysis and particularly in deriving calculus rules by the geometric approach. e first resultconcerns the strict separation of a nonempty, closed, convex set and an out-of-set point.

Proposition 2.1 Let ˝ be a nonempty, closed, convex subset of Rn and let Nx … ˝. en there is anonzero element v 2 Rn such that

sup˚hv; xi

ˇx 2 ˝

< hv; Nxi: (2.1)

Proof. Denote Nw WD ˘. NxI˝/ and let v WD Nx � Nw ¤ 0. Applying Proposition 1.78 gives us

hv; x � Nwi D h Nx � Nw; x � Nwi � 0 for any x 2 ˝:

It follows furthermore that

hv; x � . Nx � v/i D hv; x � Nwi � 0;

which implies in turn that hv; xi � hv; Nxi � kvk2 and thus verifies (2.1). �

In the setting of Proposition 2.1 we choose a number b 2 R such that

sup˚hv; xi

ˇx 2 ˝

< b < hv; Nxi

and define the function A.x/ WD hv; xi for which A. Nx/ > b and A.x/ < b for all x 2 ˝. en wesay that Nx and ˝ are strictly separated by the hyperplane L WD fx 2 Rn

j A.x/ D bg.e next theorem extends the result of Proposition 2.1 to the case of two nonempty, closed,

convex sets at least one of which is bounded.

38 2. SUBDIFFERENTIALCALCULUS

eorem 2.2 Let˝1 and˝2 be nonempty, closed, convex subsets of Rn with˝1 \˝2 D ;. If˝1 is bounded or ˝2 is bounded, then there is a nonzero element v 2 Rn such that

sup˚hv; xi

ˇx 2 ˝1

< inf

˚hv; yi

ˇy 2 ˝2

:

Proof. Denote ˝ WD ˝1 �˝2. en ˝ is a nonempty, closed, convex set and 0 … ˝. ApplyingProposition 2.1 to ˝ and Nx D 0, we have

WD sup˚hv; xi

ˇx 2 ˝

< 0 D hv; Nxi with some 0 ¤ v 2 Rn :

For any x 2 ˝1 and y 2 ˝2, we have hv; x � yi � , and so hv; xi � C hv; yi. erefore,

sup˚hv; xi

ˇx 2 ˝1

� C inf

˚hv; yi

ˇy 2 ˝2

< inf

˚hv; yi

ˇy 2 ˝2

;

which completes the proof of the theorem. �

Figure 2.1: Illustration of convex separation.

Next we show that two nonempty, closed, convex sets with empty intersection in Rn canbe separated, while not strictly in general, if both sets are unbounded.

Corollary 2.3 Let ˝1 and ˝2 be nonempty, closed, convex subsets of Rn such that ˝1 \˝2 D ;.en they are in the sense that there is a nonzero element v 2 Rn with

sup˚hv; xi

ˇx 2 ˝1

� inf

˚hv; yi

ˇy 2 ˝2

: (2.2)

2.1. CONVEX SEPARATION 39

Proof. For every fixed k 2 N, we define the set �k WD ˝2 \ IB.0I k/ and observe that it is aclosed, bounded, convex set, which is also nonempty if k is sufficiently large. Employing eo-rem 2.2 allows us to find 0 ¤ uk 2 Rn such that

huk; xi < huk; yi for all x 2 ˝1; y 2 �k :

Denote vk WDuk

kukkand assume without loss of generality that vk ! v ¤ 0 as k ! 1. Fix fur-

ther x 2 ˝1 and y 2 ˝2 and take k0 2 N such that y 2 �k for all k � k0. en

hvk; xi < hvk; yi whenever k � k0:

Letting k ! 1, we have hv; xi � hv; yi, which justifies the claim of the corollary. �

Now our intention is to establish a general separation theorem for convex sets that may notbe closed. To proceed, we verify first the following useful result.

Figure 2.2: Geometric illustration of Lemma 2.4.

Lemma 2.4 Let˝ be a nonempty, convex subset of Rn such that 0 2 ˝ n˝. en there is a sequencexk ! 0 as k ! 1 with xk … ˝ for all k 2 N.

Proof. Recall that ri˝ ¤ ; by eorem 1.72(i). Fix x0 2 ri˝ and show that �tx0 … ˝ for allt > 0; see Figure 2.2. Indeed, suppose the contrary that �tx0 2 ˝ and get by eorem 1.72(ii)that

0 Dt

1C tx0 C

1

1C t.�tx0/ 2 ri˝ � ˝;

a contradiction. Letting then xk WD �x0

kgives us xk … ˝ for every k and xk ! 0. �


Note that Lemma 2.4 may not hold for nonconvex sets. As an example, consider the set˝ WD Rn

nf0g for which 0 2 ˝ n˝ while the conclusion of the Lemma 2.4 fails.

eorem2.5 Let˝1 and˝2 be nonempty, convex subsets of Rn with˝1 \˝2 D ;. en theyare separated in the sense of (2.2) with some v ¤ 0.

Proof. Consider the set ˝ WD ˝1 �˝2 for which 0 … ˝. We split the proof of the theorem intothe following two cases:Case 1: 0 … ˝. In this case we apply Proposition 2.1 and find v ¤ 0 such that

hv; zi < 0 for all z 2 ˝:

is gives us hv; xi < hv; yi whenever x 2 ˝1 and y 2 ˝2, which justifies the statement.Case 2: 0 2 ˝. In this case we employ Lemma 2.4 and find a sequence xk ! 0 with xk … ˝.en Proposition 2.1 produces a sequence fvkg of nonzero elements such that

hvk; zi < hvk; xki for all z 2 ˝:

Without loss of generality, we get (dividing by kvkk and extracting a convergent subsequence)that vk ! v and kvkk D kvk D 1. is gives us by letting k ! 1 that

hv; zi � 0 for all z 2 ˝:

To complete the proof, we just repeat the arguments of Case 1. �

Figure 2.3: Extremal system of sets.

eorem 2.5 ensures the separation of convex sets, which do not intersect. To proceed nowwith separation of convex sets which may have common points as in Figure 2.3, we introducethe notion of set extremality borrowed from variational analysis that deals with both convex andnonconvex sets while emphasizing intrinsic optimality/extremality structures of problems underconsideration; see, e.g., [14] for more details.

2.1. CONVEX SEPARATION 41

Definition 2.6 Given two nonempty, convex sets˝1; ˝2 � Rn, we say that the set system f˝1;˝2g

is if for any � > 0 there is a 2 Rn such that kak < � and

.˝1 � a/ \˝2 D ;:

e next proposition gives us an important example of extremal systems of sets.

Proposition 2.7 Let ˝ be a nonempty, convex subset of Rn and let Nx 2 ˝ be a boundary point of˝. en the sets˝ and f Nxg form an extremal system.

Proof. Since Nx 2 bd˝, for any number � > 0 we get IB. NxI �=2/ \˝c ¤ ;. us choosing b 2

IB. NxI �=2/ \˝c and denoting a WD b � Nx yield kak < � and .˝ � a/ \ f Nxg D ;. �

e following result is a separation theorem for extremal systems of convex sets. It admitsa far-going extension to nonconvex settings, which is known as the extremal principle and is atthe heart of variational analysis and its applications [14]. It is worth mentioning that, in con-trast to general settings of variational analysis, we do not impose here any closedness assumptionson the sets in question. is will allow us to derive in the subsequent sections of this chaptervarious calculus rules for normals to convex sets and subgradients of convex functions by usinga simple variational device without imposing closedness and lower semicontinuity assumptionsconventional in variational analysis.

eorem 2.8 Let ˝1 and ˝2 be two nonempty, convex subsets of Rn. en ˝1 and ˝2 forman extremal system if and only if there is a nonzero element v 2 Rn which separates the sets ˝1

and ˝2 in the sense of (2.2).

Proof. Suppose that ˝1 and ˝2 form an extremal system of convex sets in Rn. en there is asequence fakg in Rn with ak ! 0 and

.˝1 � ak/ \˝2 D ;; k 2 N :

For every k 2 N, apply eorem 2.5 to the convex sets ˝1 � ak and ˝2 and find a sequence ofnonzero elements uk 2 Rn satisfying

huk; x � aki � huk; yi for all x 2 ˝1 and y 2 ˝2; k 2 N :

e normalization vk WDuk

kukkwith kvkk D 1 gives us

hvk; x � aki � hvk; yi for all x 2 ˝1 and y 2 ˝2: (2.3)

us we find v 2 Rn with kvk D 1 such that vk ! v as k ! 1 along a subsequence. Passing tothe limit in (2.3) verifies the validity of (2.2).


For the converse implication, suppose that there is a nonzero element v 2 Rn which sepa-rates the sets ˝1 and ˝2 in the sense of (2.2). en it is not hard to see that�

˝1 �1

kv�

\˝2 D ; for every k 2 N :

us ˝1 and ˝2 form an extremal system. �

2.2 NORMALSTOCONVEX SETSIn this section we apply the separation results obtained above (particularly, the extremal versionin eorem 2.8) to the study of generalized normals to convex sets and derive basic rules of thenormal cone calculus.

Definition 2.9 Let˝ � Rn be a convex set with Nx 2 ˝. e to˝ at Nx is

N. NxI˝/ WD˚v 2 Rn

ˇhv; x � Nxi � 0 for all x 2 ˝

: (2.4)

By convention, we put N. NxI˝/ WD ; for Nx … ˝.

Figure 2.4: Epigraph of the absolute value function and the normal cone to it at the origin.

We first list some elementary properties of the normal cone defined above.

Proposition 2.10 Let Nx 2 ˝ for a convex subset˝ of Rn. en we have:(i) N. NxI˝/ is a closed, convex cone containing the origin.(ii) If Nx 2 int˝, then N. NxI˝/ D f0g.

Proof. To verify (i), fix vi 2 N. NxI˝/ and �i � 0 for i D 1; 2. We get by the definition that0 2 N. NxI˝/ and hvi ; x � Nxi � 0 for all x 2 ˝ implying

h�1v1 C �2v2; x � Nxi � 0 whenever x 2 ˝:

2.2. NORMALSTOCONVEX SETS 43

us �1v1 C �2v2 2 N. NxI˝/, and hence N. NxI˝/ is a convex cone. To check its closedness, fixa sequence fvkg � N. NxI˝/ that converges to v. en passing to the limit in

hvk; x � Nxi � 0 for all x 2 ˝ as k ! 1

yields hv; x � Nxi � 0 whenever x 2 ˝, and so v 2 N. NxI˝/. is completes the proof of (i).To check (ii), pick v 2 N. NxI˝/ with Nx 2 int˝ and find ı > 0 such that Nx C ıIB � ˝.

us for every e 2 IB we have the inequality

hv; Nx C ıe � Nxi � 0;

which shows that ıhv; ei � 0, and hence hv; ei � 0. Choose t > 0 so small that tv 2 IB . enhv; tvi � 0 and therefore tkvk2 D 0, i.e., v D 0. �

We start calculus rules with deriving rather simple while very useful results.

Proposition 2.11 (i) Let ˝1 and ˝2 be nonempty, convex subsets of Rn and Rp, respectively. For. Nx1; Nx2/ 2 ˝1 �˝2, we have

N�. Nx1; Nx2/I˝1 �˝2

�D N. Nx1I˝1/ �N. Nx2I˝2/:

(ii) Let˝1 and˝2 be convex subsets of Rn with Nxi 2 ˝i for i D 1; 2. en

N. Nx1 C Nx2I˝1 C˝2/ D N. Nx1I˝1/ \N. Nx2I˝2/:

Proof. To verify (i), fix .v1; v2/ 2 N.. Nx1; Nx2/I˝1 �˝2/ and get by the definition that

h.v1; v2/; .x1; x2/ � . Nx1; Nx2/i D hv1; x1 � Nx1i C hv2; x2 � Nx2i � 0 (2.5)

whenever .x1; x2/ 2 ˝1 �˝2. Putting x2 WD Nx2 in (2.5) gives us

hv1; x1 � Nx1i � 0 for all x1 2 ˝1;

which means that v1 2 N. Nx1I˝1/. Similarly, we obtain v2 2 N. Nx2I˝2/ and thus justify the in-clusion “�” in (2.5). e opposite inclusion is obvious.Let us now verify (ii). Fix v 2 N. Nx1 C Nx2I˝1 C˝2/ and get by the definition that

hv; x1 C x2 � . Nx1 C Nx2/i � 0 whenever x1 2 ˝1; x2 2 ˝2:

Putting there x1 WD Nx1 and x2 WD Nx2 gives us v 2 N. Nx1I˝1/ \N. Nx2I˝2/.e opposite inclusionin (ii) is also straightforward. �


Consider further a linear mapping A W Rn! Rp and associate it with a p � n matrix as

usual. e adjoint mapping A� W Rp! Rn is defined by

hAx; yi D hx;A�yi for all x 2 Rn; y 2 Rp;

which corresponds to the matrix transposition. e following calculus result describes normals tosolution sets for systems of linear equations.

Proposition2.12 LetBW Rn! Rp be defined byB.x/ WD Ax C b, whereA W Rn

! Rp is a linearmapping and b 2 Rp. Given c 2 R, consider the solution set

˝ WD˚x 2 Rn

ˇBx D c

:

For any Nx 2 ˝ with B. Nx/ D c, we have

N. NxI˝/ D imA�D

˚v 2 Rn

ˇv D A�y; y 2 Rp

: (2.6)

Proof. Take v 2 N. NxI˝/ and get hv; x � Nxi � 0 for all x such that Ax C b D c. Fixing anyu 2 kerA, we see that A. Nx � u/ D A Nx D c � b, i.e., A. Nx � u/C b D c, which yields hv; ui � 0

whenever u 2 kerA. Since �u 2 kerA, it follows that hv; ui D 0 for all u 2 kerA. Arguing bycontradiction, suppose that there is no y 2 Rp with v D A�y. us v … W WD A� Rp, where theset W � Rn is obviously nonempty, closed, and convex. Employing Proposition 2.1 gives us anonzero element Nu 2 Rn satisfying

sup˚h Nu;wi

ˇw 2 W

< h Nu; vi;

and implying that 0 < h Nu; vi since 0 2 W . Furthermore

th Nu;A�yi D h Nu;A�.ty/i < h Nu; vi for all t 2 R; y 2 Rp :

Hence h Nu;A�yi D 0, which shows that hA Nu; yi D 0 whenever y 2 Rp, i.e., A Nu D 0. us Nu 2

kerA while hv; Nui > 0. is contradiction justifies the inclusion “�” in (2.6).To verify the opposite inclusion in (2.6), fix any v 2 Rn with v D A�y for some y 2 Rp.

For any x 2 ˝, we have

hv; x � Nxi D hA�y; x � Nxi D hy;Ax � A Nxi D 0;

which means that v 2 N. NxI˝/ and thus completes the proof of the proposition. �

e following proposition allows us to characterize the separation of convex (generally non-closed) sets in terms of their normal cones.

Proposition 2.13 Let ˝1 and ˝2 be convex subsets of Rn with Nx 2 ˝1 \˝2. en the followingassertions are equivalent:


(i) f˝1;˝2g forms an extremal system of two convex sets in Rn.(ii)e sets˝1 and˝2 are separated in the sense of (2.2) with some v ¤ 0.(iii) N. NxI˝1/ \

��N. NxI˝2/

�¤ f0g.

Proof. e equivalence of (i) and (ii) follows from eorem 2.8. Suppose that (ii) is satisfied.Since Nx 2 ˝2, we have

sup˚hv; xi

ˇx 2 ˝1

� hv; Nxi;

and thus hv; x � Nxi � 0 for all x 2 ˝1, i.e., v 2 N. NxI˝1/. Similarly, we can verify the inclusion�v 2 N. NxI˝2/, and hence arrive at (iii).

Now suppose that (iii) is satisfied and let 0 ¤ v 2 N. NxI˝1/ \�

�N. NxI˝2/�. en (2.2)

follows directly from the definition of the normal cone. Indeed, for any x 2 ˝1 and y 2 ˝2 onehas

hv; x � Nxi � 0 and h�v; y � Nxi � 0:

is implies hv; x � Nxi � 0 � hv; y � Nxi, and hence hv; xi � hv; yi. �

e following corollary provides a normal cone characterization of boundary points.

Corollary 2.14 Let ˝ be a nonempty, convex subset of Rn and let Nx 2 ˝. en Nx 2 bd˝ if andonly if the normal cone to˝ is nontrivial, i.e., N. NxI˝/ ¤ f0g.

Proof. e “only if ” part follows from Proposition 2.10(ii). To justify the “if ” part, take Nx 2 bd˝and consider the sets˝1 WD ˝ and˝2 WD f Nxg, which form an extremal system by Proposition 2.7.Note that N. NxI˝2/ D Rn. us the claimed result is a direct consequence of Proposition 2.13since N. NxI˝1/ \ Œ�N. NxI˝2/� D N. NxI˝1/ ¤ f0g. �

Now we are ready to establish a major result of the normal cone calculus for convex setsproviding a relationship between normals to set intersections and normals to each set in the in-tersection. Its proof is also based on convex separation in extremal systems of sets. Note that thisapproach allows us to capture general convex sets, which may not be closed.

eorem 2.15 Let ˝1 and ˝2 be convex subsets of Rn with Nx 2 ˝1 \˝2. en for any v 2

N. NxI˝1 \˝2/ there exist � � 0 and vi 2 N. NxI˝i /, i D 1; 2, such that

�v D v1 C v2; .�; v1/ ¤ .0; 0/: (2.7)

Proof. Fix v 2 N. NxI˝1 \˝2/ and get by the definition that

hv; x � Nxi � 0 for all x 2 ˝1 \˝2:


Denote �1 WD ˝1 � Œ0;1/ and �2 WD f.x; �/ j x 2 ˝2; � � hv; x � Nxig. ese convex setsform an extremal system since we obviously have . Nx; 0/ 2 �1 \�2 and

�1 \ .�2 � .0; a// D ; whenever a > 0:

By eorem 2.8, find 0 ¤ .w; / 2 Rn � R such that

hw; xi C �1 � hw; yi C �2 for all .x; �1/ 2 �1; .y; �2/ 2 �2: (2.8)

It is easy to see that � 0. Indeed, assuming > 0 and taking into account that . Nx; k/ 2 �1

when k > 0 and . Nx; 0/ 2 �2, we get

hw; Nxi C k � hw; Nxi;

which leads to a contradiction. ere are two possible cases to consider.Case 1: D 0. In this case we have w ¤ 0 and

hw; xi � hw; yi for all x 2 ˝1; y 2 ˝2:

en following the proof of Proposition 2.13 gives us w 2 N. NxI˝1/ \ Œ�N. NxI˝2/�, and thus(2.7) holds for � D 0, v1 D w, and v2 D �w.Case 2: < 0. In this case we let � WD � > 0 and deduce from (2.8), by using .x; 0/ 2 �1 whenx 2 ˝1, and . Nx; 0/ 2 �2, that

hw; xi � hw; Nxi for all x 2 ˝1:

is implies thatw 2 N. NxI˝1/, and hence w�

2 N. NxI˝1/. To proceed further, we get from (2.8),due to . Nx; 0/ 2 �1 and .y; ˛/ 2 �2 for all y 2 ˝2 with ˛ WD hv; y � Nxi, thatD

w; NxE

�

Dw; y

EC hv; y � Nxi whenever y 2 ˝2:

Dividing both sides by gives usDw ; Nx

E�

Dw ; y

EC hv; y � Nxi whenever y 2 ˝2:

is clearly implies the inequalityDw

C v; y � NxE

� 0 for all y 2 ˝2;

and thus w

C v D �w

�C v 2 N. NxI˝2/. Letting finally v1 WD

w

�2 N. NxI˝1/ and v2 WD �

w

�C

v 2 N. NxI˝2/ gives us v D v1 C v2, which ensures the validity of (2.7) with � D 1. �


Figure 2.5: Normal cone to set intersection.

Observe that representation (2.7) does not provide a calculus rule for representing normalsto set intersections unless � ¤ 0. However, it is easy to derive such an intersection rule directly fromeorem 2.15 while imposing the following basic qualification condition, which plays a crucial rolein the subsequent calculus results; see Figure 2.5.

Corollary 2.16 Let ˝1 and ˝2 be convex subsets of Rn with Nx 2 ˝1 \˝2. Assume that the basicqualification condition (BQC) is satisfied:

N. NxI˝1/ \�

�N. NxI˝2/�

D f0g: (2.9)

en we have the normal cone intersection rule

N. NxI˝1 \˝2/ D N. NxI˝1/CN. NxI˝2/: (2.10)

Proof. For any v 2 N. NxI˝1 \˝2/, we find by eorem 2.15 a number � � 0 and elements vi 2

N. NxI˝i / as i D 1; 2 such that the conditions in (2.7) hold. Assuming that � D 0 in (2.7) gives usthat 0 ¤ v1 D �v2 2 N. NxI˝1/ \ Œ�N. NxI˝2/�, which contradicts BQC (2.9). us � > 0 andv D v1=�C v2=� 2 N. NxI˝1/CN. NxI˝2/. Hence the inclusion “�” in (2.10) is satisfied. eopposite inclusion in (2.10) can be proved easily using the definition of the normal cone. Indeed,fix any v 2 N. NxI˝1/CN. NxI˝2/ with v D v1 C v2, where vi 2 N. NxI˝i / for i D 1; 2. For anyx 2 ˝1 \˝2, one has

hv; x � Nxi D hv1 C v2; x � Nxi D hv1; x � Nxi C hv2; x � Nxi � 0;

which implies v 2 N. NxI˝1 \˝2/ and completes the proof of the corollary. �


e following example shows that the intersection rule (2.10) does not hold generally with-out the validity of the basic qualification condition (2.9).

Example 2.17 Let˝1 WD f.x; �/ 2 R2j � � x2g and let˝2 WD f.x; �/ 2 R2

j � � �x2g. enfor Nx D .0; 0/ we have N. NxI˝1 \˝2/ D R2 while N. NxI˝1/CN. NxI˝2/ D f0g � R.

Next we present an easily verifiable condition ensuring the validity of BQC (2.9). Moregeneral results related to relative interior are presented in Exercise 2.13 and its solution.

Corollary 2.18 Let˝1; ˝2 � Rn be such that int˝1 \˝2 ¤ ; or int˝2 \˝1 ¤ ;. en for anyNx 2 ˝1 \˝2 both BQC (2.9) and intersection rule (2.10) are satisfied.

Proof. Consider for definiteness the case where int˝1 \˝2 ¤ ;. Let us fix Nx 2 ˝1 \˝2 andshow that BQC (2.9) holds, which ensures the intersection rule (2.10) by Corollary 2.16. Arguingby contradiction, suppose the opposite

N. NxI˝1/ \�

�N. NxI˝2/�

¤ f0g

and then find by Proposition 2.13 a nonzero element v 2 Rn that separates the sets˝1 and˝2 asin (2.2). Fixing now Nu 2 int˝1 \˝2 gives us a number ı > 0 such that NuC ıIB � ˝1. It ensuresby (2.2) that

hv; xi � hv; Nui for all x 2 NuC ıIB:

is implies in turn the inequality

hv; Nui C ıhv; ei � hv; Nui whenever e 2 IB

and tells us that hv; ei � 0 for all e 2 IB . Choosing finally e WDv

kvk, we arrive at a contradiction

and thus complete the proof of this corollary. �

Employing induction arguments allows us to derive a counterpart of Corollary 2.16 (and,similarly, of Corollary 2.18) for finitely many sets.

Corollary 2.19 Let ˝i � Rn for i D 1; : : : ; m be convex sets and let Nx 2Tm

iD1˝i . Assume thevalidity of the following qualification condition:h mX

iD1

vi D 0; vi 2 N. NxI˝i /i

H)�vi D 0 for all i D 1; : : : ; m

�:

en we have the intersection formula

N�

NxI

m\iD1

˝i

�D

mXiD1

N. NxI˝i /:

2.3. LIPSCHITZCONTINUITYOFCONVEXFUNCTIONS 49

2.3 LIPSCHITZCONTINUITYOFCONVEXFUNCTIONSIn this section we study the fundamental notion of Lipschitz continuity for convex functions. isnotion can be treated as “continuity at a linear rate” being highly important for many aspects ofconvex and variational analysis and their applications.

Definition 2.20 A function f W Rn! R is L on a set˝ � dom f if there

is a constant ` � 0 such that

jf .x/ � f .y/j � `kx � yk for all x; y 2 ˝: (2.11)

We say that f is L around Nx 2 dom f if there are constants ` � 0

and ı > 0 such that (2.11) holds with˝ D IB. NxI ı/.

Here we give several verifiable conditions that ensure Lipschitz continuity and characterizeits localized version for general convex functions. One of the most effective characterizations oflocal Lipschitz continuity, holding for nonconvex functions as well, is given via the followingnotion of the singular (or horizon) subdifferential, which was largely underinvestigated in convexanalysis and was properly recognized only in the framework of variational analysis; see [14, 27].Note that, in contrast to the (usual) subdifferential of convex analysis defined in the next section,the singular subdifferential does not reduce to the classical derivative for differentiable functions.

Definition 2.21 Let f W Rn! R be a convex function and let Nx 2 domf . e -

of the function f at Nx is defined by

@1f . Nx/ WD˚v 2 Rn

ˇ.v; 0/ 2 N

�. Nx; f . Nx/

�I epif

�: (2.12)

e following example illustrates the calculation of @1f . Nx/ based on definition (2.12).

Example 2.22 Consider the function

f .x/ WD

(0 if jxj � 1;

1 otherwise:

en epi f D Œ�1; 1� � Œ0;1/ and for Nx D 1 we have

N�. Nx; f . Nx//I epif

�D Œ0;1/ � .�1; 0�;

which shows that @1f . Nx/ D Œ0;1/; see Figure 2.6.

e next proposition significantly simplifies the calculation of @1f . Nx/.

Proposition 2.23 For a convex function f W Rn! R with Nx 2 domf , we have

@1f . Nx/ D N. NxI dom f /:


Figure 2.6: Singular subgradients.

Proof. Fix any v 2 @1f . Nx/ with x 2 dom f and observe that .x; f .x// 2 epif . en using(2.12) and the normal cone construction (2.4) give us

hv; x � Nxi D hv; x � Nxi C 0�f .x/ � f . Nx/

�� 0;

which shows that v 2 N. NxI dom f /. Conversely, suppose that v 2 N. NxI domf / and fix any.x; �/ 2 epif . en we have f .x/ � �, and so x 2 domf . us

hv; x � Nxi C 0.� � f . Nx// D hv; x � Nxi � 0;

which implies that .v; 0/ 2 N.. Nx; f . Nx//I epif /, i.e., v 2 @1f . Nx/. �

To proceed with the study of Lipschitz continuity of convex functions, we need the follow-ing two lemmas, which are of their own interest.

Lemma 2.24 Let fei j i D 1; : : : ; ng be the standard orthonormal basis of Rn. Denote

A WD˚

Nx ˙ �ei

ˇi D 1; : : : ; n

; � > 0:

en the following properties hold:(i) Nx C ei 2 coA for j j � � and i D 1; : : : ; n.(ii) IB. NxI �=n/ � coA.

Proof. (i) For j j � �, find t 2 Œ0; 1� with D t .��/C .1 � t/�. en Nx ˙ �ei 2 A gives us

Nx C ei D t. Nx � �ei /C .1 � t /. Nx C �ei / 2 coA:

2.3. LIPSCHITZCONTINUITYOFCONVEXFUNCTIONS 51

(ii) For x 2 IB. NxI �=n/, we have x D Nx C .�=n/u with kuk � 1. Represent u via feig by

u D

nXiD1

�iei ;

where j�i j �

qPniD1 �

2i D kuk � 1 for every i . is gives us

x D Nx C�

nu D Nx C

nXiD1

��i

nei D

nXiD1

1

n

�Nx C ��iei

�:

Denoting i WD ��i yields j i j � �. It follows from (i) that Nx C ��iei D Nx C iei 2 coA, andthus x 2 coA since it is a convex combination of elements in coA. �

Lemma 2.25 If a convex function f W Rn! R is bounded above on IB. NxI ı/ for some element Nx 2

domf and number ı > 0, then f is bounded on IB. NxI ı/.

Proof. Denote m WD f . Nx/ and takeM > 0 with f .x/ � M for all x 2 IB. NxI ı/. Picking any u 2

IB. NxI ı/, consider the element x WD 2 Nx � u. en x 2 IB. NxI ı/ and

m D f . Nx/ D f�x C u

2

��1

2f .x/C

1

2f .u/;

which shows that f .u/ � 2m � f .x/ � 2m �M and thus f is bounded on IB. NxI ı/. �

e next major result shows that the local boundedness of a convex function implies itslocal Lipschitz continuity.

eorem 2.26 Let f W Rn! R be convex with Nx 2 dom f . If f is bounded above on IB. NxI ı/

for some ı > 0, then f is Lipschitz continuous on IB. NxI ı=2/.

Proof. Fix x; y 2 IB. NxI ı=2/ with x ¤ y and consider the element

u WD x Cı

2kx � yk

�x � y

�:

Since e WDx � y

kx � yk2 IB , we have u D x C

ı

2e 2 Nx C

ı

2IB C

ı

2IB � Nx C ıIB . Denoting further

˛ WDı

2kx � ykgives us u D x C ˛.x � y/, and thus

x D1

˛ C 1uC

˛

˛ C 1y:


It follows from the convexity of f that

f .x/ �1

˛ C 1f .u/C

˛

˛ C 1f .y/;

which implies in turn the inequalities

f .x/ � f .y/ �1

˛ C 1.f .u/ � f .y// � 2M

1

˛ C 1D 2M

2kx � yk

ı C 2kx � yk�4M

ıkx � yk

with M WD supfjf .x/j j x 2 IB. NxI ı/g < 1 by Lemma 2.25. Interchanging the role of x and yabove, we arrive at the estimate

jf .x/ � f .y/j �4M

ıkx � yk

and thus verify the Lipschitz continuity of f on IB. NxI ı=2/. �

e following two consequences of eorem 2.26 ensure the automatic local Lipschitz con-tinuity of a convex function on appropriate sets.

Corollary 2.27 Any extended-real-valued convex function f W Rn! R is locally Lipschitz contin-

uous on in the interior of its domain int.domf /.

Proof. Pick Nx 2 int.dom f / and choose � > 0 such that Nx ˙ �ei 2 dom f for every i . Consider-ing the setA from Lemma 2.24, we get IB. NxI �=n/ � coA by assertion (ii) of this lemma. DenoteM WD maxff .a/ j a 2 Ag < 1 over the finite set A. Using the representation

x D

mXiD1

�iai with �i � 0

mXiD1

�i D 1; ai 2 A

for any x 2 IB. NxI �=n/ shows that

f .x/ �

mXiD1

�if .ai / �

mXiD1

�iM D M;

and so f is bounded above on IB. NxI �=n/. en eorem 2.26 tells us that f is Lipschitz con-tinuous on IB. NxI �=2n/ and thus it is locally Lipschitz continuous on int .domf /. �

It immediately follows from Corollary 2.27 that any finite convex function on Rn is alwayslocally Lipschitz continuous on the whole space.

Corollary 2.28 If f W Rn! R is convex, then it is locally Lipschitz continuous on Rn.

2.4. SUBGRADIENTSOFCONVEXFUNCTIONS 53

e final result of this section provides several characterizations of the local Lipschitz con-tinuity of extended-real-valued convex functions.

eorem 2.29 Let f W Rn! R be convex with Nx 2 domf . e following are equivalent:

(i) f is continuous at Nx.(ii) Nx 2 int.domf /.(iii) f is locally Lipschitz continuous around Nx.(iv) @1f . Nx/ D f0g.

Proof. To verify (i)H)(ii), by the continuity of f at Nx, for any � > 0 find ı > 0 such that

jf .x/ � f . Nx/j < � whenever x 2 IB. NxI ı/:

Since f . Nx/ < 1, this implies that IB. NxI ı/ � domf , and so Nx 2 int.domf /.Implication (ii)H)(iii) follows directly from Corollary 2.27 while (iii)H)(i) is obvious. usconditions (i), (ii), and (iii) are equivalent. To show next that (ii)H)(iv), take Nx 2 int.domf /

for which N. NxI dom f / D f0g. en Proposition 2.23 yields @1f . Nx/ D f0g.To verify finally (iv)H)(ii), we argue by contradiction and suppose that Nx … int.dom f /.

en Nx 2 bd.domf / and it follows from Corollary 2.14 thatN. NxI dom f / ¤ f0g, which contra-dicts the condition @1f . Nx/ D f0g by Proposition 2.23. �

2.4 SUBGRADIENTSOFCONVEXFUNCTIONSIn this section we define and start studying the concept of subdifferential (or the collection of sub-gradients) for convex extended-real-valued functions. is notion of generalized derivative is oneof the most fundamental in analysis, with a variety of important applications. e revolutionaryidea behind the subdifferential concept, which distinguishes it from other notions of generalizedderivatives in mathematics, is its set-valuedness as the indication of nonsmoothness of a functionaround the reference individual point. It not only provides various advantages and flexibility butalso creates certain difficulties in developing its calculus and applications. e major techniquesof subdifferential analysis revolve around convex separation of sets.

Definition 2.30 Let f W Rn! R be a convex function and let Nx 2 dom f . An element v 2 Rn is

called a of f at Nx if

hv; x � Nxi � f .x/ � f . Nx/ for all x 2 Rn : (2.13)

e collection of all the subgradients of f at Nx is called the of the function at thispoint and is denoted by @f . Nx/.


Figure 2.7: Subdifferential of the absolute value function at the origin.

Note that it suffices to take only x 2 dom f in the subgradient definition (2.13). Observealso that @f . Nx/ ¤ ; under more general conditions; see, e.g., Proposition 2.47 as well as otherresults in this direction given in [26].

e next proposition shows that the subdifferential @f . Nx/ can be equivalently defined ge-ometrically via the normal cone to the epigraph of f at . Nx; f . Nx//.

Proposition 2.31 For any convex function f W Rn! R and any Nx 2 dom f , we have

@f . Nx/ D˚v 2 Rn

ˇ.v;�1/ 2 N

�. Nx; f . Nx//I epif

�: (2.14)

Proof. Fix v 2 @f . Nx/ and .x; �/ 2 epif . Since � � f .x/, we deduce from (2.13) that˝.v;�1/; .x; �/ � . Nx; f . Nx//

˛D hv; x � Nxi � .� � f . Nx//

� hv; x � Nxi � .f .x/ � f . Nx// � 0:

is readily implies that .v;�1/ 2 N.. Nx; f . Nx//I epif /.To verify the opposite inclusion in (2.14), take an arbitrary element v 2 Rn such that

.v;�1/ 2 N.. Nx; f . Nx//I epif /. For any x 2 domf , we have .x; f .x// 2 epif . us

hv; x � Nxi � .f .x/ � f . Nx// D˝.v;�1/; .x; f .x// � . Nx; f . Nx//

˛� 0;

which shows that v 2 @f . Nx/ and so justifies representation (2.14). �

Along with representing the subdifferential via the normal cone in (2.14), there is the fol-lowing simple way to express the normal cone to a set via the subdifferential (as well as the singularsubdifferential (2.12)) of the extended-real-indicator function.


Example 2.32 Given a nonempty, convex set ˝ � Rn, define its indicator function by

f .x/ D ı.xI˝/ WD

(0 if x 2 ˝;

1 otherwise:(2.15)

en epi f D ˝ � Œ0;1/, and we have by Proposition 2.11(i) that


�D N. NxI˝/ � .�1; 0�;

which readily implies the relationships

@f . Nx/ D @1f . Nx/ D N. NxI˝/: (2.16)

A remarkable feature of convex functions is that every local minimizer for a convex functiongives also the global/absolute minimum to the function.

Proposition 2.33 Let f W Rn! R convex and let Nx 2 dom f be a local minimizer of f . en f

attains its global minimum at this point.

Proof. Since Nx is a local minimizer of f , there is ı > 0 such that

f .u/ � f . Nx/ for all u 2 IB. NxI ı/:

Fix x 2 Rn and construct a sequence of xk WD .1 � k�1/ Nx C k�1x as k 2 N. us we have xk 2

IB. NxI ı/ when k is sufficiently large. It follows from the convexity of f that

f . Nx/ � f .xk/ � .1 � k�1/f . Nx/C k�1f .x/;

which readily implies that k�1f . Nx/ � k�1f .x/, and hence f . Nx/ � f .x/ for all x 2 Rn. �

Next we recall the classical notion of derivative for functions on Rn, which we understandthroughout the book in the sense of Fréchet unless otherwise stated; see Section 3.1.

Definition 2.34 We say that f W Rn! R is .Fréchet/ at Nx 2 int.dom f / if there

exists an element v 2 Rn such that

limx! Nx

f .x/ � f . Nx/ � hv; x � Nxi

kx � NxkD 0:

In this case the element v is uniquely defined and is denoted by rf . Nx/ WD v.

e Fermat stationary rule of classical analysis tells us that if f W Rn! R is differentiable

at Nx and attains its local minimum at this point, then rf . Nx/ D 0. e following proposition


gives a subdifferential counterpart of this result for general convex functions. Its proof is a directconsequence of the definitions and Proposition 2.33.

Proposition 2.35 Let f W Rn! R be convex and let Nx 2 domf . en f attains its local/global

minimum at Nx if and only if 0 2 @f . Nx/.

Proof. Suppose that f attains its global minimum at Nx. en

f . Nx/ � f .x/ for all x 2 Rn;

which can be rewritten as

0 D h0; x � Nxi � f .x/ � f . Nx/ for all x 2 Rn :

e definition of the subdifferential shows that this is equivalent to 0 2 @f . Nx/. �

Now we show that the subdifferential (2.13) is indeed a singleton for differentiable func-tions reducing to the classical derivative/gradient at the reference point and clarifying the notionof differentiability in the case of convex functions.

Proposition 2.36 Let f W Rn! R be convex and differentiable at Nx 2 int.domf /. en we have

@f . Nx/ D frf . Nx/g and

hrf . Nx/; x � Nxi � f .x/ � f . Nx/ for all x 2 Rn : (2.17)

Proof. It follows from the differentiability of f at Nx that for any � > 0 there is ı > 0 with

� �kx � Nxk � f .x/ � f . Nx/ � hrf . Nx/; x � Nxi � �kx � Nxk whenever kx � Nxk < ı: (2.18)

Consider further the convex function

'.x/ WD f .x/ � f . Nx/ � hrf . Nx/; x � Nxi C �kx � Nxk; x 2 Rn;

and observe that '.x/ � '. Nx/ D 0 for all x 2 IB. NxI ı/. e convexity of ' ensures that '.x/ �

'. Nx/ for all x 2 Rn. us

hrf . Nx/; x � Nxi � f .x/ � f . Nx/C �kx � Nxk whenever x 2 Rn;

which yields (2.17) by letting � # 0.It follows from (2.17) that rf . Nx/ 2 @f . Nx/. Picking now v 2 @f . Nx/, we get

hv; x � Nxi � f .x/ � f . Nx/:

en the second part of (2.18) gives us that

hv � rf . Nx/; x � Nxi � �kx � Nxk whenever kx � Nxk < ı:

Finally, we observe that kv � rf . Nx/k � �, which yields v D rf . Nx/ since � > 0 was chosen ar-bitrarily. us @f . Nx/ D frf . Nx/g. �


Corollary 2.37 Let f W Rn! R be a strictly convex function. en the differentiability of f at

Nx 2 int.domf / implies the strict inequality

hrf . Nx/; x � Nxi < f .x/ � f . Nx/ whenever x ¤ Nx: (2.19)

Proof. Since f is convex, we get from Proposition 2.36 that

hrf . Nx/; u � Nxi � f .u/ � f . Nx/ for all u 2 Rn :

Fix x ¤ Nx and let u WD .x C Nx/=2. It follows from the above thatDrf . Nx/;

x C Nx

2� Nx

E� f

�x C Nx

2

�� f . Nx/ <

1

2f .x/C

1

2f . Nx/ � f . Nx/:

us we arrive at the strict inequality (2.19) and complete the proof. �

A simple while important example of a nonsmooth convex function is the norm functionon Rn. Here is the calculation of its subdifferential.

Example 2.38 Let p.x/ WD kxk be the Euclidean norm function on Rn. en we have

@p.x/ D

8<:IB if x D 0;n x

kxk

ootherwise:

To verify this, observe first that the Euclidean norm function p is differentiable at any nonzeropoint with rp.x/ D

x

kxkas x ¤ 0. It remains to calculate its subdifferential at x D 0. To proceed

by definition (2.13), we have that v 2 @p.0/ if and only if

hv; xi D hv; x � 0i � p.x/ � p.0/ D kxk for all x 2 Rn :

Letting x D v gives us hv; vi � kvk, which implies that kvk � 1, i.e., v 2 IB . Now take v 2 IB

and deduce from the Cauchy-Schwarz inequality that

hv; x � 0i D hv; xi � kvk � kxk � kxk D p.x/ � p.0/ for all x 2 Rn

and thus v 2 @p.0/, which shows that @p.0/ D IB .

e next theorem calculates the subdifferential of the distance function.

eorem 2.39 Let ˝ be a nonempty, closed, convex subset of Rn. en we have

@d. NxI˝/ D

8<:N. NxI˝/ \ IB if Nx 2 ˝;nNx �˘. NxI˝/

d. NxI˝/

ootherwise:


Proof. Consider the case where Nx 2 ˝ and fix any element w 2 @d. NxI˝/. It follows from thedefinition of the subdifferential that

hw; x � Nxi � d.xI˝/ � d. NxI˝/ D d.xI˝/ for all x 2 Rn : (2.20)

Since the distance function d.�I˝/ is Lipschitz continuous with constant ` D 1, we have

hw; x � Nxi � kx � Nxk for all x 2 Rn :

is implies that kwk � 1, i.e., w 2 IB . It also follows from (2.20) that

hw; x � Nxi � 0 for all x 2 ˝:

us w 2 N. NxI˝/, which verifies that w 2 N. NxI˝/ \ IB .To prove the opposite inclusion, fix any w 2 N. NxI˝/ \ IB . en kwk � 1 and

hw; u � Nxi � 0 for all u 2 ˝:

us for every x 2 Rn and every u 2 ˝ we have

hw; x � Nxi D hw; x � uC u � Nxi D hw; x � ui C hw; u � Nxi

� hw; x � ui � kwk � kx � uk � kx � uk:

Since u is taken arbitrarily in ˝, this implies hw; x � Nxi � d.xI˝/ D d.xI˝/ � d. NxI˝/, andhence w 2 @d. NxI˝/.

Suppose now that Nx … ˝, take Nz WD ˘. NxI˝/ 2 ˝, and fix any w 2 @d. NxI˝/. en

hw; x � Nxi � d.xI˝/ � d. NxI˝/ D d.xI˝/ � k Nx � Nzk

� kx � Nzk � k Nx � Nzk for all x 2 Rn :

Denoting p.x/ WD kx � Nzk, we have

hw; x � Nxi � p.x/ � p. Nx/ for all x 2 Rn;

which ensures that w 2 @p. Nx/ D

nNx � Nz

k Nx � Nzk

o. Let us show that w D

Nx � Nz

k Nx � Nzkis a subgradient of

d.�I˝/ at Nx. Indeed, for any x 2 Rn denote px WD ˘.xI˝/ and get

hw; x � Nxi D hw; x � Nzi C hw; Nz � Nxi D hw; x � Nzi � k Nx � Nzk

D hw; x � pxi C hw;px � Nzi � k Nx � Nzk:

Since Nz D ˘. NxI˝/, it follows from Proposition 1.78 that h Nx � Nz; px � Nzi � 0, and so we havehw;px � Nzi � 0. Using the fact that kwk D 1 and the Cauchy-Schwarz inequality gives us

hw; x � Nxi D hw; x � pxi C hw;px � Nzi � k Nx � Nzk

� kwk � kx � pxk � k Nx � Nzk D kx � pxk � k Nx � Nzk

D d.xI˝/ � d. NxI˝/ for all x 2 Rn :

us we arrive at w 2 @d. NxI˝/ and complete the proof of the theorem. �


We conclude this section by providing first-order and second-order differential charac-terizations of convexity for differentiable and twice continuously differentiable .C 2/ functions,respectively, on given convex regions of Rn.

eorem 2.40 Let f W Rn! R be a differentiable function on its domainD, which is an open

convex set. en f is convex if and only if

hrf .u/; x � ui � f .x/ � f .u/ for all x; u 2 D: (2.21)

Proof. e “only if ” part follows from Proposition 2.36. To justify the converse, suppose that(2.21) holds and then fix any x1; x2 2 D and t 2 .0; 1/. Denoting xt WD tx1 C .1 � t/x2, we havext 2 D by the convexity of D. en

hrf .xt /; x1 � xt i � f .x1/ � f .xt /; hrf .xt /; x2 � xt i � f .x2/ � f .xt /:

It follows furthermore that

thrf .xt /; x1 � xt i � tf .x1/ � tf .xt / and.1 � t /hrf .xt /; x2 � xt i � .1 � t/f .x2/ � .1 � t /f .xt /:

Summing up these inequalities, we arrive at

0 � tf .x1/C .1 � t/f .x2/ � f .xt /;

which ensures that f .xt / � tf .x1/C .1 � t /f .x2/, and so verifies the convexity of f . �

Lemma 2.41 Let f W Rn! R be convex and twice continuously differentiable on an open subset V

of its domain containing Nx. en we have

hr2f . Nx/u; ui � 0 for all u 2 Rn;

where r2f . Nx/ is the Hessian matrix of f at Nx.

Proof. Let A WD r2f . Nx/, which is symmetric matrix. en

limh!0

f . Nx C h/ � f . Nx/ � hrf . Nx/; hi �1

2hAh; hi

khk2D 0: (2.22)

It follows from (2.22) that for any � > 0 there is ı > 0 such that

��khk2

� f . Nx C h/ � f . Nx/ � hrf . Nx/; hi �1

2hAh; hi � �khk

2 for all khk � ı:


By Proposition 2.36, we readily have

f . Nx C h/ � f . Nx/ � hrf . Nx/; hi � 0:

Combining the above inequalities ensures that

� �khk2

�1

2hAh; hi whenever khk � ı: (2.23)

Observe further that for any 0 ¤ u 2 Rn the element h WD ıu

kuksatisfies khk � ı and, being

substituted into (2.23), gives us the estimate

��ı2�1

2ı2

DAu

kuk;u

kuk

E:

It shows therefore (since the case where u D 0 is trivial) that

�2�kuk2

� hAu; ui whenever u 2 Rn;

which implies by letting � # 0 that 0 � hAu; ui for all u 2 Rn. �

Now we are ready to justify the aforementioned second-order characterization of convexityfor twice continuously differentiable functions.

eorem 2.42 Let f W Rn! R be a function twice continuously differentiable on its domain

D, which is an open convex subset of Rn. en f is convex if and only if r2f . Nx/ is positivesemidefinite for every Nx 2 D.

Proof. Taking Lemma 2.41 into account, we only need to verify that if r2f . Nx/ is positivesemidefinite for every Nx 2 D, then f is convex. To proceed, for any x1; x2 2 D define xt WD

tx1 C .1 � t /x2 and consider the function

'.t/ WD f�tx1 C .1 � t /x2

�� tf .x1/ � .1 � t /f .x2/; t 2 R :

It is clear that ' is well defined on an open interval I containing .0; 1/. en '0.t/ D

hrf .xt /; x1 � x2i � f .x1/C f .x2/ and '00.t/ D hr2f .xt /.x1 � x2/; x1 � x2i � 0 for every t 2

I since r2f .xt / is positive semidefinite. It follows from Corollary 1.46 that ' is convex on I .Since '.0/ D '.1/ D 0, for any t 2 .0; 1/ we have

'.t/ D '�t.1/C .1 � t /0

�� t'.1/C .1 � t /'.0/ D 0;

which implies in turn the inequality

f�tx1 C .1 � t /x2

�� tf .x1/C .1 � t/f .x2/;

is justifies that the function f is convex on its domain and thus on Rn. �

2.5. BASICCALCULUSRULES 61

2.5 BASICCALCULUSRULESHere we derive basic calculus rules for the convex subdifferential (2.13) and its singular counter-part (2.12). First observe the obvious subdifferential rule for scalar multiplication.

Proposition 2.43 For a convex function f W Rn! R with Nx 2 domf , we have

@. f /. Nx/ D ˛@f . Nx/ and @1. f /. Nx/ D ˛@1f . Nx/ whenever ˛ > 0:

Let us now establish the fundamental subdifferential sum rules for both constructions (2.13)and (2.12) that play a crucial role in convex analysis. We give a geometric proof for this result re-ducing it to the intersection rule for normals to sets, which in turn is based on convex separation ofextremal set systems. Observe that our variational approach does not require any closedness/lowersemicontinuity assumptions on the functions in question.

eorem 2.44 Let fi W Rn! R for i D 1; 2 be convex functions and let Nx 2 dom f1 \ domf2.

Impose the singular subdifferential qualification condition

@1f1. Nx/ \�

� @1f2. Nx/�

D f0g: (2.24)

en we have the subdifferential sum rules

@.f1 C f2/. Nx/ D @f1. Nx/C @f2. Nx/; @1.f1 C f2/. Nx/ D @1f1. Nx/C @1f2. Nx/: (2.25)

Proof. For definiteness we verify the first rule in (2.25); the proof of the second result is similar.Fix an arbitrary subgradient v 2 @.f1 C f2/. Nx/ and define the sets

˝1 WD f.x; �1; �2/ 2 Rn� R � R j �1 � f1.x/g;

˝2 WD f.x; �1; �2/ 2 Rn� R � R j �2 � f2.x/g:

Let us first verify the inclusion

.v;�1;�1/ 2 N�. Nx; f1. Nx/; f2. Nx//I˝1 \˝2

�: (2.26)

For any .x; �1; �2/ 2 ˝1 \˝2, we have �1 � f1.x/ and �2 � f2.x/. en

hv; x � Nxi C .�1/.�1 � f1. Nx//C .�1/.�2 � f2. Nx//

� hv; x � Nxi ��f1.x/C f2.x/ � f1. Nx/ � f2. Nx/

�� 0;

where the last estimate holds due to v 2 @.f1 C f2/. Nx/. us (2.26) is satisfied.To proceed with employing the intersection rule in (2.26), let us first check that the assumed

qualification condition (2.24) yields

N�. Nx; f1. Nx/; f2. Nx//I˝1

�\

��N

�. Nx; f1. Nx/; f2. Nx//I˝2

��D f0g; (2.27)


which is the basic qualification condition (2.9) needed for the application of Corollary 2.16 in thesetting of (2.26). Indeed, we have from ˝1 D epif1 � R that

N�. Nx; f1. Nx/; f2. Nx//I˝1

�D N

�. Nx; f1. Nx//I epif1

�� f0g

and observe similarly the representation

N�. Nx; f1. Nx/; f2. Nx//I˝2

�D

˚.v; 0; / 2 Rn

� R � Rˇ.v; / 2 N


�:

Further, fix any element

.v; 1; 2/ 2 N�. Nx; f1. Nx/; f2. Nx//I˝1

�\

��N

�. Nx; f1. Nx/; f2. Nx//I˝2

��:

en 1 D 2 D 0, and hence

.v; 0/ 2 N�. Nx; f1. Nx//I epif1

�and .�v; 0/ 2 N


�:

It follows by the definition of the singular subdifferential that

v 2 @1f1. Nx/ \�

� @1f2. Nx/� D f0g:

It yields .v; 1; 2/ D .0; 0; 0/ 2 Rn� R � R. Applying Corollary 2.16 in (2.26) shows that

N�. Nx; f1. Nx/; f2. Nx//I˝1 \˝2

�D N

�. Nx; f1. Nx/; f2. Nx//I˝1

�CN

�. Nx; f1. Nx/; f2. Nx//I˝2

�;


.v;�1;�1/ D .v1;� 1; 0/C .v2; 0;� 2/

with .v1;� 1/ 2 N.. Nx; f1. Nx//I epif1/ and .v2;� 2/ 2 N.. Nx; f1. Nx//I epif2/. en we get 1 D

2 D 1 and v D v1 C v2, where vi 2 @fi . Nx/ for i D 1; 2. is verifies the inclusion “�” in the firstsubdifferential sum rule of (2.25).

e opposite inclusion therein follows easily from the definition of the subdifferential. In-deed, any v 2 @f1. Nx/C @f2. Nx/ can be represented as v D v1 C v2, where vi 2 @fi . Nx/ for i D 1; 2.en we have

hv; x � Nxi D hv1; x � Nxi C hv2; x � Nxi

� f1.x/ � f1. Nx/C f2.x/ � f2. Nx/ D .f1 C f1/.x/ � .f1 C f2/. Nx/

for all x 2 Rn, which readily implies that v 2 @.f1 C f2/. Nx/ and therefore completes the proofof the theorem. �

Note that by Proposition 2.23 the qualification condition (2.24) can be written as

N. NxI dom f1/ \�

�N. NxI dom f2/�

D f0g:


Let us present some useful consequences of eorem 2.44. e sum rule for the (basic) subdif-ferential (2.13) in the following corollary is known as the Moreau-Rockafellar theorem.

Corollary 2.45 Let fi W Rn! R for i D 1; 2 be convex functions such that there exists u 2

domf1 \ domf2 for which f1 is continuous at u or f2 is continuous at u. en

@.f1 C f2/.x/ D @f1.x/C @f2.x/ (2.28)

whenever x 2 domf1 \ domf2. Consequently, if both functions fi are finite-valued on Rn, then thesum rule (2.28) holds for all x 2 Rn.

Proof. Suppose for definiteness that f1 is continuous at u 2 domf1 \ domf2, which impliesthat u 2 int.domf1/ \ dom f2. en the proof of Corollary 2.18 shows that

N.xI domf1/ \�

�N.xI domf2/�

D f0g whenever x 2 dom f1 \ dom f2:

us the conclusion follows directly from eorem 2.44. �

e next corollary for sums of finitely many functions can be derived from eorem 2.44 byinduction, where the validity of the qualification condition in the next step of induction followsfrom the singular subdifferential sum rule in (2.25).

Corollary 2.46 Consider finitely many convex functions fi W Rn! R for i D 1; : : : ; m. e fol-

lowing assertions hold:(i) Let Nx 2

TmiD1 domfi and let the implicationh mX

iD1

vi D 0; vi 2 @1fi . Nx/i

H)�vi D 0 for all i D 1; : : : ; m

�(2.29)

hold. en we have the subdifferential sum rules

@� mX

iD1

fi

�. Nx/ D

mXiD1

@fi . Nx/; @1� mX

iD1

fi

�. Nx/ D

mXiD1

@fi . Nx/:

(ii) Suppose that there isu 2Tm

iD1 domfi such that all (except possibly one) functions fi are continuousat u. en we have the equality

@� mX

iD1

fi

�.x/ D

mXiD1

@fi .x/

for any common point of the domains x 2Tm

iD1 dom fi .


enext result not only ensures the subdifferentiability of convex functions, but also justifiesthe compactness of its subdifferential at interior points of the domain.

Proposition 2.47 For any convex function f W Rn! R and any Nx 2 int.domf /, we have that

the subgradient set @f . Nx/ is nonempty and compact in Rn.

Proof. Let us first verify the boundedness of the subdifferential. It follows from Corollary 2.27that f is locally Lipschitz continuous around Nx with some constant `, i.e., there exists a positivenumber � such that

jf .x/ � f .y/j � `kx � yk for all x; y 2 IB. NxI �/:

us for any subgradient v 2 @f . Nx/ and any element h 2 Rn with khk � ı we have

hv; hi � f . Nx C h/ � f . Nx/ � `khk:

is implies the bound kvk � `. e closedness of @f . Nx/ follows directly from the definition, andso we get the compactness of the subgradient set.

It remains to verify that @f . Nx/ ¤ ;. Since . Nx; f . Nx// 2 bd .epif /, we deduce fromCorollary 2.14 that N.. Nx; f . Nx//I epif / ¤ f0g. Take such an element .0; 0/ ¤ .v;� / 2

N.. Nx; f . Nx//I epif / and get by the definition that

hv; x � Nxi � .� � f . Nx// � 0 for all .x; �/ 2 epif:

Using this inequality with x WD Nx and � D f . Nx/C 1 yields � 0. If D 0, then v 2 @1f . Nx/ D

f0g by eorem 2.29, which is a contradiction. us > 0 and .v= ;�1/ 2 N.. Nx; f . Nx//I epif /.Hence v= 2 @f . Nx/, which completes the proof of the proposition. �

Given f W Rn! R and any ˛ 2 R, define the level set

˝˛ WD˚x 2 Rn

ˇf .x/ � ˛

:

Proposition 2.48 Let f W Rn! R be convex and let Nx 2 ˝˛ with f . Nx/ D ˛ for some ˛ 2 R. If

0 … @f . Nx/, then the normal cone to the level set˝˛ at Nx is calculated by

N. NxI˝˛/ D @1f . Nx/ [ R> @f . Nx/; (2.30)

where R> denotes the set of all positive real numbers.


Proof. To proceed, consider the set � WD Rn�f˛g � RnC1 and observe that

˝˛ � f˛g D epif \�:

Let us check the relationship

N�. Nx; ˛/I epif

�\

��N

�. Nx; ˛/I�

��D f0g; (2.31)

i.e., the qualification condition (2.9) holds for the sets epif and �. Indeed, take .v;��/ fromthe set on left-hand side of (2.31) and deduce from the structure of� that v D 0 and that � � 0.If we suppose that � > 0, then

1

�.v;��/ D .0;�1/ 2 N

�. Nx; ˛/I epif

�;

and so 0 2 @f . Nx/, which contradicts the assumption of this proposition. us we have .v;��/ D

.0; 0/, and then applying the intersection rule of Corollary 2.16 gives us

N. Nx;˝˛/ � R D N�. Nx; ˛/I˝˛ � f˛g

�D N

�. Nx; ˛/I epif

�C .f0g � R/:

Fix now any v 2 N. NxI˝˛/ and get .v; 0/ 2 N. Nx;˝˛/ � R. is implies the existence of � � 0

such that .v;��/ 2 N.. Nx; ˛/I epif / and .v; 0/ D .v;��/C .0; �/. If � D 0, then v 2 @1f . Nx/.In the case when � > 0 we obtain v 2 �@f . Nx/ and thus verify the inclusion “�” in (2.30).

Let us finally prove the opposite inclusion in (2.30). Since ˝˛ � domf , we get

@1f . Nx/ D N. NxI domf / � N. NxI˝˛/:

Take any v 2 R> @f . Nx/ and find � > 0 such that v 2 �@f . Nx/. Taking any x 2 ˝˛ gives us f .x/ �

˛ D f . Nx/. erefore, we have

h��1v; x � Nxi � f .x/ � f . Nx/ � 0;

which implies that v 2 N. NxI˝˛/ and thus completes the proof. �

Corollary 2.49 In the setting of Proposition 2.48 suppose that f is continuous at Nx and that 0 …

@f . Nx/. en we have the representation

N. NxI˝˛/ D RC @f . Nx/ D[˛�0

˛@f . Nx/; ˛ 2 R :

Proof. Since f is continuous at Nx, it follows from eorem 2.29 that @1f . Nx/ D f0g. en theresult of this corollary follows directly from Proposition 2.48. �


We now obtain some other calculus rules for subdifferentiation of convex function. Forbrevity, let us only give results for the basic subdifferential (2.13) leaving their singular subdiffer-ential counterparts as exercises; see Section 2.11.

To proceed with deriving chain rules, we present first the following geometric lemma.

Lemma 2.50 Let B W Rn! Rp be an affine mapping given by B.x/ D A.x/C b, where A W

Rn! Rp is a linear mapping and b 2 Rp. For any . Nx; Ny/ 2 gphB , we have

N�. Nx; Ny/I gphB

�D

˚.u; v/ 2 Rn

� Rpˇu D �A�v

:

Proof. It is clear that the set gphB is convex and .u; v/ 2 N.. Nx; Ny/ gphB/ if and only if

hu; x � Nxi C hv; B.x/ � B. Nx/i � 0 whenever x 2 Rn : (2.32)

It follows directly from the definitions that

hu; x � Nxi C hv;B.x/ � B. Nx/i D hu; x � Nxi C hv;A.x/ � A. Nx/i

D hu; x � Nxi C hA�v; x � Nxi

D huC A�v; x � Nxi;

which implies the equivalence of (2.32) to huC A�v; x � Nxi � 0, and so to u D �A�v. �

eorem2.51 Let f W Rp! R be a convex function and letB W Rn

! Rp be as in Lemma 2.50with B. Nx/ 2 domf for some Nx 2 Rp. Denote Ny WD B. Nx/ and assume that

kerA�\ @1f . Ny/ D f0g: (2.33)

en we have the subdifferential chain rule

@.f ı B/. Nx/ D A��@f . Ny/

�D

˚A�v

ˇv 2 @f . Ny/

: (2.34)

Proof. Fix v 2 @.f ı B/. Nx/ and form the subsets of Rn� Rp

� R by

˝1 WD gphB � R and ˝2 WD Rn�epif:

It follows from the definition of the subdifferential and the normal cone that .v; 0;�1/ 2

N.. Nx; Ny; Nz/I˝1 \˝2/, where Nz WD f . Ny/. Indeed, fix any .x; y; �/ 2 ˝1 \˝2. en y D B.x/

and � � f .y/, and so � � f .B.x//. us

hv; x � Nxi C 0.y � Ny/C .�1/.� � Nz/ � hv; x � Nxi � Œf .B.x// � f .B. Nx//� � 0:


Employing the intersection rule from Corollary 2.19 under (2.33) gives us

.v; 0;�1/ 2 N�. Nx; Ny; Nz/I˝1

�CN

�. Nx; Ny; Nz/I˝2

�;

which reads that .v; 0;�1/ D .v;�w; 0/C .0; w;�1/ with .v;�w/ 2 N.. Nx; Ny/I gphB/ and.w;�1/ 2 N.. Ny; Nz/I epif /. en we get

v D A�w and w 2 @f . Ny/;

which implies in turn that v 2 A�.@f . Ny// and thus verifies the inclusion “�” in (2.34). e op-posite inclusion follows directly from the definition of the subdifferential. �

In the corollary below we present some conditions that guarantee that the qualification(2.33) in eorem 2.51 is satisfied.

Corollary 2.52 In the setting of eorem 2.51 suppose that either A is surjective or f is continuousat the point Ny 2 Rn; the latter assumption is automatic when f is finite-valued. en we have thesubdifferential chain rule (2.34).

Proof. Let us check that the qualification condition (2.33) holds under the assumptions ofthis corollary. e surjectivity of A gives us kerA� D f0g, and in the second case we haveNy 2 int.domf /, so @1f . Ny/ D f0g. e validity of (2.33) is obvious. �

e last consequence of eorem 2.51 presents a chain rule for normals to sets.

Corollary 2.53 Let ˝ � Rp be a convex set and let B W Rn! Rp be an affine mapping as in

Lemma 2.50 satisfying Ny WD B. Nx/ 2 ˝. en we have

N. NxIB�1.˝// D A�.N. NyI˝//

under the validity of the qualification condition

kerA�\N. NyI˝/ D f0g:

Proof. is follows from eorem 2.51 with f .x/ D ı.xI˝/. �

Given fi W Rn! R for i D 1; : : : ; m, consider the maximum function

f .x/ WD maxiD1;:::;m

fi .x/; x 2 Rn; (2.35)

and for Nx 2 Rn define the set

I. Nx/ WD˚i 2 f1; : : : ; mg

ˇfi . Nx/ D f . Nx/

:


In the proposition below we assume that each function fi for i D 1; : : : ; m is continuousat the reference point. A more general version follows afterward.

Proposition 2.54 Let fi W Rn! R, i D 1; : : : ; m, be convex functions. Take any point Nx 2Tm

iD1 dom fi and assume that each fi is continuous at Nx. en we have the maximum rule

@�maxfi /. Nx/ D co

[i2I. Nx/

@fi . Nx/:

Proof. Let f .x/ be the maximum function defined in (2.35). We obviously have

epif D

m\iD1

epifi :

Since fi . Nx/ < f . Nx/ DW N for any i … I. Nx/, there exist a neighborhood U of Nx and ı > 0 suchthat fi .x/ < ˛ whenever .x; ˛/ 2 U � . N � ı; N C ı/. It follows that . Nx; N / 2 int epi fi , and soN.. Nx; N /I epifi / D f.0; 0/g for such indices i .

Let us show that the qualification condition from Corollary 2.19 is satisfied for the sets˝i WD epifi , i D 1; : : : ; m, at . Nx; N /. Fix .vi ;��i / 2 N

�. Nx; N /I epifi

�with

mXiD1

.vi ; �i / D 0:

en .vi ;��i / D .0; 0/ for i … I. Nx/, and henceXi2I. Nx/

.vi ;��i / D 0:

Note that fi . Nx/ D f . Nx/ D N for i 2 I. Nx/. e proof of Proposition 2.47 tells us that �i � 0

for i 2 I. Nx/. us the equalityP

i2I. Nx/ �i D 0 yields �i D 0, and so vi 2 @1fi . Nx/ D f0g forevery i 2 I. Nx/ by eorem 2.29. is verifies that .vi ;��i / D .0; 0/ for i D 1; : : : ; m. It followsfurthermore that


�D

mXiD1

N�. Nx; N /I epifi

�D

Xi2I. Nx/

N�. Nx; fi . Nx//I epifi

�:

For any v 2 @f . Nx/, we have .v;�1/ 2 N.. Nx; f . Nx//I epif /, which allows us to find .vi ;��i / 2

N.. Nx; fi . Nx//I epifi / for i 2 I. Nx/ such that

.v;�1/ DX

i2I. Nx/

.vi ;��i /:

is yieldsP

i2I. Nx/ �i D 1 and v DP

i2I. Nx/ vi . If �i D 0, then vi 2 @1fi . Nx/ D f0g. Hence weassume without loss of generality that �i > 0 for every i 2 I. Nx/. Since N.. Nx; fi . Nx//I epifi / is a


cone, it gives us .vi=�i ;�1/ 2 N.. Nx; fi . Nx//I epifi /, and so vi 2 �i@fi . Nx/. We get the represen-tation v D

Pi2I. Nx/ �iui , where ui 2 @fi . Nx/ and

Pi2I. Nx/ �i D 1. us

v 2 co[

i2I. Nx/

@fi . Nx/:

Note that the opposite inclusion follows directly from

@fi . Nx/ � @f . Nx/ for all i 2 I. Nx/;

which in turn follows from the definition. Indeed, for any i 2 I. Nx/ and v 2 @fi . Nx/ we have

hv; x � Nxi � fi .x/ � fi . Nx/ D fi .x/ � f . Nx/ � f .x/ � f . Nx/ whenever x 2 Rn;

which implies that v 2 @f . Nx/ and thus completes the proof. �

To proceed with the general case, define

�. Nx/ WD˚.�1; : : : ; �m/

ˇ�i � 0;

mXiD1

�i D 1; �i

�fi . Nx/ � f . Nx/

�D 0

and observe from the definition of �. Nx/ hat

�. Nx/ D˚.�1; : : : ; �m/

ˇ�i � 0;

Xi2I. Nx/

�i D 1; �i D 0 for i … I. Nx/:

eorem 2.55 Let fi W Rn! R be convex functions for i D 1; : : : ; m. Take any point Nx 2Tm

iD1 dom fi and assume that each fi is continuous at Nx for any i … I. Nx/ and that the quali-fication conditionh X

i2I. Nx/

vi D 0; vi 2 @1fi . Nx/i

H)�vi D 0 for all i 2 I. Nx/

�holds. en we have the maximum rule

@�max fi

�. Nx/ D

[ n Xi2I. Nx/

�i ı @fi . Nx/ˇ.�1; : : : ; �m/ 2 �. Nx/

o;

where �i ı @fi . Nx/ WD �i@fi . Nx/ if �i > 0, and �i ı @fi . Nx/ WD @1fi . Nx/ if �i D 0.

Proof. Let f .x/ be the maximum function from (2.35). Repeating the first part of the proof ofProposition 2.54 tells us thatN.. Nx; f . Nx//I epifi / D f.0; 0/g for i … I. Nx/. e imposed qualifica-tion condition allows us to employ the intersection rule from Corollary 2.19 to get


�D

mXiD1

N�. Nx; f . Nx//I epifi

�D

Xi2I. Nx/

N�. Nx; fi . Nx//I epifi

�: (2.36)


For any v 2 @f . Nx/, we have .v;�1/ 2 N.. Nx; f . Nx//I epi f /, and thus find by using (2.36) suchpairs .vi ;��i / 2 N.. Nx; fi . Nx//I epifi / for i 2 I. Nx/ that

.v;�1/ DX

i2I. Nx/

.vi ;��i /:

is shows thatP

i2I. Nx/ �i D 1 and v DP

i2I. Nx/ vi with vi 2 �i ı @fi . Nx/ and �i � 0 for all i 2

I. Nx/. erefore, we arrive at

v 2[ n X

i2I. Nx/

�i ı @fi . Nx/ˇ.�1; : : : ; �m/ 2 �. Nx/

o;

which justifies the inclusion “�” in the maximum rule. e opposite inclusion can also be verifiedeasily by using representations (2.36). �

2.6 SUBGRADIENTSOFOPTIMALVALUEFUNCTIONSis section is mainly devoted to the study of the class of optimal value/marginal functions definedin (1.4). To proceed, we first introduce and examine the notion of coderivatives for set-valuedmappings, which is borrowed from variational analysis and plays a crucial role in many aspects ofthe theory and applications of set-valued mappings; in particular, those related to subdifferenti-ation of optimal value functions considered below.

Definition 2.56 Let F W Rn!! Rp be a set-valued mapping with convex graph and let . Nx; Ny/ 2

gphF . e of F at . Nx; Ny/ is the set-valued mappingD�F. Nx; Ny/W Rp!! Rn with the

valuesD�F. Nx; Ny/.v/ WD

˚u 2 Rn

ˇ.u;�v/ 2 N

�. Nx; Ny/I gphF

�; v 2 Rp : (2.37)

It follows from (2.37) and definition (2.4) of the normal cone to convex sets that thecoderivative values of convex-graph mappings can be calculated by

D�F. Nx; Ny/.v/ D

nu 2 Rn

ˇhu; Nxi � hv; Nyi D max

.x;y/2gph F

�hu; xi � hv; yi

�o:

Now we present explicit calculations of the coderivative in some particular settings.

Proposition 2.57 LetF W Rn! Rp be a single-valued mapping given byF.x/ WD Ax C b, where

A W Rn! Rp is a linear mapping, and where b 2 Rp. For any pair . Nx; Ny/ such that Ny D A Nx C b,

we have the representation

D�F. Nx; Ny/.v/ D fA�vg whenever v 2 Rp :

2.6. SUBGRADIENTSOFOPTIMALVALUEFUNCTIONS 71

Proof. Lemma 2.50 tells us that

N�. Nx; Ny/I gphF / D

˚.u; v/ 2 Rn

� Rpˇu D �A�v

:

us u 2 D�F. Nx; Ny/.v/ if and only if u D A�v. �

Proposition 2.58 Given a convex function f W Rn! R with Nx 2 domf , define the set-valued

mapping F W Rn!! R by

F.x/ WD�f .x/;1

�:

Denoting Ny WD f . Nx/, we have

D�F. Nx; Ny/.�/ D

8<:�@f . Nx/ if � > 0;@1f . Nx/ if � D 0;

; if � < 0:

In particular, if f is finite-valued, thenD�F. Nx; Ny/.�/ D �@f . Nx/ for all � � 0.

Proof. It follows from the definition of F that

gphF D˚.x; �/ 2 Rn

� Rˇ� 2 F.x/

D

˚.x; �/ 2 Rn

� Rˇ� � f .x/

D epif:

us the inclusion v 2 D�F. Nx; Ny/.�/ is equivalent to .v;��/ 2 N.. Nx; Ny/I epif /. If � > 0, then.v=�;�1/ 2 N.. Nx; Ny/I epif /, which implies v=� 2 @f . Nx/, and hence v 2 �@f . Nx/. e formulain the case where � D 0 follows from the definition of the singular subdifferential. If � < 0, thenD�F. Nx; Ny/.�/ D ; because � � 0 whenever .v;��/ 2 N.. Nx; Ny/I epif / by the proof of Propo-sition 2.47. To verify the last part of the proposition, observe that in this case D�F. Nx; Ny/.0/ D

@1f . Nx/ D f0g by Proposition 2.23, so we arrive at the conclusion. �

e next simple example is very natural and useful in what follows.

Example 2.59 Define F W Rn!! Rp by F.x/ D K for every x 2 Rn, where K � Rp

is a nonempty, convex set. en gphF D Rn�K, and Proposition 2.11(i) tells us that

N.. Nx; Ny/I gphF / D f0g �N. NyIK/ for any . Nx; Ny/ 2 Rn�K. us

D�F. Nx; Ny/.v/ D

(f0g if � v 2 N. NyIK/;

; otherwise:

In particular, for the case where K D Rp we have

D�F. Nx; Ny/.v/ D

(f0g if v D 0;

; otherwise:


Now we derive a formula for calculating the subdifferential of the optimal value/marginal function(1.4) given in the form


ˇy 2 F.x/

via the coderivative of the set-valued mapping F and the subdifferential of the function '. Firstwe derive the following useful estimate.

Lemma 2.60 Let �.�/ be the optimal value function (1.4) generated by a convex-graph mappingF W Rn

!! Rp and a convex function ' W Rn� Rp

! R. Suppose that �.x/ > �1 for all x 2 Rn,fix some Nx 2 dom�, and consider the solution set

S. Nx/ WD˚

Ny 2 F. Nx/ˇ�. Nx/ D '. Nx; Ny/

:

If S. Nx/ ¤ ;, then for any Ny 2 S. Nx/ we have[.u;v/2@'. Nx; Ny/

�uCD�F. Nx; Ny/.v/

�� @�. Nx/: (2.38)

Proof. Pick w from the left-hand side of (2.38) and find .u; v/ 2 @'. Nx; Ny/ with

w � u 2 D�F. Nx; Ny/.v/:

It gives us .w � u;�v/ 2 N.. Nx; Ny/I gphF / and thus

hw � u; x � Nxi � hv; y � Nyi � 0 for all .x; y/ 2 gphF;

which shows that whenever y 2 F.x/ we have

hw; x � Nxi � hu; x � Nxi C hv; y � Nyi � '.x; y/ � '. Nx; Ny/ D '.x; y/ � �. Nx/:

is implies in turn the estimate

hw; x � Nxi � infy2F .x/

'.x; y/ � �. Nx/ D �.x/ � �. Nx/;

which justifies the inclusion w 2 @�. Nx/, and hence completes the proof of (2.38). �

eorem2.61 Consider the optimal value function (1.4) under the assumptions of Lemma 2.60.For any Ny 2 S. Nx/, we have the equality

@�. Nx/ D[

.u;v/2@'. Nx; Ny/

�uCD�F. Nx; Ny/.v/

�(2.39)

provided the validity of the qualification condition

@1'. Nx; Ny/ \�

�N.. Nx; Ny/I gphF /�

D f.0; 0/g; (2.40)

which holds, in particular, when ' is continuous at . Nx; Ny/.

2.6. SUBGRADIENTSOFOPTIMALVALUEFUNCTIONS 73

Proof. Taking Lemma 2.60 into account, it is only required to verify the inclusion “�” in (2.39).To proceed, pick w 2 @�. Nx/ and Ny 2 S. Nx/. For any x 2 Rn, we have

hw; x � Nxi � �.x/ � �. Nx/ D �.x/ � '. Nx; Ny/

� '.x; y/ � '. Nx; Ny/ for all y 2 F.x/:

is implies that, whenever .x; y/ 2 Rn� Rp, the following inequality holds:

hw; x � Nxi C h0; y � Nyi � '.x; y/C ı�.x; y/I gphF / �

�'. Nx; Ny/C ı

�. Nx; Ny/I gphF

��:

Denote further f .x; y/ WD '.x; y/C ı..x; y/I gphF / and deduce from the subdifferential sumrule in eorem 2.44 under the qualification condition (2.40) that the inclusion

.w; 0/ 2 @f . Nx; Ny/ D @'. Nx; Ny/CN�. Nx; Ny/I gphF

�is satisfied. is shows that .w; 0/ D .u1; v1/C .u2; v2/ with .u1; v1/ 2 @'. Nx; Ny/ and .u2; v2/ 2

N.. Nx; Ny/I gphF /. is yields v2 D �v1, so .u2;�v1/ 2 N.. Nx; Ny/I gphF /. Finally, we get u2 2

D�F. Nx; Ny/.v1/ and therefore

w D u1 C u2 2 u1 CD�F. Nx; Ny/.v1/;


Let us present several important consequences of eorem 2.61. e first one concerns thefollowing chain rule for convex compositions.

Corollary 2.62 Let f W Rn! R be a convex function and let ' W R ! R be a nondecreasing convex

function. Take Nx 2 Rn and denote Ny WD f . Nx/ assuming that Ny 2 dom'. en

@.' ı f /. Nx/ D[

�2@'. Ny/

�@f . Nx/

under the assumption that either @1'. Ny/ D f0g or 0 … @f . Nx/.

Proof. e composition ' ı f is a convex function by Proposition 1.39. Define the set-valuedmapping F.x/ WD Œf .x/;1/ and observe that

.' ı f /.x/ D infy2F .x/

'.y/

since ' is nondecreasing. Note each of the assumptions @1'. Ny/ D f0g and 0 … @f . Nx/ ensure thevalidity of the qualification condition (2.40). It follows from eorem 2.61 that

@.' ı f /. Nx/ D[

�2@'. Ny/

D�F. Nx; Ny/.�/:


Taking again into account that ' is nondecreasing yields � � 0 for every � 2 @'. Ny/. is tells usby Proposition 2.58 that

@.' ı f /. Nx/ D[

�2@'. Ny/

D�F. Nx; Ny/.�/ D[

�2@'. Ny/

�@f . Nx/;

which verifies the claimed chain rule of this corollary. �

e next corollary addresses calculating subgradients of the optimal value functions overparameter-independent constraints.

Corollary 2.63 Let ' W Rn� Rp

! R, where the function '.x; �/ is bounded below on Rp for everyx 2 Rn. Define


ˇy 2 Rp

; S.x/ WD

˚y 2 Rp

ˇ'.x; y/ D �.x/

:

For Nx 2 dom� and for every Ny 2 S. Nx/, we have

@�. Nx/ D˚u 2 Rn

ˇ.u; 0/ 2 @'. Nx; Ny/

:

Proof. Follows directly from eorem 2.61 for F.x/ � Rp and Example 2.59. Note that thequalification condition (2.40) is satisfied since N.. Nx; Ny/I gphF / D f.0; 0/g. �

e next corollary specifies the general result for problems with affine constraints.

Corollary 2.64 LetB W Rp! Rn be given byB.y/ WD Ay C b via a linearmappingAW Rp

! Rn

and an element b 2 Rn and let ' W Rp! R be a convex function bounded below on the inverse image

set B�1.x/ for all x 2 Rn. Define

�.x/ WD inf˚'.y/

ˇB.y/ D xg; S.x/ WD

˚y 2 B�1.x/

ˇ'.y/ D �.x/

and fix Nx 2 Rn with �. Nx/ < 1 and S. Nx/ ¤ ;. For any Ny 2 S. Nx/, we have

@�. Nx/ D .A�/�1�@'. Ny/

�:

Proof. e setting of this corollary corresponds to eorem 2.61 with F.x/ D B�1.x/ and'.x; y/ � '.y/. en we have

N�. Nx; Ny/I gphF

�D

˚.u; v/ 2 Rn

� Rpˇ

� A�u D v;

which therefore implies the coderivative representation

D�F. Nx; Ny/.v/ D˚u 2 Rn

ˇA�u D v

D .A�/�1.v/; v 2 Rp :

2.7. SUBGRADIENTSOF SUPPORTFUNCTIONS 75

It is easy to see that the qualification condition (2.40) is automatic in this case. is yields

@�. Nx/ D[

v2@'. Ny/

�D�F. Nx; Ny/.v/

�D .A�/�1.@'. Ny//;

and thus we complete the proof. �

Finally in this section, we present a useful application of Corollary 2.64 to deriving a cal-culus rule for subdifferentiation of infimal convolutions important in convex analysis.

Corollary 2.65 Given convex functions f1; f2W Rn! R, consider their infimal convolution

.f1 ˚ f2/.x/ WD inf˚f1.x1/C f2.x2/

ˇx1 C x2 D x

; x 2 Rn;

and suppose that .f1 ˚ f2/.x/ > �1 for all x 2 Rn. Fix Nx 2 dom .f1 ˚ f2/ and Nx1; Nx2 2 Rn sat-isfying Nx D Nx1 C Nx2 and .f1 ˚ f2/. Nx/ D f1. Nx1/C f2. Nx2/. en we have

@.f1 ˚ f2/. Nx/ D @f1. Nx1/ \ @f2. Nx2/:

Proof. Define 'W Rn� Rn

! R and AW Rn� Rn

! Rn by

'.x1; x2/ WD f1.x1/C f2.x2/; A.x1; x2/ WD x1 C x2:

It is easy to see that A�v D .v; v/ for any v 2 Rn and that @'. Nx1; Nx2/ D .@f1. Nx1/; @f2. Nx2// for. Nx1; Nx2/ 2 Rn

� Rn. en v 2 @.f1 ˚ f2/. Nx/ if and only if we have A�v D .v; v/ 2 @'. Nx1; Nx2/,which means that v 2 @f1. Nx1/ \ @f2. Nx2/. us the claimed calculus result follows from Corol-lary 2.64. �

2.7 SUBGRADIENTSOF SUPPORTFUNCTIONSis short section is devoted to the study of subgradient properties for an important class ofnonsmooth convex functions known as support functions of convex sets.

Definition 2.66 Given a nonempty .not necessarily convex/ subset˝ ofRn, the of˝ is defined by

�˝.x/ WD sup˚hx; !i

ˇ! 2 ˝

: (2.41)

First we present some properties of (2.41) that can be deduced from the definition.

Proposition 2.67 e following assertions hold:(i) For any nonempty subset˝ of Rn, the support function �˝ is subadditive, positively homogeneous,and hence convex.


(ii) For any nonempty subsets˝1; ˝2 � Rn and any x 2 Rn, we have

�˝1C˝2.x/ D �˝1

.x/C �˝2.x/; �˝1[˝2

.x/ D max˚�˝1

.x/; �˝2.x/

:

e next result calculates subgradients of support functions via convex separation.

eorem 2.68 Let ˝ be a nonempty, closed, convex subset of Rn and let

S.x/ WD˚p 2 ˝

ˇhx; pi D �˝.x/

:

For any Nx 2 dom �˝ , we have the subdifferential formula

@�˝. Nx/ D S. Nx/:

Proof. Fix p 2 @�˝. Nx/ and get from definition (2.13) that

hp; x � Nxi � �˝.x/ � �˝. Nx/ whenever x 2 Rn : (2.42)

Taking this into account, it follows for any u 2 Rn that

hp; ui D hp; uC Nx � Nxi � �˝.uC Nx/ � �˝. Nx/ � �˝.u/C �˝. Nx/ � �˝. Nx/ D �˝.u/: (2.43)

Assuming that p 62 ˝, we use the separation result of Proposition 2.1 and find u 2 Rn with

�˝.u/ D sup˚hu; !i

ˇ! 2 ˝

< hp; ui;

which contradicts (2.43) and so verifies p 2 ˝. From (2.43) we also get hp; Nxi � �˝. Nx/. Lettingx D 0 in (2.42) gives us �˝. Nx/ � hp; Nxi. Hence hp; Nxi D �˝. Nx/ and thus p 2 S. Nx/.

Now fix p 2 S. Nx/ and get from p 2 ˝ and hp; Nxi D �˝. Nx/ that

hp; x � Nxi D hp; xi � hp; Nxi � �˝.x/ � �˝. Nx/ for all x 2 Rn :

is shows that p 2 @�˝. Nx/ and thus completes the proof. �

Note that eorem 2.68 immediately implies that

@�˝.0/ D S.0/ D˚! 2 ˝

ˇh0; !i D �˝.0/ D 0

D ˝ (2.44)

for any nonempty, closed, convex set ˝ � Rn.e last result of this section justifies a certain monotonicity property of support functions

with respect to set inclusions.

Proposition 2.69 Let ˝1 and ˝2 be a nonempty, closed, convex subsets of Rn. en the inclusion˝1 � ˝2 holds if and only if �˝1

.v/ � �˝2.v/ for all v 2 Rn.

2.8. FENCHELCONJUGATES 77

Proof. Taking ˝1 � ˝2 as in the proposition, for any v 2 Rn we have

�˝1.v/ D sup

˚hv; xi

ˇx 2 ˝1

� sup

˚hv; xi

ˇx 2 ˝2

D �˝2

.v/:

Conversely, suppose that �˝1.v/ � �˝2

.v/whenever v 2 Rn. Since �˝1.0/ D �˝2

.0/ D 0, it fol-lows from definition (2.13) of the subdifferential that

@�˝1.0/ � @�˝2

.0/;

which yields ˝1 � ˝2 by formula (2.44). �

2.8 FENCHELCONJUGATESMany important issues of convex analysis and its applications (in particular, to optimization) arebased on duality. e following notion plays a crucial role in duality considerations.

Definition 2.70 Given a function f W Rn! R .not necessarily convex/, its F

f � W Rn! Œ�1;1� is

f �.v/ WD sup˚hv; xi � f .x/

ˇx 2 Rn

D sup

˚hv; xi � f .x/

ˇx 2 domf

: (2.45)

Note that f �.v/ D �1 is allowed in (2.45) while f �.v/ > �1 for all v 2 Rn if domf ¤

;. It follows directly from the definitions that for any nonempty set ˝ � Rn the conjugate to itsindicator function is the support function of ˝:

ı�˝.v/ D sup

˚hv; xi

ˇx 2 ˝

D �˝.v/; v 2 ˝: (2.46)

e next two propositions can be easily verified.

Proposition 2.71 Let f W Rn! R be a function, not necessarily convex, with domf ¤ ;. en

its Fenchel conjugate f � is convex on Rn.

Proof. Function (2.45) is convex as the supremum of a family of affine functions. �

Proposition 2.72 Let f; g W Rn! R be such that f .x/ � g.x/ for all x 2 Rn. en we have

f �.v/ � g�.v/ for all v 2 Rn.

Proof. For any fixed v 2 Rn, it follows from (2.45) that

hv; xi � f .x/ � hv; xi � g.x/; x 2 Rn :

is readily implies the relationships

f �.v/ D sup˚hv; xi � f .x/

ˇx 2 Rn

� sup

˚hv; xi � g.x/

ˇx 2 Rn

D g�.v/

for all v 2 Rn, and therefore f � � g� on Rn. �


Figure 2.8: Fenchel conjugate.

e following two examples illustrate the calculation of conjugate functions.

Example 2.73 (i) Given a 2 Rn and b 2 R, consider the affine function

f .x/ WD ha; xi C b; x 2 Rn :

en it can be seen directly from the definition that

f �.v/ D

(�b if v D a;

1 otherwise:

(ii) Given any p > 1, consider the power function

f .x/ WD

8<:xp

pif x � 0;

1 otherwise:

For any v 2 R, the conjugate of this function is given by

f �.v/ D supnvx �

xp

p

ˇx � 0

oD � inf

nxp

p� vx

ˇx � 0

o:

It is clear that f �.v/ D 0 if v � 0 since in this case vx � p�1xp � 0 when x � 0. Consideringthe case of v > 0, we see that function v.x/ WD p�1xp � vx is convex and differentiable on


.0;1/ with 0v.x/ D xp�1 � v. us 0

v.x/ D 0 if and only if x D v1=.p�1/, and so v attainsits minimum at x D v1=.p�1/. erefore, the conjugate function is calculated by

f �.v/ D

�1 �

1

p

�vp=.p�1/; v 2 Rn :

Taking q from q�1 D 1 � p�1, we express the conjugate function as

f �.v/ D

8<:0 if v � 0;

vq

qotherwise:

Note that the calculation in Example 2.73(ii) shows that

vx �xp

pCvq

qfor any x; v � 0:

e first assertion of the next proposition demonstrates that such a relationship in a more generalsetting. To formulate the second assertion below, we define the biconjugate of f as the conjugateof f �, i.e., f ��.x/ WD .f �/�.x/.

Proposition 2.74 Let f W Rn! R be a function with domf ¤ ;. en we have:

(i) hv; xi � f .x/C f �.v/ for all x; v 2 Rn.(ii) f ��.x/ � f .x/ for all x 2 Rn.

Proof. Observe first that (i) is obvious if f .x/ D 1. If x 2 domf , we get from (2.45) thatf �.v/ � hv; xi � f .x/, which verifies (i). It implies in turn that

sup˚hv; xi � f �.v/

ˇv 2 Rn

� f .x/ for all x; v 2 Rn;

which thus verifies (ii) and completes the proof. �

e following important result reveals a close relationship between subgradients andFenchel conjugates of convex functions.

eorem 2.75 For any convex function f W Rn! R and any Nx 2 domf , we have that v 2

@f . Nx/ if and only iff . Nx/C f �.v/ D hv; Nxi: (2.47)


Proof. Taking any v 2 @f . Nx/ and using definition (2.13) gives us

f . Nx/C hv; xi � f .x/ � hv; Nxi for all x 2 Rn :

is readily implies the inequality

f . Nx/C f �.v/ D f . Nx/C sup˚hv; xi � f .x/

ˇx 2 Rn

� hv; Nxi:

Since the opposite inequality holds by Proposition 2.74(i), we arrive at (2.47).Conversely, suppose that f . Nx/C f �.v/ D hv; Nxi. Applying Proposition 2.74(i), we get the

estimate f �.v/ � hv; xi � f .x/ for every x 2 Rn. is shows that v 2 @f . Nx/. �

e result obtained allows us to find conditions ensuring that the biconjugate f �� of aconvex function agrees with the function itself.

Proposition 2.76 Let Nx 2 domf for a convex function f W Rn! R. Suppose that @f . Nx/ ¤ ;.en

we have the equality f ��. Nx/ D f . Nx/.

Proof. By Proposition 2.74(ii) it suffices to verify the opposite inequality therein. Fix v 2 @f . Nx/

and get hv; Nxi D f . Nx/C f �.v/ by the preceding theorem. is shows that

f . Nx/ D hv; Nxi � f �.v/ � sup˚h Nx; vi � f �.v/

ˇv 2 Rn

D f ��. Nx/;

which completes the proof of this proposition. �

Taking into account Proposition 2.47, we arrive at the following corollary.

Corollary 2.77 Let f W Rn! R be a convex function and let Nx 2 int.domf /. en we have the

equality f ��. Nx/ D f . Nx/ .

Finally in this section, we prove a necessary and sufficient condition for the validity of thebiconjugacy equality f D f �� known as the Fenchel-Moreau theorem.

eorem 2.78 Let f W Rn! R be a function with domf ¤ ; and let A be the set of all affine

functions of the form '.x/ D ha; xi C b for x 2 Rn, where a 2 Rn and b 2 R. Denote

A.f / WD˚' 2 A

ˇ'.x/ � f .x/ for all x 2 Rn

:

en A.f / ¤ ; whenever epif is closed and convex. Moreover, the following are equivalent:(i) epif is closed and convex.(ii) f .x/ D sup'2A.f / '.x/ for all x 2 Rn.(iii) f ��.x/ D f .x/ for all x 2 Rn.


Proof. Let us first show that A.f / ¤ ;. Fix any x0 2 domf and choose �0 < f .x0/. en.x0; �0/ … epif . By Proposition 2.1, there exist . Nv; N / 2 Rn

� R and � > 0 such that

h Nv; xi C N � < h Nv; x0i C N �0 � � whenever .x; �/ 2 epif: (2.48)

Since .x0; f .x0/C ˛/ 2 epif for all ˛ � 0, we get

N .f . Nx/C ˛/ < N �0 � � whenever ˛ � 0:

is implies N < 0 since if not, we can let ˛ ! 1 and arrive at a contradiction. For any x 2

domf , it follows from (2.48) as .x; f .x// 2 epif that

h Nv; xi C N f .x/ < h Nv; x0i C N �0 � � for all x 2 dom f:

is allows us to conclude that

f .x/ > hNv

N ; x0 � xi C �0 �

�

N if x 2 dom f:

Define now '.x/ WD hNv

N ; x0 � xi C �0 �

�

N . en ' 2 A.f /, and so A.f / ¤ ;.

Let us next prove that (i)H)(ii). By definition we need to show that for any �0 < f .x0/

there is ' 2 A.f / such that �0 < '.x0/. Since .x0; �0/ … epif , we apply again Proposition 2.1to obtain (2.48). In the case where x0 2 dom f , it was proved above that ' 2 A.f /. Moreover,we have '.x0/ D �0 �

�

N < �0 since N < 0. Consider now the case where x0 … domf . It follows

from (2.48) by taking any x 2 dom f and letting � ! 1 that N � 0. If N < 0, we can apply thesame procedure and arrive at the conclusion. Hence we only need to consider the case whereN D 0. In this case

h Nv; x � x0i C � < 0 whenever x 2 domf:

Since A.f / ¤ ;, choose '0 2 A.f / and define

'k.x/ WD '0.x/C k.h Nv; x � x0i C �/; k 2 N :

It is obvious that 'k 2 A.f / and 'k.x0/ D '0.x0/C k� > �0 for large k. is justifies (ii).Let us now verify implication (ii)H)(iii). Fix any ' 2 A.f /. en ' � f , and hence '�� f ��.Applying Proposition 2.76 ensures that ' D '�� f ��. It follows therefore that

f .x/ D sup˚'.x/

ˇ' 2 A.f /g � f �� for every x 2 Rn :

e opposite inequality f �� f holds by Proposition 2.74(ii), and thus f �� D f .e last implication (iii)H)(i) is obvious because the set epig� is always closed and convex

for any function g W Rn! R. �


2.9 DIRECTIONALDERIVATIVESOur next topic is directional differentiability of convex functions and its relationships with subdif-ferentiation. In contrast to classical analysis, directional derivative constructions in convex analysisare one-sided and related to directions with no classical plus-minus symmetry.

Definition 2.79 Let f W Rn! R be an extended-real-valued function and let Nx 2 dom f . e

of the function f at the point Nx in the direction d 2 Rn is the followinglimit—if it exists as either a real number or ˙1:

f 0. NxI d/ WD limt!0C

f . Nx C td / � f . Nx/

t: (2.49)

Note that construction (2.49) is sometimes called the right directional derivative f at Nx inthe direction d . Its left counterpart is defined by

f 0�. NxI d/ WD lim

t!0�

f . Nx C td / � f . Nx/

t:

It is easy to see from the definitions that

f 0�. NxI d/ D �f 0. NxI �d/ for all d 2 Rn;

and thus properties of the left directional derivative f 0�. NxI d/ reduce to those of the right one

(2.49), which we study in what follows.

Lemma 2.80 Given a convex function f W Rn! R with Nx 2 dom f and given d 2 Rn, define

'.t/ WDf . Nx C td / � f . Nx/

t; t > 0:

en the function ' is nondecreasing on .0;1/.

Proof. Fix any numbers 0 < t1 < t2 and get the representation

Nx C t1d Dt1

t2

�Nx C t2d

�C

�1 �

t1

t2

�Nx:

It follows from the convexity of f that

f . Nx C t1d/ �t1

t2f . Nx C t2d/C

�1 �

t1

t2

�f . Nx/;

which implies in turn the inequality

'.t1/ Df . Nx C t1d/ � f . Nx/

t1�f . Nx C t2d/ � f . Nx/

t2D '.t2/:

is verifies that ' is nondecreasing on .0;1/. �

2.9. DIRECTIONALDERIVATIVES 83

e next proposition establishes the directional differentiability of convex functions.

Proposition 2.81 For any convex function f W Rn! R and any Nx 2 dom f , the directional

derivative f 0. NxI d/ .and hence its left counterpart/ exists in every direction d 2 Rn. Furthermore,it admits the representation via the function ' is defined in Lemma 2.80:

f 0. NxI d/ D inft>0

'.t/; d 2 Rn

Proof. Lemma 2.80 tells us that the function ' is nondecreasing. us we have

f 0. NxI d/ D limt!0C

f . Nx C td / � f . Nx/

tD lim

t!0C'.t/ D inf

t>0'.t/;

which verifies the results claimed in the proposition. �

Corollary 2.82 If f W Rn! R is a convex function, then f 0. NxI d/ is a real number for any Nx 2

int.dom f / and d 2 Rn.

Proof. It follows from eorem 2.29 that f is locally Lipschitz continuous around Nx, i.e., thereis ` � 0 such that ˇf . Nx C td / � f . Nx/

t

ˇ�`tkdk

tD `kdk for all small t > 0;

which shows that jf 0. NxI d/j � `kdk < 1. �

To establish relationships between directional derivatives and subgradients of general con-vex functions, we need the following useful observation.

Lemma 2.83 For any convex function f W Rn! R and Nx 2 domf , we have

f 0. NxI d/ � f . Nx C d/ � f . Nx/ whenever d 2 Rn :

Proof. Using Lemma 2.80, we have for the function ' therein that

'.t/ � '.1/ D f . Nx C d/ � f . Nx/ for all t 2 .0; 1/;

which justifies the claimed property due to f 0. NxI d/ D inft>0 '.t/ � '.1/. �

eorem 2.84 Let f W Rn! R be convex with Nx 2 domf . e following are equivalent:

(i) v 2 @f . Nx/.(ii) hv; d i � f 0. NxI d/ for all d 2 Rn.(iii) f 0

�. NxI d/ � hv; d i � f 0. NxI d/ for all d 2 Rn.


Proof. Picking any v 2 @f . Nx/ and t > 0, we get

hv; td i � f . Nx C td / � f . Nx/ whenever d 2 Rn;

which verifies the implication (i)H)(ii) by taking the limit as t ! 0C. Assuming now that as-sertion (ii) holds, we get by Lemma 2.83 that

hv; d i � f 0. NxI d/ � f . Nx C d/ � f . Nx/ for all d 2 Rn :

It ensures by definition (2.13) that v 2 @f . Nx/, and thus assertions (i) and (ii) are equivalent.It is obvious that (iii) yields (ii). Conversely, if (ii) is satisfied, then for d 2 Rn we have

hv;�d i � f 0. NxI �d/, and thus

f 0�. NxI d/ D �f 0. NxI �d/ � hv; d i for any d 2 Rn :

is justifies the validity of (iii) and completes the proof of the theorem. �

Let us next list some properties of (2.49) as a function of the direction.

Proposition 2.85 For any convex function f W Rn! R with Nx 2 domf , we define the directional

function .d/ WD f 0. NxI d/, which satisfies the following properties:(i) .0/ D 0.(ii) .d1 C d2/ � .d1/C .d2/ for all d1; d2 2 Rn.(iii) .˛d/ D ˛ .d/ whenever d 2 Rn and ˛ > 0.(iv) If furthermore Nx 2 int.domf /, then is finite on Rn.

Proof. It is straightforward to deduce properties (i)–(iii) directly from definition (2.49). For in-stance, (ii) is satisfied due to the relationships

.d1 C d2/ D limt!0C

f . Nx C t .d1 C d2// � f . Nx/

t

D limt!0C

f .Nx C 2td1 C Nx C 2td2

2/ � f . Nx/

t

� limt!0C

f . Nx C 2td1/ � f . Nx/

2tC lim

t!0C

f . Nx C 2td2/ � f . Nx/

2tD .d1/C .d2/:

us it remains to check that .d/ is finite for every d 2 Rn when Nx 2 int.domf /. To proceed,choose ˛ > 0 so small that Nx C ˛d 2 dom f . It follows from Lemma 2.83 that

.˛d/ D f 0. NxI˛d/ � f . Nx C ˛d/ � f . Nx/ < 1:

Employing (iii) gives us .d/ < 1. Further, we have from (i) and (ii) that

0 D .0/ D �d C .�d/

�� .d/C .�d/; d 2 Rn;

which implies that .d/ � � .�d/. is yields .d/ > �1 and so verifies (iv). �

2.9. DIRECTIONALDERIVATIVES 85

To derive the second major result of this section, yet another lemma is needed.

Lemma 2.86 We have the following relationships between a convex function f W Rn! R and the

directional function defined in Proposition 2.85:(i) @f . Nx/ D @ .0/.(ii) �.v/ D ı˝.v/ for all v 2 Rn, where˝ WD @ .0/.

Proof. It follows from eorem 2.84 that v 2 @f . Nx/ if and only if

hv; d � 0i D hv; d i � f 0. NxI d/ D .d/ D .d/ � .0/; d 2 Rn :

is is equivalent to v 2 @ .0/, and hence (i) holds.To justify (ii), let us first show that �.v/ D 0 for all v 2 ˝ D @ .0/. Indeed, we have

�.v/ D sup˚hv; d i � .d/

ˇd 2 Rn

� hv; 0i � .0/ D 0:

Picking now any v 2 @ .0/ gives us

hv; d i D hv; d � 0i � .d/ � .0/ D .d/; d 2 Rn;

which implies therefore that


ˇd 2 Rn

� 0

and so ensures the validity of �.v/ D 0 for any v 2 @ .0/.It remains to verify that �.v/ D 1 if v … @ .0/. For such an element v, find d0 2 Rn

with hv; d0i > .d0/. Since is positively homogeneous by Proposition 2.85, it follows that


ˇd 2 Rn

� sup

t>0

.hv; td0i � .td0//

D supt>0

t .hv; d0i � .d0// D 1;

which completes the proof of the lemma. �

Now we are ready establish a major relationship between the directional derivative and thesubdifferential of an arbitrary convex function.

eorem 2.87 Given a convex function f W Rn! R and a point Nx 2 int.domf /, we have

f 0. NxI d/ D max˚hv; d i

ˇv 2 @f . Nx/

for any d 2 Rn :


Proof. It follows from Proposition 2.76 that

f 0. NxI d/ D .d/ D ��.d/; d 2 Rn;

by the properties of the function .d/ D f 0. NxI d/ from Proposition 2.85. Note that is a finiteconvex function in this setting. Employing now Lemma 2.86 tells us that �.v/ D ı˝.v/, where˝ D @ .0/ D @f . Nx/. Hence we have by (2.46) that

��.d/ D ı�˝.d/ D sup

˚hv; d i

ˇv 2 ˝

:

Since the set ˝ D @f . Nx/ is compact by Proposition 2.47, we complete the proof. �

2.10 SUBGRADIENTSOF SUPREMUMFUNCTIONSLet T be a nonempty subset of Rp and let g W T � Rn

! R. For convenience, we also use thenotation gt .x/ WD g.t; x/. e supremum function f W Rn

! R for gt over T is

f .x/ WD supt2T

g.t; x/ D supt2T

gt .x/; x 2 Rn : (2.50)

If the supremum in (2.50) is attained (this happens, in particular, when the index set T is compactand g.�; x/ is continuous), then (2.50) reduces to the maximum function, which can be written inform (2.35) when T is a finite set.

e main goal of this section is to calculate the subdifferential (2.13) of (2.50) when thefunctions gt are convex. Note that in this case the supremum function (2.50) is also convex byProposition 1.43. In what follows we assume without mentioning it that the functions gt W Rn

! Rare convex for all t 2 T .

For any point Nx 2 Rn, define the active index set

S. Nx/ WD˚t 2 T

ˇgt . Nx/ D f . Nx/

; (2.51)

which may be empty if the supremum in (2.50) is not attained for x D Nx. We first present a simplelower subdifferential estimate for (2.66).

Proposition 2.88 Let dom f ¤ ; for the supremum function (2.50) over an arbitrary index set T .For any Nx 2 domf , we have the inclusion

clco[

t2S. Nx/

@gt . Nx/ � @f . Nx/: (2.52)

Proof. Inclusion (2.52) obviously holds if S. Nx/ D ;. Supposing now that S. Nx/ ¤ ;, fix t 2 S. Nx/

and v 2 @gt . Nx/. en we get gt . Nx/ D f . Nx/ and therefore

hv; x � Nxi � gt .x/ � gt . Nx/ D gt .x/ � f . Nx/ � f .x/ � f . Nx/;

which shows that v 2 @f . Nx/. Since the subgradient set @f . Nx/ is a closed and convex, we concludethat inclusion (2.52) holds in this case. �

2.10. SUBGRADIENTSOF SUPREMUMFUNCTIONS 87

Further properties of the supremum/maximum function (2.50) involve the following usefulextensions of continuity. Its lower version will be studied and applied in Chapter 4.

Definition 2.89 Let˝ be a nonempty subset of Rn. A function f W ˝ ! R is - Nx 2 ˝ if for every sequence fxkg in˝ converging to Nx it holds

lim supk!1

f .xk/ � f . Nx/:

We say that f is ˝ if it is upper semicontinuous at every point of this set.

Proposition 2.90 Let T be a nonempty, compact subset ofRp and let g.�I x/ be upper semicontinuouson T for every x 2 Rn. en f in (2.50) is a finite-valued convex function.

Proof. We need to check that f .x/ < 1 for every x 2 Rn. Suppose by contradiction that f . Nx/ D

1 for some Nx 2 Rn. en there exists a sequence ftkg � T such that g.tk; Nx/ ! 1. Since Tis compact, assume without loss of generality that tk ! Nt 2 T . By the upper semicontinuity ofg.�; Nx/ on T we have

1 D lim supk!1

g.tk; Nx/ � g.Nt ; Nx/ < 1;

which is a contradiction that completes the proof of the proposition. �

Proposition 2.91 Let the index set ; ¤ T � Rp be compact and let g.�I x/ be upper semicontinuousfor every x 2 Rn. en the active index set S. Nx/ � T is nonempty and compact for every Nx 2 Rn.Furthermore, we have the compactness in Rn of the convex hull

C WD co[

t2S. Nx/

@gt . Nx/:

Proof. Fix Nx 2 Rn and get from Proposition 2.90 that f . Nx/ < 1 for the supremum function(2.50). Consider an arbitrary sequence ftkg � T with g.tk; Nx/ ! f . Nx/ as k ! 1. By the com-pactness of T we find Nt 2 T such that tk ! Nt along some subsequence. It follows from the as-sumed upper semicontinuity of g that

f . Nx/ D lim supk!1

g.tk; Nx/ � g.Nt ; Nx/ � f . Nx/;

which ensures that Nt 2 S. Nx/ by the construction ofS in (2.51), and thusS. Nx/ ¤ ;. Since g.t; Nx/ �

f . Nx/, for any t 2 T we get the representation

S. Nx/ D˚t 2 T

ˇg.t; Nx/ � f . Nx/


implying the closedness of S. Nx/ (and hence its compactness since S. Nx/ � T ), which is an imme-diate consequence of the upper semicontinuity of g.�I Nx/.

It remains to show that the subgradient set

Q WD[

t2S. Nx/

@gt . Nx/

is compact in Rn; this immediately implies the claimed compactness of C D coQ, since theconvex hull of a compact set is compact (show it, or see Corollary 3.7 in the next chapter). Toproceed with verifying the compactness ofQ, we recall first thatQ � @f . Nx/ by Proposition 2.88.is implies that the setQ is bounded in Rn, since the subgradient set of the finite-valued convexfunction is compact by Proposition 2.47. To check the closedness ofQ, take a sequence fvkg � Q

converging to some Nv and for each k 2 N and find tk 2 S. Nx/ such that vk 2 @gtk . Nx/. Since S. Nx/

is compact, we may assume that tk ! Nt 2 S. Nx/. en g.tk; Nx/ D g.Nt ; Nx/ D f . Nx/ and

hvk; x � Nxi � g.tk; x/ � g.tk; Nx/ D g.tk; x/ � g.Nt ; Nx/ for all x 2 Rn

by definition (2.13). is gives us the relationships

h Nv; x � Nxi D lim supk!1

hvk; x � Nxi � lim supk!1

g.tk; x/ � g.Nt ; Nx/ � g.Nt ; x/ � g.Nt ; Nx/ D gNt .x/ � gNt . Nx/;

which verify that Nv 2 @gNt . Nx/ � Q and thus complete the proof. �

e next result is of its own interest while being crucial for the main subdifferential resultof this section.

eorem 2.92 In the setting of Proposition 2.91 we have

f 0. NxI v/ � �C .v/ for all Nx 2 Rn; v 2 Rn

via the support function of the set C defined therein.

Proof. It follows from Proposition 2.81 and Proposition 2.82 that �1 < f 0. NxI v/ D

inf�>0 '.�/ < 1, where the function ' is defined in Lemma 2.80 and is proved to be nonde-creasing on .0;1/. us there is a strictly decreasing sequence �k # 0 as k ! 1 with

f 0. NxI v/ D limk!1

f . Nx C �kv/ � f . Nx/

�k

; k 2 N :

For every k, we select tk 2 T such that f . Nx C �kv/ D g.tk; Nx C �kv/. e compactness of Tallows us to find Nt 2 T such that tk ! Nt 2 T along some subsequence. Let us show that Nt 2 S. Nx/.Since g.tk; �/ is convex and �k < 1 for large k, we have

g.tk; Nx C �kv/ D g�tk; �k. Nx C v/C .1 � �k/ Nx

�� kg.tk; Nx C v/C .1 � �k/g.tk; Nx/; (2.53)

2.10. SUBGRADIENTSOF SUPREMUMFUNCTIONS 89

which implies by the continuity of f (any finite-valued convex function is continuous) and theupper semicontinuity of g.�I x/ that

f . Nx/ D limk!1

f . Nx C �kv/ D limk!1

g.tk; Nx C �kv/ � g.Nt ; Nx/:

is yields f . Nx/ D g.Nt ; Nx/, and so Nt 2 S. Nx/. Moreover, for any � > 0 and large k, we get

g.tk; Nx C v/ � C � with WD g.Nt ; Nx C v/

by the upper semicontinuity of g. It follows from (2.53) that

g.tk; Nx/ �g.tk; Nx C �kv/ � �k. C �/

1 � �k

! f . Nx/;

which tells us that limk!1 g.tk; Nx/ D f . Nx/ due to

f . Nx/ � lim infk!1

g.tk; Nx/ � lim supk!1

g.tk; Nx/ � g.Nt ; Nx/ D f . Nx/:

Fix further any � > 0 and, taking into account that �k < � for all k sufficiently large, deduce fromLemma 2.80 that

f . Nx C �kv/ � f . Nx/

�k

Dg.tk; Nx C �kv/ � g.Nt ; Nx/

�k

�g.tk; Nx C �kv/ � g.tk; Nx/

�k

�g.tk; Nx C �v/ � g.tk; Nx/

�

is ensures in turn the estimates

lim supk!1

f . Nx C �kv/ � f . Nx/

�k

� lim supk!1

g.tk; Nx C �v/ � g.tk; Nx/

�

�g.Nt ; Nx C �v/ � g.Nt ; Nx/

�

DgNt . Nx C �v/ � gNt . Nx/

�;

which implies the relationships

f 0. NxI v/ � inf�>0

gNt . Nx C �v/ � gNt . Nx/

�D g0

Nt. NxI v/:

Using finally eorem 2.87 and the inclusion @gNt . Nx/ � C from Proposition 2.88 give us

f 0. NxI v/ � g0Nt. NxI v/ D sup

w2@gNt . Nx/

hw; vi � supw2C

hw; vi D �C .v/;

and justify therefore the statement of the theorem. �


Now we are ready to establish the main result of this section.

eorem 2.93 Let f be defined in (2.50), where the index set ; ¤ T � Rp is compact, andwhere the function g.t; �/ is convex on Rn for every t 2 T . Suppose in addition that the functiong.�; x/ is upper semicontinuous on T for every x 2 Rn. en we have

@f . Nx/ D co[

t2S. Nx/

@gt . Nx/:

Proof. Taking any w 2 @f . Nx/, we deduce from eorem 2.87 and eorem 2.92 that

hw; vi � f 0. Nx; v/ � �C .v/ for all v 2 Rn;

where C is defined in Proposition 2.91. us w 2 @�C .0/ D C by formula (2.44) for the set˝ D C , which is a nonempty, closed, convex subset of Rn. e opposite inclusion follows fromProposition 2.88. �


Exercise 2.1 (i) Let ˝ � Rn be a nonempty, closed, convex cone and let Nx … ˝. Show thatthere exists a nonzero element v 2 Rn such that hv; xi � 0 for all x 2 ˝ and hv; Nxi > 0.(ii) Let ˝ be a subspace of Rn and let Nx … ˝. Show that there exists a nonzero element v 2 Rn

such that hv; xi D 0 for all x 2 ˝ and hv; Nxi > 0.

Exercise 2.2 Let ˝1 and ˝2 be two nonempty, convex subsets of Rn satisfying the conditionri˝1 \ ri˝2 D ;. Show that there is a nonzero element v 2 Rn which separates the sets˝1 and˝2 in the sense of (2.2).

Exercise 2.3 Two nonempty, convex subsets of Rn are called properly separated if there exists anonzero element v 2 Rn such that

sup˚hv; xi

ˇx 2 ˝1

� inf

˚hv; yi

ˇy 2 ˝2

;

inf˚hv; xi

ˇx 2 ˝1

< sup

˚hv; yi

ˇy 2 ˝2

:

Prove that ˝1 and ˝2 are properly separated if and only if ri˝1 \ ri˝2 D ;.

Exercise 2.4 Let f W Rn! R be a convex function attaining its minimum at some point Nx 2

domf . Show that the sets f˝1;˝2g with˝1 WD epif and˝2 WD Rn�ff . Nx/g form an extremal

system.


Exercise 2.5 Let ˝i for i D 1; 2 be nonempty, convex subsets of Rn with at least one point incommon and satisfy

int˝1 ¤ ; and int˝1 \˝2 D ;:

Show that there are elements v1; v2 in Rn, not equal to zero simultaneously, and real numbers˛1; ˛2 such that

hvi ; xi � ˛i for all x 2 ˝i ; i D 1; 2;

v1 C v2 D 0; and ˛1 C ˛2 D 0:

Exercise 2.6 Let ˝ WD fx 2 Rnˇx � 0g. Show that

N. NxI˝/ D˚v 2 Rn

ˇv � 0; hv; Nxi D 0

for any Nx 2 ˝:

Exercise 2.7 Let f W Rn! R be a convex function differentiable at Nx. Show that


�D

˚�

�rf . Nx/;�1

� ˇ� � 0

:

Exercise 2.8 Use induction to prove Corollary 2.19.

Exercise 2.9 Calculate the subdifferentials and the singular subdifferentials of the followingconvex functions at every point of their domains:(i) f .x/ D jx � 1j C jx � 2j; x 2 R.(ii) f .x/ D ejxj; x 2 R.

(iii) f .x/ D

(x2 � 1 if jxj � 1;

1 otherwise on R :

(iv) f .x/ D

(�

p1 � x2 if jxj � 1;

1 otherwise on R :

(v) f .x1; x2/ D jx1j C jx2j; .x1; x2/ 2 R2.(vi) f .x1; x2/ D max

˚jx1j; jx2j

; .x1; x2/ 2 R2.

(vii) f .x/ D max˚ha; xi � b; 0

; x 2 Rn, where a 2 Rn and b 2 R.

Exercise 2.10 Let f W R ! R be a nondecreasing convex function and let Nx 2 R. Show thatv � 0 for v 2 @f . Nx/.

Exercise 2.11 Use induction to prove Corollary 2.46.

Exercise 2.12 Let f W Rn! R be a convex function. Give an alternative proof for the part of

Proposition 2.47 stating that @f . Nx/ ¤ ; for any element Nx 2 int.domf /. Does the conclusionstill holds if Nx 2 ri.dom f /?


Exercise 2.13 Let˝1 and˝2 be nonempty, convex subsets of Rn. Suppose that ri˝1 \ ri˝2 ¤

;. Prove that

N. NxI˝1 \˝2/ D N. NxI˝1/CN. NxI˝2/ for all Nx 2 ˝1 \˝2:

Exercise 2.14 Let f1; f2W Rn! R be convex functions satisfying the condition

ri .domf1/ \ ri .domf2/ ¤ ;:

Prove that for all Nx 2 dom f1 \ dom f2 we have

@.f1 C f2/. Nx/ D @f1. Nx/C @f2. Nx/; @1.f1 C f2/. Nx/ D @1f1. Nx/C @1f2. Nx/:

Exercise 2.15 In the setting of eorem 2.51 show that

N.. Nx; Ny; Nz/I˝1/ \ Œ�N.. Nx; Ny; Nz/I˝2/

under the qualification condition (2.33).

Exercise 2.16 Calculate the singular subdifferential of the composition f ı B in the setting ofeorem 2.51.

Exercise 2.17 Consider the composition g ı h in the setting of Proposition 1.54 and take anyNx 2 dom .g ı h/. Prove that

@.g ı h/. Nx/ D

n pXiD1

�ivi

ˇ.�1; : : : ; �p/ 2 @g. Ny/; vi 2 @fi . Nx/; i D 1; : : : ; p

ounder the validity of the qualification condition

h0 2

pXiD1

�i@fi . Nx/; .�1; : : : ; �p/ 2 @1g. Ny/i

H)��i D 0 for i D 1; : : : ; p

�:

Exercise 2.18 Calculate the singular subdifferential of the optimal value functions in the settingof eorem 2.61.

Exercise 2.19 Calculate the support functions for the following sets:(i) ˝1 WD

˚x D .x1; : : : ; xn/ 2 Rn

ˇkxk1 WD

PniD1 jxi j � 1

:

(ii) ˝2 WD˚x D .x1; : : : ; xn/ 2 Rn

ˇkxk1 WD maxn

iD1 jxi j � 1:


Exercise 2.20 Find the Fenchel conjugate for each of the following functions on R:(i) f .x/ D ex .

(ii) f .x/ D

�� ln.x/ x > 0;

1 x � 0:

(iii) f .x/ D ax2 C bx C c, where a > 0.(iv) f .x/ D ıIB.x/, where IB D Œ�1; 1�.(v) f .x/ D jxj.

Exercise 2.21 Let ˝ be a nonempty subset of Rn. Find the biconjugate function ı��˝ .

Exercise 2.22 Prove the following statements for the listed functions on Rn:

(i) .�f /�.v/ D �f ��v�

�, where � > 0.

(ii) .f C �/�.v/ D f �.v/ � �, where � 2 R.(iii) f �

u .v/ D f �.v/C hu; vi, where u 2 Rn and fu.x/ WD f .x � u/.

Exercise 2.23 Let f W Rn! R be a convex and positively homogeneous function. Show that

f �.v/ D ı˝.v/ for all v 2 Rn, where ˝ WD @f .0/.

Exercise 2.24 Let f W Rn! R be a function with domf D ;. Show that f ��.x/ D f .x/ for

all x 2 Rn.

Exercise 2.25 Let f W R ! R be convex. Prove that @f .x/ D Œf 0�.x/; f

0C.x/�, where f 0

�.x/

and f 0C.x/ stand for the left and right derivative of f at x, respectively.

Exercise 2.26 Define the function f W R ! R by

f .x/ WD

(�

p1 � x2 if jxj � 1;

1 otherwise:

Show that f is a convex function and calculate its directional derivatives f 0.�1I d/ and f 0.1I d/

in the direction d D 1.

Exercise 2.27 Let f W Rn! R be convex and let Nx 2 dom f . For an element d 2 Rn, consider

the function ' defined in Lemma 2.80 and show that ' is nondecreasing on .�1; 0/.

Exercise 2.28 Let f W Rn! R be convex and let Nx 2 domf . Prove the following:

(i) f 0�. NxI d/ � f 0. NxI d/ for all d 2 Rn.

(ii) f . Nx/ � f . Nx � d/ � f 0�. NxI d/ � f 0. NxI d/ � f . Nx C d/ � f . Nx/, d 2 Rn.

Exercise 2.29 Prove the assertions of Proposition 2.67.


Exercise 2.30 Let ˝ � Rn be a nonempty, compact set. Using eorem 2.93, calculate thesubdifferential of the support function �˝.0/.

Exercise 2.31 Let ˝ be a nonempty, compact subset of Rn. Define the function

�˝.x/ WD sup˚kx � !k

ˇ! 2 ˝g; x 2 Rn :

Prove that �˝ is a convex function and find its subdifferential @�˝.x/ at any x 2 Rn.

Exercise 2.32 Using the representation

kxk D sup˚hv; xi

ˇkvk � 1

and eorem 2.93, calculate the subdifferential @p. Nx/ of the norm function p.x/ WD kxk at anypoint Nx 2 Rn.

Exercise 2.33 Let d.xI˝/ be the distance function (1.5) associated with some nonempty,closed, convex subset ˝ of Rn.(i) Prove the representation

d.xI˝/ D sup˚hx; vi � �˝.v/

ˇkvk � 1

via the support function of ˝.(ii) Calculate the subdifferential @d. NxI˝/ at any point Nx 2 Rn.

95

C H A P T E R 3

Remarkable Consequences ofConvexity

is chapter is devoted to remarkable topics of convex analysis that, together with subdifferentialcalculus and related results of Chapter 2, constitute the major achievements of this discipline mostimportant for many applications—first of all to convex optimization.

3.1 CHARACTERIZATIONSOFDIFFERENTIABILITYIn this section we define another classical notion of differentiability, the Gâteaux derivative, andshow that for convex functions it can be characterized, together with the Fréchet one, via a single-element subdifferential of convex analysis.

Definition 3.1 Let f W Rn! R be finite around Nx. We say that f is G

at Nx if there is v 2 Rn such that for any d 2 Rn we have

limt!0

f . Nx C td / � f . Nx/ � thv; d i

tD 0:

en we call v the G of f at Nx and denote by f 0G. Nx/.

It follows from the definition and the equality f 0�. NxI d/ D �f 0. NxI �d/ that f 0

G. Nx/ D v ifand only if the directional derivative f 0. NxI d/ exists and is linearwith respect to d with f 0. NxI d/ D

hv; d i for all d 2 Rn; see Exercise 3.1 and its solution. Moreover, if f is Gâteaux differentiableat Nx, then

hf 0G. Nx/; d i D lim

t!0

f . Nx C td / � f . Nx/

tfor all d 2 Rn :

Proposition 3.2 Let f W Rn! R be a function (not necessarily convex) which is locally Lipschitz

continuous around Nx 2 dom f . en the following assertions are equivalent:(i) f is Fréchet differentiable at Nx.(ii) f is Gâteaux differentiable at Nx.

96 3. REMARKABLECONSEQUENCESOFCONVEXITY

Proof. Let f be Fréchet differentiable at Nx and let v WD rf . Nx/. For a fixed 0 ¤ d 2 Rn, takinginto account that k Nx C td � Nxk D jt j � kdk ! 0 as t ! 0, we have

limt!0

f . Nx C td / � f . Nx/ � thv; d i

tD kdk lim

t!0

f . Nx C td / � f . Nx/ � hv; td i

tkdkD 0;

which implies the GOateaux differentiablity of f at Nx with f 0G. Nx/ D rf . Nx/. Note that the local

Lipschitz continuity of f around Nx is not required in this implication.Assuming now that f is GOateaux differentiable at Nx with v WD f 0

G. Nx/, let us show that

limh!0

f . Nx C h/ � f . Nx/ � hv; hi

khkD 0:

Suppose by contradiction that it does not hold. en we can find �0 > 0 and a sequence of hk ! 0

with hk ¤ 0 as k 2 N such thatˇf . Nx C hk/ � f . Nx/ � hv; hkij

khkk� �0: (3.1)

Denote tk WD khkk and dk WDhk

khkk. en tk ! 0, and we can assume without loss of generality

that dk ! d with kdk D 1. is gives us the estimate

�0kdkk �

ˇf . Nx C tkdk/ � f . Nx/ � hv; tkdkij

tk:

Let ` � 0 be a Lipschitz constant of f around Nx. e triangle inequality ensures that

�0kdkk �

ˇf . Nx C tkdk/ � f . Nx/ � hv; tkdkij

tk

�

ˇf . Nx C tkdk/ � f . Nx C tkd/j C jhv; tkd i � hv; tkdkij

tkC

ˇf . Nx C tkd/ � f . Nx/ � hv; tkd ij

tk

� `kdk � dk C kvk � kdk � dk C

ˇf . Nx C tkd/ � f . Nx/ � hv; tkd ij

tk! 0 as k ! 1:

is contradicts (3.1) since �0kdkk ! �0, so f is Fréchet differentiable at Nx. �

Now we are ready to derive a nice subdifferential characterization of differentiability.

eorem 3.3 Let f W Rn! R be a convex function with Nx 2 int(domf /. e following asser-

tions are equivalent:(i) f is Fréchet differentiable at Nx.(ii) f is Gâteaux differentiable at Nx.(iii) @f . Nx/ is a singleton.

3.2. CARATHÉODORYTHEOREMANDFARKAS LEMMA 97

Proof. It follows from eorem 2.29 that f is locally Lipschitz continuous around Nx, andhence (i) and (ii) are equivalent by Proposition 3.2. Suppose now that (ii) is satisfied and letv WD f 0

G. Nx/. en f 0�. NxI d/ D f 0. NxI d/ D hv; d i for all d 2 Rn. Observe that @f . Nx/ ¤ ; since

Nx 2 int(domf /. Fix any w 2 @f . Nx/. It follows from eorem 2.84 that

f 0�. NxI d/ � hw; d i � f 0. NxI d/ whenever d 2 Rn;

and thus hw; d i D hv; d i, so hw � v; d i D 0 for all d . Plugging there d D w � v gives us w D v

and thus justifies that @f . Nx/ is a singleton.Suppose finally that (iii) is satisfied with @f . Nx/ D fvg and then show that (ii) holds. By

eorem 2.87 we have the representation

f 0. NxI d/ D supw2@f . Nx/

hw; d i D hv; d i;

for all d 2 Rn. is implies that f is GOateaux and hence Fréchet differentiable at Nx. �

3.2 CARATHÉODORYTHEOREMANDFARKAS LEMMA

We begin with preliminary results on conic and convex hulls of sets that lead us to one of the mostremarkable results of convex geometry established by Constantin Carathéodory in 1911. Recallthat a set˝ is called a cone if �x 2 ˝ for all � � 0 and x 2 ˝. It follows from the definition that0 2 ˝ if ˝ is a nonempty cone. Given a nonempty set ˝ � Rn, we say that K˝ is the convexcone generated by ˝, or the convex conic hull of ˝, if it is the intersection of all the convex conescontaining ˝.

Figure 3.1: Convex cone generated by a set.

98 3. REMARKABLECONSEQUENCESOFCONVEXITYProposition 3.4 Let˝ be a nonempty subset of Rn. en the convex cone generated by˝ admits thefollowing representation:

K˝ D

n mXiD1

�iai

ˇ�i � 0; ai 2 ˝; m 2 N

o: (3.2)

If, in particular, the set˝ is convex, thenK˝ D RC˝.

Proof. Denote by C the set on the right-hand side of (3.2) and observe that it is a convex conewhich contains ˝. It remains to show that C � K for any convex cone K containing ˝. Tosee it, fix such a cone and form the combination x D

PmiD1 �iai with any �i � 0, ai 2 ˝, and

m 2 N. If �i D 0 for all i , then x D 0 2 K. Otherwise, suppose that �i > 0 for some i and denote� WD

PmiD1 �i > 0. is tells us that

x D �� mX

iD1

�i

�ai

�2 K;

which yields C � K, so K˝ D C . e last assertion of the proposition is obvious. �

Proposition 3.5 For any nonempty set˝ � Rn and any x 2 K˝ n f0g, we have

x D

mXiD1

�iai with �i > 0; ai 2 ˝ as i D 1; : : : ; m and m � n:

Proof. Let x 2 K˝ n f0g. It follows from Proposition 3.4 that x DPm

iD1 �iai with some �i > 0

and ai 2 ˝ as for i D 1; : : : ; m and m 2 N. If the elements a1; : : : ; am are linearly dependent,then there are numbers i 2 R, not all zeros, such that

mXiD1

iai D 0:

We can assume that I WD fi D 1; : : : ; m j i > 0g ¤ ; and get for any � > 0 that

x D

mXiD1

�iai D

mXiD1

�iai � �� mX

iD1

iai

�D

mXiD1

��i � � i

�ai :

Let � WD minn�i

i

ˇi 2 I

oD�i0

i0

, where i0 2 I , and denote ˇi WD �i � � i for i D 1; : : : ; m.

en we have the relationships

x D

mXiD1

ˇiai with ˇi0 D 0 and ˇi � 0 for any i D 1; : : : ; m:


Continuing this procedure, after a finite number of steps we represent x as a positive linear com-bination of linearly independent elements faj j j 2 J g, where J � f1; : : : ; mg. us the indexset J cannot contain more than n elements. �

e following remarkable result is known as the Carathéodory theorem. It relates convexhulls of sets with the dimension of the space; see an illustration in Figure 3.2.

Figure 3.2: Carathéodory theorem.

eorem 3.6 Let ˝ be a nonempty subset of Rn. en every x 2 co˝ can be represented as aconvex combination of no more than nC 1 elements of the set ˝.

Proof. Denoting B WD f1g �˝ � RnC1, it is easy to observe that coB D f1g � co˝ and thatcoB � KB . Take now any x 2 co˝. en .1; x/ 2 coB and by Proposition 3.5 find �i � 0 and.1; ai / 2 B for i D 0; : : : ; m with m � n such that

.1; x/ D

mXiD0

�i .1; ai /:

is implies that x DPm

iD0 �iai withPm

iD0 �i D 1, �i � 0, and m � n. �

e next useful assertion is a direct consequence of eorem 3.6.

Corollary 3.7 Suppose that a subset˝ of Rn is compact. en the set co˝ is also compact.

Now we present two propositions, which lead us to another remarkable result known as theFarkas lemma; see eorem 3.10.


Proposition 3.8 Let a1; : : : ; am be linearly dependent elements in Rn, where ai ¤ 0 for all i D

1; : : : ; m. Define the sets

˝ WD˚a1; : : : ; am

and ˝i WD ˝ n

˚ai

; i D 1; : : : ; m:

en the convex cones generated by these sets are related as

K˝ D

m[iD1

K˝i: (3.3)

Proof. Obviously, [miD1K˝i

� K˝ . To prove the converse inclusion, pick x 2 K˝ and get

x D

mXiD1

�iai with �i � 0; i D 1; : : : ; m:

If x D 0, then x 2 [miD1K˝i

obviously. Otherwise, the linear dependence of ai allows us to find i 2 R not all zeros such that

mXiD1

iai D 0:

Following the proof of Proposition 3.5, we represent x as

x D

mXiD1;i¤i0

�iai with some i0 2 f1; : : : ; mg:

is shows that x 2 [miD1K˝i

and thus verifies that equality (3.3) holds. �

Proposition 3.9 Let a1; : : : ; am be elements of Rn. en the convex cone K˝ generated by the set˝ WD fa1; : : : ; amg is closed in Rn.

Proof. Assume first that the elements a1; : : : ; am are linearly independent and take any sequencefxkg � K˝ converging to x. By the construction of K˝ , find numbers ˛ki � 0 for each i D

1; : : : ; m and k 2 N such that

xk D

mXiD1

˛kiai :

Letting ˛k WD .˛k1; : : : ; ˛km/ 2 Rm and arguing by contradiction, it is easy to check that thesequence f˛kg is bounded. is allows us to suppose without loss of generality that ˛k !

.˛1; : : : ; ˛m/ as k ! 1 with ˛i � 0 for all i D 1; : : : ; m. us we get by passing to the limitthat xk ! x D

PmiD1 ˛iai 2 K˝ , which verifies the closedness of K˝ .


Considering next arbitrary elements a1; : : : ; am, assumewithout loss of generality that ai ¤

0 for all i D 1; : : : ; m. By the above, it remains to examine the case where a1; : : : ; am are linearlydependent. en by Proposition 3.8 we have the cone representation (3.3), where each set ˝i

containsm � 1 elements. If at least one of the sets˝i consists of linearly dependent elements, wecan represent the corresponding cone K˝i

in a similar way via the sets of m � 2 elements. isgives us after a finite number of steps that

K˝ D

p[j D1

KCj;

where each set Cj � ˝ contains only linear independent elements, and so the cones KCjare

closed. us the cone K˝ under consideration is closed as well. �

Now we are ready to derive the fundamental result of convex analysis and optimizationestablished by Gyula Farkas in 1894.

eorem 3.10 Let˝ WD fa1; : : : ; amg with a1; : : : ; am 2 Rn and let b 2 Rn. e following as-sertions are equivalent:(i) b 2 K˝ .(ii) For any x 2 Rn, we have the implication�

hai ; xi � 0; i D 1; : : : ; m�

H)�hb; xi � 0

�:

Proof. To verify .i/ H) .ii/, take b 2 K˝ and find �i � 0 for i D 1; : : : ; m such that

b D

mXiD1

�iai :

en, given x 2 Rn, the inequalities hai ; xi � 0 for all i D 1; : : : ; m imply that

hb; xi D

mXiD1

�i hai ; xi � 0;

which is (ii). To prove the converse implication, suppose that b … K˝ and show that (ii) doesnot hold in this case. Indeed, since the setK˝ is closed and convex, the strict separation result ofProposition 2.1 yields the existence of Nx 2 Rn satisfying

sup˚hu; Nxi

ˇu 2 K˝g < hb; Nxi:

Since K˝ is a cone, we have 0 D h0; Nxi < hb; Nxi and tu 2 K˝ for all t > 0, u 2 K˝ . us

t sup˚hu; Nxi

ˇu 2 K˝

< hb; Nxi whenever t > 0:


Dividing both sides of this inequality by t and letting t ! 1 give us

sup˚hu; Nxi

ˇu 2 K˝

� 0:

It follows that hai ; Nxi � 0 for all i D 1; : : : ; m while hb; Nxi > 0. is is a contradiction, whichtherefore completes the proof of the theorem. �

3.3 RADONTHEOREMANDHELLYTHEOREMIn this section we present a celebrated theorem of convex geometry discovered by Edward Hellyin 1913. Let us start with the following simple lemma.

Lemma 3.11 Consider arbitrary elements w1; : : : ; wm in Rn withm � nC 2. en these elementsare affinely dependent.

Proof. Form the elements w2 � w1; : : : ; wm � w1 in Rn and observe that these elements are lin-early dependent sincem � 1 > n. us the elementsw1; : : : ; wm are affinely dependent by Propo-sition 1.63. �

We continue with a theorem by Johann Radon established in 1923 and used by himself togive a simple proof of the Helly theorem.

eorem 3.12 Consider the elements w1; : : : ; wm of Rn with m � nC 2 and let I WD

f1; : : : ; mg. en there exist two nonempty, disjoint subsets I1 and I2 of I with I D I1 [ I2

such that for ˝1 WD fwi j i 2 I1g, ˝2 WD fwi j i 2 I2g one has

co˝1 \ co˝2 ¤ ;:

Proof. Since m � nC 2, it follows from Lemma 3.11 that the elements w1; : : : ; wm are affinelydependent. us there are real numbers �1; : : : ; �m, not all zeros, such that

mXiD1

�iwi D 0;

mXiD1

�i D 0:

Consider the index sets I1 WD fi D 1; : : : ; m j �i � 0g and I2 WD fi D 1; : : : ; m j �i < 0g. SincePmiD1 �i D 0, both sets I1 and I2 are nonempty and

Pi2I1

�i D �P

i2I2�i . Denoting � WDP

i2I1�i , we have the equalitiesX

i2I1

�iwi D �Xi2I2

�iwi andXi2I1

�i

�wi D

Xi2I2

��i

�wi :

3.4. TANGENTSTOCONVEX SETS 103

Define ˝1 WD fwi j i 2 I1g, ˝2 WD fwi j i 2 I2g and observe that co˝1 \ co˝2 ¤ ; sinceXi2I1

�i

�wi D

Xi2I2

��i

�wi 2 co˝1 \ co˝2:

is completes the proof of the Radon theorem. �

Now we are ready to formulate and prove the fundamental Helly theorem.

eorem 3.13 Let O WD˚˝1; : : : ; ˝m

be a collection of convex sets in Rn with m � nC 1.

Suppose that the intersection of any subcollection of nC 1 sets from O is nonempty. en wehave

TmiD1˝i ¤ ;.

Proof. Let us prove the theorem by using induction with respect to the number of elements inO. e conclusion of the theorem is obvious ifm D nC 1. Suppose that the conclusion holds forany collection of m convex sets of Rn with m � nC 1. Let

˚˝1; : : : ; ˝m;˝mC1

be a collection

of convex sets in Rn such that the intersection of any subcollection of nC 1 sets is nonempty. Fori D 1; : : : ; mC 1, define

�i WD

mC1\j D1;j ¤i

j

and observe that �i � j whenever j ¤ i and i; j D 1; : : : ; mC 1. It follows from the in-duction hypothesis that �i ¤ ;, and so we can pick wi 2 �i for every i D 1; : : : ; mC 1. Ap-plying the above Radon theorem for the elements w1; : : : wmC1, we find two nonempty, dis-joint subsets index sets I1 and I2 of I WD f1; : : : ; mC 1g with I D I1 [ I2 such that for W1 WD

fwi j i 2 I1g and W2 WD fwi j i 2 I2g one has coW1 \ coW2 ¤ ;. is allows us to pick an ele-ment w 2 coW1 \ coW2. We finally show that w 2 \

mC1iD1 ˝i and thus arrive at the conclusion.

Since i ¤ j for i 2 I1 and j 2 I2, it follows that �i � j whenever i 2 I1 and j 2 I2. Fixany i D 1; : : : ; mC 1 and consider the case where i 2 I1. en wj 2 �j � ˝i for every j 2 I2.Hence w 2 coW2 D co fwj j j 2 I2g � ˝i due the convexity of ˝i . It shows that w 2 ˝i forevery i 2 I1. Similarly, we can verify that w 2 ˝i for every i 2 I2 and thus complete the proofof the theorem. �

3.4 TANGENTSTOCONVEX SETSIn Chapter 2 we studied the normal cone to convex sets and learned that the very definition andmajor properties of it were related to convex separation. In this section we define the tangentcone to convex sets independently while observing that it can be constructed via duality from thenormal cone to the set under consideration.

We start with cones. Given a nonempty, convex cone C � Rn, the polar of C is

C ıWD

˚v 2 Rn

ˇhv; ci � 0 for all c 2 C

:


Proposition 3.14 For any nonempty, convex cone C � Rn, we have

.C ı/ı D clC:

Proof. e inclusion C � .C ı/ı follows directly from the definition, and hence clC � .C ı/ı

since .C ı/ı is a closed set. To verify the converse inclusion, take any c 2 .C ı/ı and suppose bycontradiction that c … clC . Since C is a cone, by convex separation from Proposition 2.1, findv ¤ 0 such that

hv; xi � 0 for all x 2 C and hv; ci > 0;

which yields v 2 C ı thus contradicts to the fact that c 2 .C ı/ı. �

Definition 3.15 Let˝ be a nonempty, convex set. e to˝ at Nx 2 ˝ is

T . NxI˝/ WD clKf˝� Nxg D cl RC.˝ � Nx/:

Figure 3.3: Tangent cone and normal cone.

Now we get the duality correspondence between normals and tangents to convex sets; see ademonstration in Figure 3.3.

eorem 3.16 Let ˝ be a convex set and let Nx 2 ˝. en we have the equalities

N. NxI˝/ D�T . NxI˝/

�ı and T . NxI˝/ D�N. NxI˝/

�ı: (3.4)

3.4. TANGENTSTOCONVEX SETS 105

Proof. To verify first the inclusion N. NxI˝/ � ŒT . NxI˝/�ı, pick any v 2 N. NxI˝/ and get by def-inition that hv; x � Nxi � 0 for all x 2 ˝. is implies that˝

v; t.x � Nx/˛� 0 whenever t � 0 and x 2 ˝:

Fix w 2 T . NxI˝/ and find tk � 0 and xk 2 ˝ with tk.xk � Nx/ ! w as k ! 1. en

hv;wi D limk!1

hv; tk.xk � Nx/i � 0 and thus v 2�T . NxI˝/

�ı:

To verify the opposite inclusion in (3.4), fix any v 2 ŒT . NxI˝/�ı and deduce from the relationshv;wi � 0 as w 2 T . NxI˝/ and x � Nx 2 T . NxI˝/ as x 2 ˝ that

hv; x � Nxi � 0 for all x 2 ˝:

is yields w 2 N. NxI˝/ and thus completes the proof of the first equality in (3.4). e secondequality in (3.4) follows from the first one due to�

N. NxI˝/�ı

D�T . NxI˝/

�ııD clT . NxI˝/

by Proposition 3.14 and the closedness of T . NxI˝/ by Definition 3.15. �

e next result, partly based on the Farkas lemma from eorem 3.10, provides effectiverepresentations of the tangent and normal cones to linear inequality systems.

eorem 3.17 Let ai 2 Rn, let bi 2 R for i D 1; : : : ; m, and let

˝ WD˚x 2 Rn

ˇhai ; xi � bi

:

Suppose that Nx 2 ˝ and define the active index set I. Nx/ WD fi D 1; : : : ; m j hai ; Nxi D big. enwe have the relationships

T . NxI˝/ D˚v 2 Rn

ˇhai ; vi � 0 for all i 2 I. Nx/

; (3.5)

N. NxI˝/ D cone˚ai

ˇi 2 I. Nx/

: (3.6)

Here we use the convention that cone ; WD f0g.

Proof. To justify (3.5), consider the convex cone

K WD˚v 2 Rn

ˇhai ; vi � 0 for all i 2 I. Nx/

:

It is easy to see that for any x 2 ˝, any i 2 I. Nx/, and any t � 0 we have˝ai ; t .x � Nx/

˛D t

�hai ; xi � hai ; Nxi

�� t.bi � bi / D 0;


and hence RC.˝ � Nx/ � K. us it follows from the closedness of K that

T . NxI˝/ D cl RC.˝ � Nx/ � K:

To verify the opposite inclusion in (3.5), observe that hai ; Nxi < bi for i … I. Nx/. Using this andpicking an arbitrary element v 2 K give us

hai ; Nx C tvi � bi for all i … I. Nx/ and small t > 0:

Since hai ; vi � 0 for all i 2 I. Nx/, it tells us that hai ; Nx C tvi � bi as i D 1; : : : ; m yielding

Nx C tv 2 ˝; or v 21

t

�˝ � Nx

�� T . NxI˝/:

is justifies the tangent cone representation (3.5).Next we prove (3.6) for the normal cone. e first equality in (3.4) ensures that

w 2 N. NxI˝/ if and only if hw; vi � 0 for all v 2 T . NxI˝/:

en we see from (3.5) that w 2 N. NxI˝/ if and only if�hai ; vi � 0 for all i 2 I. Nx/

�H)

�hw; vi � 0

�whenever v 2 Rn. e Farkas lemma from eorem 3.10 tells us that w 2 N. NxI˝/ if and only ifw 2 conefai j i 2 I. Nx/g, and thus (3.6) holds. �

3.5 MEANVALUETHEOREMSe mean value theorem for differentiable functions is one of the central results of classical realanalysis. Now we derive its subdifferential counterparts for general convex functions.

eorem 3.18 Let f W R ! R be convex, let a < b, and let the interval Œa; b� belong to thedomain of f , which is an open interval in R. en there is c 2 .a; b/ such that

f .b/ � f .a/

b � a2 @f .c/: (3.7)

Proof. Define the real-valued function gW Œa; b� ! R by

g.x/ WD f .x/ �

hf .b/ � f .a/

b � a.x � a/C f .a/

ifor which g.a/ D g.b/. is implies, by the convexity and hence continuity of g on its opendomain, that g has a local minimum at some c 2 .a; b/. e function

h.x/ WD �

hf .b/ � f .a/

b � a.x � a/C f .a/

i

3.5. MEANVALUETHEOREMS 107

Figure 3.4: Subdifferential mean value theorem.

is obviously differentiable at c, and hence

@h.c/ D˚h0.c/

D

n�f .b/ � f .a/

b � a

o:

e subdifferential Fermat rule and the subdifferential sum rule from Chapter 2 yield

0 2 @g.c/ D @f .c/ �

nf .b/ � f .a/

b � a

o;

which implies (3.7) and completes the proof of the theorem. �

To proceed further with deriving a counterpart of the mean value theorem for convex func-tions on Rn (eorem 3.20), we first present the following lemma.

Lemma 3.19 Let f W Rn! R be a convex function and letD be the domain of f , which is an open

subset of Rn. Given a; b 2 D with a ¤ b, define the function 'W R ! R by

'.t/ WD f�tb C .1 � t /a

�; t 2 R : (3.8)

en for any number t0 2 .0; 1/ we have

@'.t0/ D˚hv; b � ai

ˇv 2 @f .c0/

with c0 WD t0aC .1 � t0/b:

Proof. Define the vector-valued functions AW R ! Rn and BW R ! Rn by

A.t/ WD .b � a/t and B.t/ WD tb C .1 � t/a D t .b � a/C a D A.t/C a; t 2 R :


It is clear that '.t/ D .f ı B/.t/ and that the adjoint mapping to A is A�v D hv; b � ai, v 2 Rn.By the subdifferential chain rule from eorem 2.51, we have

@'.t0/ D A�@f .c0/ D˚hv; b � ai

ˇv 2 @f .c0/

;

which verifies the result claimed in the lemma. �

eorem 3.20 Let f W Rn! R be a convex function in the setting of Lemma 3.19. en there

exists an element c 2 .a; b/ such that

f .b/ � f .a/ 2˝@f .c/; b � a

˛WD

˚hv; b � ai

ˇv 2 @f .c/

:

Proof. Consider the function 'W R ! R from (3.8) for which '.0/ D f .a/ and '.1/ D f .b/.Applying first eorem 3.18 and then Lemma 3.19 to ', we find t0 2 .0; 1/ such that

f .b/ � f .a/ D '.1/ � '.0/ 2 @'.t0/ D˚hv; b � ai

ˇv 2 @f .c/

with c D t0aC .1 � t0/b. is justifies our statement. �

3.6 HORIZONCONESis section concerns behavior of unbounded convex sets at infinity.

Definition 3.21 Given a nonempty, closed, convex subset F of Rn and a point x 2 F , the of F at x is defined by the formula

F1.x/ WD˚d 2 Rn

ˇx C td 2 F for all t > 0

:

Note that the horizon cone is also known in the literature as the asymptotic cone of F at the pointin question. Another equivalent definition of F1.x/ is given by

F1.x/ D\t>0

F � x

t;

which clearly implies that F1.x/ is a closed, convex cone in Rn. e next proposition shows thatF1.x/ is the same for any x 2 F , and so it can be simply denoted by F1.

Proposition 3.22 Let F be a nonempty, closed, convex subset of Rn. en we have the equalityF1.x1/ D F1.x2/ for any x1; x2 2 F .

3.6. HORIZONCONES 109

Figure 3.5: Horizon cone.

Proof. It suffices to verify that F1.x1/ � F1.x2/ whenever x1; x2 2 F . Taking any directiond 2 F1.x1/ and any number t > 0, we show that x2 C td 2 F . Consider the sequence

xk WD1

k

�x1 C ktd

�C

�1 �

1

k

�x2; k 2 N :

en xk 2 F for every k because d 2 F1.x1/ and F is convex. We also have xk ! x2 C td , andso x2 C td 2 F since F is closed. us d 2 F1.x2/. �

Corollary 3.23 Let F be a closed, convex subset of Rn that contains the origin. en

F1 D\t>0

tF:

Proof. It follows from the construction of F1 and Proposition 3.22 that

F1 D F1.0/ D\t>0

F � 0

tD

\t>0

1

tF D

\t>0

tF;

which therefore justifies the claim. �

Proposition 3.24 Let F be a nonempty, closed, convex subset of Rn. For a given element d 2 Rn,the following assertions are equivalent:(i) d 2 F1.(ii)ere are sequences ftkg � Œ0;1/ and ffkg � F such that tk ! 0 and tkfk ! d .


Proof. To verify (i)H)(ii), take d 2 F1 and fix Nx 2 F . It follows from the definition that

Nx C kd 2 F for all k 2 N :

For each k 2 N, this allows us to find fk 2 F such that

Nx C kd D fk; or equivalently, 1k

Nx C d D1

kfk :

Letting tk WD1

k, we see that tkfk ! d as k ! 1.

To prove (ii)H)(i), suppose that there are sequences ftkg � Œ0;1/ and ffkg � F withtk ! 0 and tkfk ! d . Fix x 2 F and verify that d 2 F1 by showing that

x C td 2 F for all t > 0:

Indeed, for any fixed t > 0 we have 0 � t � tk < 1 when k is sufficiently large. us

.1 � t � tk/x C t � tkfk ! x C td as k ! 1:

It follows from the convexity of F that every element .1 � t � tk/x C t � tkfk belongs to F . Hencex C td 2 F by the closedness, and thus we get d 2 F1. �

Finally in this section, we present the expected characterization of the set boundedness interms of its horizon cone.

eorem 3.25 Let F be a nonempty, closed, convex subset of Rn. en the set F is bounded ifand only if its horizon cone is trivial, i.e., F1 D f0g.

Proof. Suppose F is bounded and take any d 2 F1. By Proposition 3.24, there exist sequencesftkg � Œ0;1/ with tk ! 0 and ffkg � F such that tkfk ! d as k ! 1. It follows from theboundedness of F that tkfk ! 0, which shows that d D 0.

To prove the converse implication, suppose by contradiction that F is unbounded whileF1 D f0g. en there is a sequence fxkg � F with kxkk ! 1. is allows us to form the se-quence of (unit) directions

dk WDxk

kxkk; k 2 N :

It ensures without loss of generality that dk ! d as k ! 1 with kdk D 1. Fix any x 2 F andobserve that for all t > 0 and k 2 N sufficiently large we have

uk WD

�1 �

t

kxkk

�x C

t

kxkkxk 2 F

by the convexity of F . Furthermore, x C td 2 F since uk ! x C td and F is closed. us d 2

F1, which is a contradiction that completes the proof of the theorem. �

3.7. MINIMALTIMEFUNCTIONSANDMINKOWSKIGAUGE 111

3.7 MINIMALTIMEFUNCTIONSANDMINKOWSKIGAUGE

e main object of this and next sections is a remarkable class of convex functions known asthe minimal time function. ey are relatively new in convex analysis and highly important inapplications. Some of their recent applications to location problems are given in Chapter 4. Wealso discuss relationships between minimal time functions and Minkowski gauge functions well-recognized in convex analysis and optimization.

Definition 3.26 Let F be a nonempty, closed, convex subset of Rn and let ˝ � Rn be a nonemptyset, not necessarily convex. e associated with the sets F and˝ is definedby

T F˝ .x/ WD inf

˚t � 0

ˇ.x C tF / \˝ ¤ ;

: (3.9)

Figure 3.6: Minimal time functions.

ename comes from the following interpretation: (3.9) signifies theminimal time needed for thepoint x to reach the target set˝ along the constant dynamicsF ; see Figure 3.6. Note that whenF D

IB , the closed unit ball in Rn, the minimal time function (3.9) reduces to the distance functiond.xI˝/ considered in Section 2.5. In the case where F D f0g, the minimal time function (3.9) isthe indicator function of ˝. If ˝ D f0g, we get T F

˝ .x/ D �F .�x/, where �F is the Minkowskigauge defined later in this section. e minimal time function (3.9) also covers another interestingclass of functions called the directional minimal time function, which corresponds to the case whereF is a nonzero vector.

In this section we study general properties of the minimal time function (3.9) used in whatfollows. ese properties are certainly of independent interest.

Observe that T F˝ .x/ D 0 if x 2 ˝, and that T F

˝ .x/ is finite if there exists t � 0 with

.x C tF / \˝ ¤ ;;

which holds, in particular, when 0 2 intF . If no such t exists, then T F˝ .x/ D 1. us the min-

imal time function (3.9) is an extended-real-valued function in general.


e sets F and ˝ in (3.9) are not necessarily bounded. e next theorem reveals behaviorof T F

˝ .x/ in the case where ˝ is bounded, while F is not necessarily bounded. It is an exerciseto formulate and prove related results for the case where F is bounded, while˝ is not necessarilybounded.

eorem3.27 LetF be a nonempty, closed, convex subset of Rn and let˝ � Rn be a nonempty,closed set. Assume that ˝ is bounded. en

T F˝ .x/ D 0 implies x 2 ˝ � F1:

e converse holds if we assume additionally that 0 2 F .

Proof. Suppose that T F˝ .x/ D 0 and find tk ! 0, tk � 0, such that

.x C tkF / \˝ ¤ ;; k 2 N :

is gives us qk 2 ˝ and fk 2 F with x C tkfk D qk for each k. By the boundedness andclosedness of ˝, we can select a subsequence of fqkg (no relabeling) that converges to someq 2 ˝, and thus tkfk ! q � x. It follows from Proposition 3.24 that q � x 2 F1, which justi-fies x 2 ˝ � F1.

To prove the opposite implication, pick any x 2 ˝ � F1 and find q 2 ˝ and d 2 F1 suchthat x D q � d . Since 0 2 F and d 2 F1, we get from Corollary 3.23 the inclusions k.q � x/ D

kd 2 F for all k 2 N. is shows that

q � x 21

kF and so

�x C

1

kF

�\˝ ¤ ;; k 2 N :

e latter ensures that 0 � T F˝ .x/ �

1

kfor all k 2 N, which holds only when T F

˝ .x/ D 0 andthus completes the proof of the theorem. �

e following example illustrates the result of eorem 3.27.

Example 3.28 Let F D R �Œ�1; 1� � R2 and let ˝ be the disk centered at .1; 0/ with radiusr D 1. Using the representation of F1 in Corollary 3.23, we get F1 D R �f0g and ˝ � F1 D

R �Œ�1; 1�. us T F˝ .x/ D 0 if and only if x 2 R �Œ�1; 1�.

e next result characterizes the convexity of the minimal time function (3.9).

Proposition 3.29 Let F be a nonempty, closed, convex subset of Rn. If ˝ � Rn is nonempty andconvex, then the minimal time function (3.9) is convex. If furthermore F is bounded and ˝ is closed,then the converse holds as well.


Proof. Take any x1; x2 2 domT F˝ and let � 2 .0; 1/. We now show that

T F˝

��x1 C .1 � �/x2

�� T F

˝ .x1/C .1 � �/T F˝ .x2/ (3.10)

provided that both sets F and ˝ are convex. Denote i WD T F˝ .xi / for i D 1; 2 and, given any

� > 0, select numbers ti so that

i � ti < i C � and .xi C tiF / \˝ ¤ ;; i D 1; 2:

It follows from the convexity of F and ˝ that��x1 C .1 � �/x2 C

��t1 C .1 � �/t2

�F

�\˝ ¤ ;;

which implies in turn the estimates

T F˝

��x1 C .1 � �/x2

�� t1 C .1 � �/t2 � �T F

˝ .x1/C .1 � �/T F˝ .x2/C �:

Since � is arbitrary, we arrive at (3.10) and thus verifies the convexity of T F˝ .

Conversely, suppose that T F˝ is convex provided that F is bounded and that ˝ is closed.

en the level set ˚x 2 Rn

ˇT F

˝ .x/ � 0

is definitely convex. We can easily show set reduces to the target set ˝, which justifies the con-vexity of the latter. �

Definition3.30 LetF be a nonempty, closed, convex subset ofRn.eM associatedwith the set F is defined by

�F .x/ WD inf˚t � 0

ˇx 2 tF

; x 2 Rn : (3.11)

e following theorem summarizes the main properties of Minkowski gauge functions.

eorem 3.31 Let F be a closed, convex set that contains the origin. e Minkowski gauge �F

is an extended-real-valued function, which is positively homogeneous and subadditive. Further-more, �F .x/ D 0 if and only if x 2 F1.

Proof. We first prove that �F is subadditive meaning that

�F .x1 C x2/ � �F .x1/C �F .x2/ for all x1; x2 2 Rn : (3.12)

is obviously holds if the right-hand side of (3.12) is equal to infinity, i.e., we have thatx1 … dom �F or x2 … dom �F . Suppose now that x1; x2 2 dom �F and fix � > 0. en there arenumbers t1; t2 � 0 such that

�F .x1/ � t1 < �F .x1/C �; �F .x2/ � t2 < �F .x2/C � and x1 2 t1F; x2 2 t2F:


It follows from the convexity of F that x1 C x2 2 t1F C t2F � .t1 C t2/F , and so

�F .x1 C x2/ � t1 C t2 < �F .x1/C �F .x2/C 2�;

which justifies (3.12) since � > 0 was chosen arbitrarily. To check the positive homogeneousproperty of �F , we take any ˛ > 0 and conclude that

�F .˛x/ D inf˚t � 0

ˇ˛x 2 tF

D inf

nt � 0

ˇx 2

t

˛F

oD ˛ inf

n t˛

� 0ˇx 2

t

˛F

oD ˛�F .x/:

It remains to verify the last statement of the theorem. Observe that �F .x/ D T �F˝ .x/ is a special

case of theminimal time function with˝ D f0g. Since 0 2 F ,eorem 3.27 tells us that �F .x/ D

0 if and only if x 2 ˝ � .�F /1 D F1, which completes the proof. �

We say for simplicity that 'W Rn! R is `�Lipschitz continuous on Rn if it is Lipschitz con-

tinuous on Rn with Lipschitz constant ` � 0. e next result calculates a constant ` of Lipschitzcontinuity of the Minkowski gauge on Rn.

Proposition 3.32 Let F be a closed, convex set that contains the origin as an interior point. en theMinkowski gauge �F is `-Lipschitz continuous on Rn with the constant

` WD infn1r

ˇIB.0I r/ � F; r > 0

o: (3.13)

In particular, we have �F .x/ � `kxk for all x 2 Rn.

Proof. e condition 0 2 intF ensures that ` < 1 for the constant ` from (3.13). Take any r > 0satisfying IB.0I r/ � F and get from the definitions that

�F .x/ D inf˚t � 0

ˇx 2 tF

� inf

˚t � 0

ˇx 2 tIB.0I r/

D inf

˚t � 0 j kxk � rt

D

kxk

r;

which implies that �F .x/ � `kxk. en the subadditivity of �F gives us that

�F .x/ � �F .y/ � �F .x � y/ � `kx � yk for all x; y 2 Rn;

and thus �F is `�Lipschitz continuous on Rn with constant (3.13). �

Next we relate the minimal time function to the Minkowski gauge.

eorem 3.33 Let F be a nonempty, closed, convex set. en for any ˝ ¤ ; we have

T F˝ .x/ D inf

˚�F .! � x/

ˇ! 2 ˝

; x 2 Rn : (3.14)


Proof. Consider first the case where T F˝ .x/ D 1. In this case we have˚

t � 0ˇ.x C tF / \˝ ¤ ;

D ;;

which implies that ft � 0 j ! � x 2 tF g D ; for every ! 2 ˝. us �F .! � x/ D 1 as ! 2 ˝,and hence the right-hand side of (3.14) is also infinity.

Considering now the case where T F˝ .x/ < 1, fix any t � 0 such that .x C tF / \˝ ¤ ;.

is gives us f 2 F and ! 2 ˝ with x C tf D !. us ! � x 2 tF , and so �F .! � x/ � t ,which yields inf!2˝ �F .! � x/ � t . It follows from definition (3.9) that

inf!2˝

�F .! � x/ � T F˝ .x/ < 1:

To verify the opposite inequality in (3.14), denote WD inf!2˝ �F .! � x/ and for any � > 0 find! 2 ˝ with �F .! � x/ < C �. By definition of the Minkowski gauge (3.11), we get t � 0 suchthat t < C � and ! � x 2 tF . is gives us ! 2 x C tF and therefore

T F˝ .x/ � t < C �:

Since � was chosen arbitrarily, we arrive at

T F˝ .x/ � D inf

!2˝�F .! � x/;

which implies the remaining inequality in (3.14) and completes the proof. �

As a consequence of the obtained result and Proposition 3.32, we establish now the`�Lipschitz continuity of minimal time function with the constant ` calculated in (3.13).

Corollary 3.34 Let F be a closed, convex set that contains the origin as an interior point. en forany nonempty set˝ the function T F

˝ is `�Lipschitz with constant (3.13).

Proof. It follows from the subadditivity of �F and the result of Proposition 3.32 that

�F .! � x/ � �F .! � y/C �F .y � x/ � �F .! � y/C `ky � xk for all ! 2 ˝

whenever x; y 2 Rn. en eorem 3.33 tells us that

T F˝ .x/ � T F

˝ .y/C `kx � yk;

which implies the `�Lipschitz property of T F˝ due to the arbitrary choice of x; y. �

Define next the enlargement of the target set ˝ via minimal time function by

˝r WD˚x 2 Rn

ˇT F

˝ .x/ � r; r > 0: (3.15)


Proposition 3.35 Let F be a nonempty, closed, convex subset of Rn and let˝ � Rn be a nonemptyset. en for any nonempty set˝ and any x … ˝r with T F

˝ .x/ < 1 we have

T F˝ .x/ D T F

˝r.x/C r:

Proof. It follows from ˝ � ˝r that T F˝r.x/ � T F

˝ .x/ < 1. Take any t � 0 with

.x C tF / \˝r ¤ ;

and find f1 2 F and u 2 ˝r such that x C tf1 D u. Since u 2 ˝r , we have T F˝ .u/ � r . For any

� > 0, there is s � 0 with s < r C � and .uC sF / \˝ ¤ ;. is implies the existence of ! 2 ˝

and f2 2 F such that uC sf2 D !. us

! D uC sf2 D .x C tf1/C sf2 2 x C .t C s/F

since F is convex. is gives us

T F˝ .x/ � t C s � t C r C � and so T F

˝ .x/ � t C r

by the arbitrary choice of � > 0. is allows us to conclude that

T F˝ .x/ � T F

˝rC r:

To verify the opposite inequality, denote WD T F˝ .x/ and observe that r < by x … ˝r . For any

� > 0, find t 2 Œ0;1/, f 2 F , and ! 2 ˝ with � t < C � and x C tf D !. Hence

! D x C tf D x C .t � r/f C rf 2 x C .t � r/f C rF and so T F˝ .x C .t � r/f / � r;

which shows that x C .t � r/f 2 ˝r . We also see that x C .t � r/f 2 x C .t � r/F . usT F

˝r.x/ � t � r � � r C �, which verifies that

r C T F˝r

� D T F˝ .x/

since � > 0 was chosen arbitrarily. is completes the proof. �

Proposition 3.36 Let F be a nonempty, closed, convex subset of Rn. en for any nonempty set ˝,any x 2 domT F

˝ , any t � 0, and any f 2 F we have

T F˝ .x � tf / � T F

˝ .x/C t:

Proof. Fix � > 0 and select s � 0 so that

T F˝ .x/ � s < T F

˝ .x/C � and .x C sF / \˝ ¤ ;:

en .x � tf C tF C sF / \˝ ¤ ;, and hence .x � tf C .t C s/F / \˝ ¤ ;. It shows that

T F˝ .x � tf / � t C s � t C T F

˝ .x/C �;

which implies the conclusion of the proposition by letting � ! 0C. �

3.8. SUBGRADIENTSOFMINIMALTIMEFUNCTIONS 117

3.8 SUBGRADIENTSOFMINIMALTIMEFUNCTIONSObserving that the minimal time function (3.9) is intrinsically nonsmooth, we study here its gen-eralized differentiation properties in the case where the target set ˝ is convex. In this case (sincethe constant dynamics F is always assumed convex) function (3.9) is convex as well by Proposi-tion 3.29. e results below present precise formulas for calculating the subdifferential @T F

˝ . Nx/

depending on the position of Nx with respect to the set ˝ � F1.

eorem 3.37 Let both sets 0 2 F � Rn and ˝ � Rn be nonempty, closed, convex. For anyNx 2 ˝ � F1, the subdifferential of T F

˝ at Nx is calculated by

@T F˝ . Nx/ D N. NxI˝ � F1/ \ C �; (3.16)

where C � WD fv 2 Rnj �F .�v/ � 1g is defined via (2.41). In particular, we have

@T F˝ . Nx/ D N. NxI˝/ \ C � for Nx 2 ˝: (3.17)

Proof. Fix any v 2 @T F˝ . Nx/. Using (2.13) and the fact that T F

˝ .x/ D 0 for all x 2 ˝ � F1, we getv 2 N. NxI˝ � F1/. Write Nx 2 ˝ � F1 as Nx D N! � d with N! 2 ˝ and d 2 F1. Fix any f 2 F

and define xt WD . Nx C d/ � tf D N! � tf as t > 0. en .xt C tF / \˝ ¤ ;, and so

hv; xt � Nxi � T F˝ .xt / � t; or equivalently

Dv;d

t� f

E� 1 for all t > 0:

By letting t ! 1, it implies that hv;�f i � 1, and hence we get v 2 C �. is verifies the inclu-sion @T F

˝ . Nx/ � N. NxI˝ � F1/ \ C �.To prove the opposite inclusion in (3.16), fix any v 2 N. NxI˝ � F1/ \ C � and any u 2

domT F˝ . For any � > 0, there exist t 2 Œ0;1/, ! 2 ˝, and f 2 F such that

T F˝ .u/ � t < T F

˝ .u/C � and uC tf D !:

Since ˝ � ˝ � F1, we have the relationships

hv; u � Nxi D hv; ! � Nxi C thv;�f i � t < T F˝ .u/C � D T F

˝ .u/ � T F˝ . Nx/C �;

which show that v 2 @T F˝ . Nx/. In this way we getN. NxI˝ � F1/ \ C � � @T F

˝ . Nx/ and thus justifythe subdifferential formula (3.16) in the general case where Nx 2 ˝ � F1.

It remains to verify the simplified representation (3.17) in the case where Nx 2 ˝. Since˝ � ˝ � F1, we obviously have the inclusion

N. NxI˝ � F1/ \ C �� N. NxI˝/ \ C �:

To show the converse inclusion, pick v 2 N. NxI˝/ \ C � and fix x 2 ˝ � F1. Taking ! 2 ˝

and d 2 F1 with x D ! � d gives us td 2 F since 0 2 F , so hv;�td i � 1 for all t > 0. ushv;�d i � 0, which readily implies that

hv; x � Nxi D hv; ! � d � Nxi D hv; ! � Nxi C hv;�d i � 0:

In this way we arrive at v 2 N. NxI˝ � F1/ \ C � and thus complete the proof. �


e next theorem provides the subdifferential calculation for minimal time functions in thecases complemented to those in eorem 3.37, i.e., when Nx … ˝ � F1 and Nx … ˝.

eorem 3.38 Let both sets 0 2 F � Rn and˝ � Rn be nonempty, closed, convex and let Nx 2

domT F˝ for the minimal time function (3.9). Suppose that˝ is bounded. en for Nx … ˝ � F1

we have@T F

˝ . Nx/ D N. NxI˝r/ \ S�; (3.18)

where the enlargement ˝r is defined in (3.15), and where

S�WD

˚v 2 Rn

ˇ�F .�v/ D 1

with r D T F

˝ . Nx/ > 0:

Proof. Pick v 2 @T F˝ . Nx/. Observe from the definition that

hv; x � Nxi � T F˝ .x/ � T F

˝ . Nx/ for all x 2 Rn : (3.19)

en T F˝ .x/ � r D T F

˝ . Nx/, and hence hv; x � Nxi � 0 for any x 2 ˝r . is yields v 2 N. NxI˝r/.Using (3.19) for x WD Nx � f with f 2 F and Proposition 3.36 ensures that �F .�v/ � 1. Fix� 2 .0; r/ and select t 2 R, f 2 F , and ! 2 ˝ so that

r � t < r C �2 and ! D Nx C tf:

We can write ! D Nx C �f C .t � �/f and then see that T F˝ . Nx C �f / � t � �. Applying (3.19)

to x WD Nx C �f gives us the estimates

hv; �f i � T F˝ . Nx C �f / � T F

˝ . Nx/ � t � � � r � �2� �:

which imply in turn that1 � � � h�v; f i � �F .�v/:

Letting now � # 0 tells us that �F .�v/ � 1, i.e., �F .�v/ D 1 in this case. us v 2 S�, whichproves the inclusion @T F

˝ . Nx/ � N. NxI˝r/ \ S�.To verify the opposite inclusion in (3.18), take any v 2 N. NxI˝r/ with �F .�v/ D 1 and

show that (3.19) holds. It follows from eorem 3.37 that v 2 @T F˝r. Nx/, and so

hv; x � Nxi � T F˝r.x/ for all x 2 Rn :

Fix x 2 Rn and deduce from Proposition 3.35 that T F˝ .x/ � r D T F

˝r.x/ in the case t WD

T F˝ .x/ > r , which ensures (3.19). Suppose now that t � r and for any � > 0 choose f 2 F

such that hv;�f i > 1 � �. en Proposition 3.36 tells us that T F˝ .x � .r � t /f / � r , and so

x � .r � t/f 2 ˝r . Since v 2 N. NxI˝r/, one has hv; x � .r � t/f � Nxi � 0, which yields

hv; x � Nxi � hv; f i.r � t / � .1 � �/.t � r/ D .1 � �/�T F

˝ .x/ � T F˝ . Nx/

�:

Since � > 0 was arbitrarily chosen, we get (3.19), and hence verify that v 2 @T F˝ . Nx/. �

3.8. SUBGRADIENTSOFMINIMALTIMEFUNCTIONS 119

e next result addresses the same setting as in eorem 3.38 while presenting anotherformula for calculating the subdifferential of the minimal time function that involves the subdif-ferential of the Minkowski gauge and generalized projections to the target set.

eorem 3.39 Let both sets 0 2 F � Rn and ˝ � Rn be nonempty, closed, convex and letNx 2 domT F

˝ . Suppose that ˝ is bounded and that Nx … ˝ � F1. en we have

@T F˝ . Nx/ D

�� @�F . N! � Nx/

�\N. N!I˝/ (3.20)

for any N! 2 ˘F . NxI˝/ via the generalized projector

˘F . NxI˝/ WD˚! 2 ˝

ˇT F

˝ . Nx/ D �F .! � Nx/: (3.21)

Proof. Fixing any v 2 @T F˝ . Nx/ and N! 2 ˘F . NxI˝/, we get T F

˝ . Nx/ D �F . N! � Nx/ and

hv; x � Nxi � T F˝ .x/ � T F

˝ . Nx/ for all x 2 Rn : (3.22)

It is easy to check the following relationships that hold for any ! 2 ˝:

hv; ! � N!i D hv; .! � N! C Nx/ � Nxi � T F˝ .! � N! C Nx/ � T F

˝ . Nx/

� �F

�! � .! � N! C Nx/

�� F . N! � Nx/ D 0:

is shows that v 2 N. N!I˝/. Denote now eu WD N! � Nx and for any t 2 .0; 1/ and u 2 Rn get

hv;�t .u � eu/i � T F˝

�Nx � t .u � eu/� � T F

˝ . Nx/

� �F

�N! � . Nx � t .u � eu//� � �F . N! � Nx/

D �F .euC t.u � eu// � �F .eu/D �F

�tuC .1 � t/eu�

� �F .eu/� t�F .u/C .1 � t /�F .eu/ � �F .eu/D t

��F .u/ � �F .eu/�

by applying (3.22) with x WD Nx � t .u � eu/. It follows that

h�v; u � eui � �F .u/ � �F .eu/ for all u 2 Rn :

us �v 2 @�F .eu/, and so v 2 �@�F . N! � Nx/, which proves the inclusion “�” in (3.20).To verify the opposite inclusion, write

�F .x/ D inf˚t � 0

ˇx 2 tF

D inf

˚t � 0

ˇ.�x C tF / \O ¤ ;

D T F

O .�x/

with O WD f0g. Since Nx … ˝ � F1, for any N! 2 ˝ we have Nx � N! … O � F1 D �F1. Further-more, it follows from �v 2 @�F . N! � Nx/ by applying eorem 3.38 that v 2 S�. Hence it remainsto justify the inclusion �

� @�F . N! � Nx/�

\N. N!I˝/ � N. NxI˝r/ (3.23)


and then to apply eorem 3.38. To proceed with the proof of (3.23), pick any vector x 2 ˝r

and a positive number � arbitrarily small. en there are t < r C �, f 2 F , and ! 2 ˝ such that! D x C tf , which yields

hv; x � Nxi D hv; ! � tf � Nxi

D th�v; f i C hv; ! � N!i C hv; N! � Nxi

� t C hv; ! � N!i C hv; N! � Nxi

� T F˝ . Nx/C � C hv; ! � N!i C hv; N! � Nxi:

Observe that hv; ! � N!i � 0 by v 2 N. N!I˝/. Moreover,

hv; N! � Nxi D h�v; 0 � . N! � Nx/i � �F .0/ � �F . N! � Nx/ D �T F˝ . Nx/

since �v 2 @�F . N! � Nx/. It follows that hv; x � Nxi � � for all x 2 ˝r . Hence v 2 N. NxI˝r/ be-cause � > 0 was chosen arbitrarily. is completes the proof of the theorem. �

3.9 NASHEQUILIBRIUMis section gives a brief introduction to noncooperative game theory and presents a simple proofof the existence of Nash equilibrium as a consequence of convexity and Brouwer’s fixed pointtheorem. For simplicity, we only consider two-person games while more general situations can betreated similarly.

Definition 3.40 Let ˝1 and ˝2 be nonempty subsets of Rn and Rp, respectively. A - in the case of two players I and II consists of two strategy sets ˝i and two real-valuedfunctions ui W ˝1 �˝2 ! R for i D 1; 2 called the . en we refer to this gameas f˝i ; uig for i D 1; 2.

e key notion of Nash equilibrium was introduced by John Forbes Nash, Jr., in 1950 whoproved the existence of such an equilibrium and was awarded the Nobel Memorial Prize in Eco-nomics in 1994.

Definition 3.41 Given a noncooperative two-person game f˝i ; uig, i D 1; 2, a N - is an element . Nx1; Nx2/ 2 ˝1 �˝2 satisfying the conditions

u1.x1; Nx2/ � u1. Nx1; Nx2/ for all x1 2 ˝1;

u2. Nx1; x2/ � u2. Nx1; Nx2/ for all x2 2 ˝2:

e conditions above mean that . Nx1; Nx2/ is a pair of best strategies that both players agreewith in the sense that Nx1 is a best response of Player I when Player II chooses the strategy Nx2, andNx2 is a best response of Player II when Player I chooses the strategy Nx1. Let us now consider twoexamples to illustrate this concept.

3.9. NASHEQUILIBRIUM 121

Example 3.42 In a two-person game suppose that each player can only choose either strategyA or B . If both players choose strategy A, they both get 4 points. If Player I chooses strategy Aand Player II chooses strategy B , then Player I gets 1 point and Player II gets 3 points. If Player Ichooses strategy B and Player II chooses strategy A, then Player I gets 3 points and Player II gets1 point. If both choose strategy B , each player gets 2 points. e payoff function of each playeris represented in the payoff matrix below.

....4; 4 ..1; 3

..3; 1 ..2; 2.

A

.

B

.A .B

.

Player II

.Pl

ayer

I

In this example, Nash equilibrium occurs when both players choose strategy A, and it also occurswhen both players choose strategy B . Let us consider, for instance, the case where both playerschoose strategy B . In this case, given that Player II chooses strategy B , Player I also wants tokeep strategy B because a change of strategy would lead to a reduction of his/her payoff from 2

to 1. Similarly, given that Player I chooses strategy B , Player II wants to keep strategy B becausea change of strategy would also lead to a reduction of his/her payoff from 2 to 1.

Example 3.43 Let us consider another simple two-person game called matching pennies. Sup-pose that Player I and Player II each has a penny. Each player must secretly turn the penny toheads or tails and then reveal his/her choices simultaneously. Player I wins the game and getsPlayer II’s penny if both coins show the same face (heads-heads or tails-tails). In the other casewhere the coins show different faces (heads-tails or tails-heads), Player II wins the game and getsPlayer I’s penny. e payoff function of each player is represented in the payoff matrix below.

....1;�1 ..�1; 1

..�1; 1 ..1;�1.

Heads

.

Tails

.Heads .Tails

.

Player II

.

Play

erI

In this game it is not hard to see that no Nash equilibrium exists.Denote by A the matrix that represents the payoff function of Player I, and denote by B

the matrix that represents the payoff function of Player II:

A D1 �1

�1 1and B D

�1 1

1 �1


Now suppose that Player I is playing with a mind reader who knows Player I’s choice of faces. IfPlayer I decides to turn heads, then Player II knows about it and chooses to turn tails and winsthe game. In the case where Player I decides to turn tails, Player II chooses to turn heads andagain wins the game. us to have a fair game, he/she decides to randomize his/her strategy by,e.g., tossing the coin instead of putting the coin down. We describe the new game as follows.

....1;�1 ..�1; 1

..�1; 1 ..1;�1.

Heads, q1

.

Tails, q2

.Heads, p1 .Tails, p2

.

Player II

.Pl

ayer

I

In this new game, Player I uses a coin randomly with probability of coming up heads p1 and prob-ability of coming up tails p2, where p1 C p2 D 1. Similarly, Player II uses another coin randomlywith probability of coming up heads q1 and probability of coming up tails q2, where q1 C q2 D 1.e new strategies are called mixed strategies while the original ones are called pure strategies.

Suppose that Player II uses mixed strategy fq1; q2g. en Player I’s expected payoff forplaying the pure strategy heads is

uH .q1; q2/ D q1 � q2 D 2q1 � 1:

Similarly, Player I’s expected payoff for playing the pure strategy tails is

uT .q1; q2/ D �q1 C q2 D �2q1 C 1:

us Player I’s expected payoff for playing mixed strategy fp1; p2g is

u1.p; q/ D p1uH .q1; q2/C p2uT .q1; q2/ D p1.q1 � q2/C p2.�q1 C q2/ D pTAq;

where p D Œp1; p2�T and q D Œq1; q2�

T .By the same arguments, if Player I chooses mixed strategy fp1; p2g, then Player II’s ex-

pected payoff for playing mixed strategy fq1; q2g is

u2.p; q/ D pTBq D �u1.p; q/:

For this new game, an element . Np; Nq/ 2 � �� is a Nash equilibrium if

u1.p; Nq/ � u1. Np; Nq/ for all p 2 �;

u2. Np; q/ � u2. Np; Nq/ for all q 2 �;

where � WD˚.p1; p2/

ˇp1 � 0; p2 � 0; p1 C p2 D 1

is a nonempty, compact, convex set.

Nash [23, 24] proved the existence of his equilibrium in the class of mixed strategies in amore general setting. His proof was based on Brouwer’s fixed point theorem with rather involved

3.9. NASHEQUILIBRIUM 123

arguments. Now we present, following Rockafellar [28], a much simpler proof of the Nash equi-librium theorem that also uses Brouwer’s fixed point theorem while applying in addition justelementary tools of convex analysis and optimization.

eorem 3.44 Consider a two-person game f˝i ; uig, where ˝1 � Rn and ˝2 � Rp arenonempty, compact, convex sets. Let the payoff functions ui W ˝1 �˝2 ! R be given by

u1.p; q/ WD pTAq; u2.p; q/ WD pTBq;

where A and B are n � p matrices. en this game admits a Nash equilibrium.

Proof. It follows from the definition that an element . Np; Nq/ 2 ˝1 �˝2 is a Nash equilibrium ofthe game under consideration if and only if

� u1.p; Nq/ � �u1. Np; Nq/ for all p 2 ˝1;

� u2. Np; q/ � �u2. Np; Nq/ for all q 2 ˝2:

It is a simple exercise to prove (see also an elementary convex optimization result of Corollary 4.15in Chapter 4) that this holds if and only if we have the normal cone inclusions

rpu1. Np; Nq/ 2 N. NpI˝1/; rqu2. Np; Nq/ 2 N. NqI˝2/: (3.24)

By using the structures of ui and defining ˝ WD ˝1 �˝2, conditions (3.24) can be expressed inone of the following two equivalent ways:

A Nq 2 N. NpI˝1/ and BTNp 2 N. NqI˝2/I

.A Nq; BTNp/ 2 N. NpI˝1/ �N. NqI˝2/ D N.. Np; Nq/I˝1 �˝2/ D N.. Np; Nq/I˝/:

By Proposition 1.78 on the Euclidean projection, this is equivalent also to

. Np; Nq/ D ˘.. Np; Nq/C .A Nq;BTNp/I˝/: (3.25)

Defining now the mapping F W˝ ! ˝ by

F.p; q/ WD ˘�.p; q/C .Aq;BTp/I˝/ with ˝ D ˝1 �˝2

and employing another elementary projection property from Proposition 1.79 allow us to con-clude that the mapping F is continuous and the set˝ is nonempty, compact, and convex. By theclassical Brouwer fixed point theorem (see, e.g., [33]), the mappingF has a fixed point . Np; Nq/ 2 ˝,which satisfies (3.25), and thus it is a Nash equilibrium of the game. �



Exercise 3.1 Let f W Rn! R be a function finite around Nx. Show that f is Gâteaux dif-

ferentiable at Nx with f 0G. Nx/ D v if and only if the directional derivative f 0. NxI d/ exists and

f 0. NxI d/ D hv; d i for all d 2 Rn.

Exercise 3.2 Let ˝ be a nonempty, closed, convex subset of Rn.(i) Prove that the function f .x/ WD Œd.xI˝/�2 is differentiable on Rn and

rf . Nx/ D 2Œ Nx �˘. NxI˝/�

for every Nx 2 Rn.(ii) Prove that the function g.x/ WD d.xI˝/ is differentiable on Rn

n˝.

Exercise 3.3 Let F be a nonempty, compact, convex subset of Rn and let A W Rn! Rp be a

linear mapping. Define

f .x/ WD supu2F

�hAx; ui �

1

2kuk

2�:

Prove that f is convex and differentiable.

Exercise 3.4 Give an example of a function f W Rn! R, which is Gâteuax differentiable but

not Fréchet differentiable. Can such a function be convex?

Exercise 3.5 Let f W Rn! R be a convex function and let Nx 2 int.domf /. Show that if @f

@xi

at Nx exists for all i D 1; : : : ; n, then f is (Fréchet) differentiable at Nx.

Exercise 3.6 Let f W Rn! R be differentiable on Rn. Show that f is strictly convex if and

only if it satisfies the condition

hrf . Nx/; x � Nxi < f .x/ � f . Nx/ for all x 2 Rn and Nx 2 Rn; x ¤ Nx:

Exercise 3.7 Let f W Rn! R be twice continuously differentiable (C 2) on Rn. Prove that if

the Hessian r2f . Nx/ is positive definite for all Nx 2 Rn, then f is strictly convex. Give an exampleshowing that the converse is not true in general.

Exercise 3.8 A convex function f W Rn! R is called strongly convex with modulus c > 0 if

for all x1; x2 2 Rn and � 2 .0; 1/ we have

f��x1 C .1 � �/x2

�� f .x1/C .1 � �/f .x2/ �

1

2c�.1 � �/kx1 � x2k

2:


Prove that the following are equivalent for any C 2 convex function f W Rn! R:

(i) f strongly convex.(ii) f �

c

2k � k2 is convex.

(iii) hr2f .x/d; d i � ckdk2 for all x; d 2 Rn.

Exercise 3.9 A set-valued mapping F W Rn!! Rn is said to be monotone if

hv2 � v1; x2 � x1i � 0 whenever vi 2 F.xi /; i D 1; 2:

F is called strictly monotone if

hv2 � v1; x2 � x1i > 0 whenever vi 2 F.xi /; x1 ¤ x2; i D 1; 2:

It is called strongly monotone with modulus ` > 0 if

hv2 � v1; x2 � x1i � `kx1 � x2k2 whenever vi 2 F.xi /; i D 1; 2:

(i) Prove that for any convex function f W Rn! R the mapping F.x/ WD @f .x/ is monotone.

(ii) Furthermore, for any convex function f W Rn! R the subdifferential mapping @f .x/ is

strictly monotone if and only if f is strictly convex, and it is strongly monotone if and only iff is strongly convex.

Exercise 3.10 Prove Corollary 3.7.

Exercise 3.11 LetA be an p � nmatrix and let b 2 Rp. Use the Farkas lemma to show that thesystem Ax D b, x � 0 has a feasible solution x 2 Rn if and only if the system AT y � 0, bT y > 0

has no feasible solution y 2 Rp.

Exercise 3.12 Deduce Carathéodory’s theorem from Radon’s theorem and its proof.

Exercise 3.13 Use Carathéodory’s theorem to prove Helly’s theorem.

Exercise 3.14 Let C be a subspace of Rn. Show that the polar of C is represented by

C ıD C?

WD˚v 2 Rn

ˇhv; ci D 0 for all c 2 C

:

Exercise 3.15 Let C be a nonempty, convex cone of Rn. Show that

C ıD

˚v 2 Rn

ˇhv; ci � 1 for all c 2 C

:

Exercise 3.16 Let ˝1 and ˝2 be convex sets with int˝1 \˝2 ¤ ;. Show that

T . NxI˝1 \˝2/ D T . NxI˝1/ \ T . NxI˝2/ for any Nx 2 ˝1 \˝2:


Exercise 3.17 Let A be p � n matrix, let

˝ WD˚x 2 Rn

ˇAx D 0; x � 0

;

and let Nx D . Nx1; : : : ; Nxn/ 2 ˝. Verify the tangent and normal cone formulas:

T . NxI˝/ D fu D .u1; : : : ; un/ 2 RnˇAu D 0; ui � 0 for any i with Nxi D 0

;

N. NxI˝/ D˚v 2 Rn

ˇv D A�y � z; y 2 Rp; z 2 Rn; zi � 0; hz; Nxi D 0

:

Exercise 3.18 Give an example of a closed, convex set ˝ � R2 with a point Nx 2 ˝ such thatthe cone RC.˝ � Nx/ is not closed.

Exercise 3.19 Let f W Rn! R be a convex function such that there is ` � 0 satisfying @f .x/ �

`IB . Show that f is `�Lipschitz continuous on Rn.

Exercise 3.20 Let f W R ! R be convex. Prove that f is nondecreasing if and only if @f .x/ �

Œ0;1/ for all x 2 R.

Exercise 3.21 Find the horizon cones of the following sets:(i) F WD f.x; y/ 2 R2

j y � x2g.(ii) F WD f.x; y/ 2 R2

j y � jxjg.(iii) F WD f.x; y/ 2 R2

j x > 0; y � 1=xg.(iv) F WD

˚.x; y/ 2 R2

j x > 0; y �px2 C 1g.

Exercise 3.22 Let vi 2 Rn and bi > 0 for i D 1; : : : ; m. Consider the set

F WD˚x 2 Rn

ˇhvi ; xi � bi for all i D 1; : : : ; mg: (3.26)

Prove the following representation of the horizon cone and Minkowski gauge for F :

F1 D˚x 2 Rn

ˇhvi ; xi � 0 for all i D 1; : : : ; m

;

�F .u/ D maxn0; max

i2f1;:::;mg

hvi ; ui

bi

o:

Exercise 3.23 Let A;B be nonempty, closed, convex subsets of Rn.(i) Show that .A � B/1 D A1 � B1.(ii) Find conditions ensuring that .AC B/1 D A1 C B1.


Exercise 3.24 Let F ¤ ; be a closed, bounded, convex set and let ˝ ¤ ; be a closed set inRn. Prove that T F

˝ .x/ D 0 if and only if x 2 ˝. is means that the condition 0 2 F in eo-rem 3.27(i) is not required in this case.

Exercise 3.25 Let F ¤ ; be a closed, bounded, convex set and let ˝ ¤ ; be a closed, convexset. Consider the sets C � and S� defined in eorem 3.37 and eorem 3.38, respectively. Provethe following:(i) If Nx 2 ˝, then

@T F˝ . Nx/ D N. NxI˝/ \ C �:

(ii) If Nx … ˝, then@T F

˝ . Nx/ D N. NxI˝r/ \ S�; r WD T F˝ . Nx/ > 0:

(iii) If Nx … ˝, then@T F

˝ . Nx/ D�

� @�F . N! � Nx/�

\N. N!I˝/

for any N! 2 ˘F . NxI˝/

Exercise 3.26 Give a direct proof of characterizations (3.24) of Nash equilibrium.

Exercise 3.27 In Example 3.43 with mixed strategies, find . Np; Nq/ that yields a Nash equilibriumin this game.

129

C H A P T E R 4

Applications to Optimizationand Location Problems

In this chapter we give some applications of the results of convex analysis from Chapters 1–3to selected problems of convex optimization and facility location. Applications to optimizationproblems are mainly classical, being presented however in a simplified and unified way. On theother hand, applications to location problems are mostly based on recent journal publications andhave not been systematically discussed from the viewpoint of convex analysis in monographs andtextbooks.

4.1 LOWERSEMICONTINUITYANDEXISTENCEOFMINIMIZERS

We first discuss the notion of lower semicontinuity for extended-real-valued functions, whichplays a crucial role in justifying the existence of absolute minimizers and related topics. Note thatmost of the results presented below do not require the convexity of f .

Definition 4.1 An extended-real-valued function f W Rn! R is .l.s.c./

Nx 2 Rn if for every ˛ 2 R with f . Nx/ > ˛ there is ı > 0 such that

f .x/ > ˛ for all x 2 IB. NxI ı/: (4.1)

We simply say that f is if it is l.s.c. at every point of Rn.

e following examples illustrate the validity or violation of lower semicontinuity.

Example 4.2 (i) Define the function f W R ! R by

f .x/ WD

(x2 if x ¤ 0;

�1 if x D 0:

It is easy to see that it is l.s.c. at Nx D 0 and in fact on the whole real line. On the other hand, thereplacement of f .0/ D �1 by f .0/ D 1 destroys the lower semicontinuity at Nx D 0.

130 4. APPLICATIONSTOOPTIMIZATIONANDLOCATIONPROBLEMS

Figure 4.1: Which function is l.s.c?

(ii) Consider the indicator function of the set Œ�1; 1� � R given by

f .x/ WD

(0 if jxj � 1;

1 otherwise:

en f is a lower semicontinuous function on R. However, the small modification

g.x/ WD

(0 if jxj < 1;

1 otherwise

violates the lower semicontinuity on R. In fact, g is not l.s.c. at x D ˙1 while it is lower semi-continuous at any other point of the real line.

e next four propositions present useful characterizations of lower semicontinuity.

Proposition 4.3 Let f W Rn! R and let Nx 2 domf . en f is l.s.c. at Nx if and only if for any

� > 0 there is ı > 0 such that

f . Nx/ � � < f .x/ whenever x 2 IB. NxI ı/: (4.2)

Proof. Suppose that f is l.s.c. at Nx. Since � WD f . Nx/ � � < f . Nx/, there is ı > 0 with

f . Nx/ � � D � < f .x/ for all x 2 IB. NxI ı/:

Conversely, suppose that for any � > 0 there is ı > 0 such that (4.2) holds. Fix any � < f . Nx/ andchoose � > 0 for which � < f . Nx/ � �. en

� < f . Nx/ � � < f .x/ whenever x 2 IB. NxI ı/

for some ı > 0, which completes the proof. �

4.1. LOWERSEMICONTINUITYANDEXISTENCEOFMINIMIZERS 131

Proposition 4.4 e function f W Rn! R is l.s.c. if and only if for any number � 2 R the level set

fx 2 Rnj f .x/ � �g is closed.

Proof. Suppose that f is l.s.c. For any fix � 2 R, denote

˝ WD˚x 2 Rn

ˇf .x/ � �

:

If Nx … ˝, we have f . Nx/ > �. By definition, find ı > 0 such that

f .x/ > � for all x 2 IB. NxI ı/;

which shows that IB. NxI ı/ � ˝c , the complement of ˝. us ˝c is open, and so ˝ is closed.To verify the converse implication, suppose that the set fx 2 Rn

j f .x/ � �g is closed forany � 2 R. Fix Nx 2 Rn and � 2 R with f . Nx/ > � and then let

� WD˚x 2 Rn

ˇf .x/ � �

:

Since � is closed and Nx … �, there is ı > 0 such that IB. NxI ı/ � �c . is implies

f .x/ > � for all x 2 IB. NxI ı/


Proposition 4.5 e function f W Rn! R is l.s.c. if and only if its epigraph is closed.

Proof. To verify the “only if ” part, fix . Nx; �/ … epif meaning � < f . Nx/. For any � > 0 withf . Nx/ > �C � > �, there is ı > 0 such that f .x/ > �C � whenever x 2 IB. NxI ı/. us

IB. NxI ı/ � .� � �; �C �/ � .epif /c ;

and hence the set epi f is closed.Conversely, suppose that epif is closed. Employing Proposition 4.4, it suffices to verify

that for any � 2 R the set ˝ WD fx 2 Rnj f .x/ � �g is closed. To see this, fix any sequence

fxkg � ˝ that converges to Nx. Since f .xk/ � �, we have .xk; �/ 2 epif for all k. It follows from.xk; �/ ! . Nx; �/ that . Nx; �/ 2 epif , and so f . Nx/ � �. us Nx 2 ˝, which justifies the closednessof the set ˝. �

Proposition 4.6 e function f W Rn! R is l.s.c. at Nx if and only if for any sequence xk ! Nx we

have lim infk!1 f .xk/ � f . Nx/.


Proof. If f is l.s.c. at Nx, take a sequence of xk ! Nx. For any � < f . Nx/, there exists ı > 0such that (4.1) holds. en xk 2 IB. NxI ı/, and so � < f .xk/ for large k. It follows that � �

lim infk!1 f .xk/, and hence f . Nx/ � lim infk!1 f .xk/ by the arbitrary choice of � < f . Nx/.To verify the converse, suppose by contradiction that f is not l.s.c. at Nx. en there is � <

f . Nx/ such that for every ı > 0 we have xı 2 IB. NxI ı/ with � � f .xı/. Applying this to ık WD1

kgives us a sequence fxkg � Rn converging to Nx with � � f .xk/. It yields

f . Nx/ > � � lim infk!1

f .xk/;

which is a contradiction that completes the proof. �

Now we are ready to obtain a significant result on the lower semicontinuity of the minimalvalue function (3.9), which is useful in what follows.

eorem4.7 In the setting of eorem 3.27 with 0 2 F we have that the minimal time functionT F

˝ is lower semicontinuous.

Proof. Pick Nx 2 Rn and fix an arbitrary sequence fxkg converging to Nx as k ! 1. Based onProposition 4.6, we check that

lim infk!1

T F˝ .xk/ � T F

˝ . Nx/:

e above inequality obviously holds if lim infk!1 T F˝ .xk/ D 1; thus it remains to consider the

case of lim infk!1 T F˝ .xk/ D 2 Œ0;1/. Since it suffices to show that � T F

˝ . Nx/, we assumewithout loss of generality that T F

˝ .xk/ ! . By the definition of the minimal time function, wefind a sequence ftkg � Œ0;1/ such that

T F˝ .xk/ � tk < T F

˝ .xk/C1

kand .xk C tkF / \˝ ¤ ; as k 2 N :

For each k 2 N, fix fk 2 F and !k 2 ˝ with xk C tkfk D !k and suppose without loss of gen-erality that !k ! ! 2 ˝. Consider the two cases: > 0 and D 0. If > 0, then

fk !! � Nx

2 F as k ! 1;

which implies that . Nx C F / \˝ ¤ ;, and hence � T F˝ . Nx/. Consider the other case where

D 0. In this case the sequence ftkg converges to 0, and so tkfk ! ! � Nx. It follows from Propo-sition 3.24 that ! � Nx 2 F1, which yields Nx 2 ˝ � F1. Employing eorem 3.27 tells us thatT F

˝ . Nx/ D 0 and also � T F˝ . Nx/. is verifies the lower semicontinuity of T F

˝ in setting of e-orem 3.27. �

4.1. LOWERSEMICONTINUITYANDEXISTENCEOFMINIMIZERS 133

e following two propositions indicate two cases of preserving lower semicontinuity underimportant operations on functions.

Proposition 4.8 Let fi W Rn! R, i D 1; : : : ; m, be l.s.c. at some point Nx. en the sum

PmiD1 fi

is l.s.c. at this point.

Proof. It suffices to verify it for m D 2. Taking xk ! Nx, we get

lim infk!1

fi .xk/ � f . Nx/ for i D 1; 2;

which immediately implies that

lim infk!1

�f1.xk/C f2.xk/

�� lim inf

k!1f1.xk/C lim inf

k!1f2.xk/ � f1. Nx/C f2. Nx/;

and thus f1 C f2 is l.s.c. at Nx by Proposition 4.6. �

Proposition 4.9 Let I be any nonempty index set and let fi W Rn! R be l.s.c. for all i 2 I . en

the supremum function f .x/ WD supi2I fi .x/ is also lower semicontinuous.

Proof. For any number � 2 R, we have directly from the definition that˚x 2 Rn

ˇf .x/ � �

D

˚x 2 Rn

ˇsupi2I

fi .x/ � �

D\i2I

˚x 2 Rn

ˇfi .x/ � �

:

us all the level sets of f are closed as intersections of closed sets. is insures the lower semi-continuity of f by Proposition 4.4. �

e next result provides two interrelated unilateral versions of the classical Weierstrass ex-istence theorem presented in eorem 1.18. e crucial difference is that now we assume thelower semicontinuity versus continuity while ensuring the existence of only minima, not togetherwith maxima as in eorem 1.18.

eorem 4.10 Let f W Rn! R be a l.s.c. function. e following hold:

(i) e function f attains its absolute minimum on any nonempty, compact subset ˝ � Rn thatintersects its domain.(ii)Assume that infff .x/ j x 2 Rn

g < 1 and that there is a real number � > infff .x/ j x 2 Rng

for which the level set fx 2 Rnj f .x/ < �g of f is bounded.en f attains its absoluteminimum

on Rn at some point Nx 2 domf .


Proof. To verify (i), show first that f is bounded below on ˝. Suppose on the contrary that forevery k 2 N there is xk 2 ˝ such that f .xk/ � �k. Since ˝ is compact, we get without loss ofgenerality that xk ! Nx 2 ˝. It follows from the lower semicontinuity that


f .xk/ D �1;

which is a contradiction showing that f is bounded below on ˝. To justify now assertion (i),observe that WD infx2˝ f .x/ < 1. Take fxkg � ˝ such that f .xk/ ! . By the compactnessof ˝, suppose that xk ! Nx 2 ˝ as k ! 1. en


f .xk/ D ;

which shows that f . Nx/ D , and so Nx realizes the absolute minimum of f on ˝.It remains to prove assertion (ii). It is not hard to show that f is bounded below, so WD

infx2Rn f .x/ is a real number. Let fxkg � Rn for which f .xk/ ! as k ! 1 and let k0 2 Nbe such that f .xk/ < � for all k � k0. Since the level set fx 2 Rn

j f .x/ < �g is bounded, weselect a subsequence of fxkg (no relabeling) that converges to some Nx. en


f .xk/ � ;

which shows that Nx 2 domf is a minimum point of f on Rn. �

In contrast to all the results given above in this section, the next theorem requires convexity.It provides effective characterizations of the level set boundedness imposed in eorem 4.10(ii).Note that condition (iv) of the theorem below is known as coercivity.

eorem 4.11 Let f W Rn! R be convex and let infff .x/ j x 2 Rn

g < 1. e following as-sertions are equivalent:(i) ere exists a real number ˇ > infff .x/ j x 2 Rn

g for which the level set Lˇ WD fx 2

Rnj f .x/ < ˇg is bounded.

(ii) All the level sets fx 2 Rnj f .x/ < �g of f are bounded.

(iii) limkxk!1 f .x/ D 1.

(iv) lim infkxk!1

f .x/

kxk> 0.

Proof. Let WD infff .x/ j x 2 Rng, which is a real number or �1. We first verify (i)H)(ii).

Suppose by contradiction that there is � 2 R such that the level set L� is not bounded. enthere exists a sequence fxkg � L� with kxkk ! 1. Fix ˛ 2 . ; ˇ/, Nx 2 L˛ and define the convexcombination

yk WD Nx C1p

kxkk

�xk � Nx

�D

1pkxkk

xk C

�1 �

1pkxkk

�Nx:

4.2. OPTIMALITYCONDITIONS 135

By the convexity of f , we have the relationships

f .yk/ �1p

kxkkf .xk/C .1 �

1pkxkk

/f . Nx/ <�pkxkk

C .1 �1p

kxkk/˛ ! ˛ < ˇ;

and hence yk 2 Lˇ for k sufficiently large, which ensures the boundedness of fykg. It followsfrom the triangle inequality that

kykk �

xkpkxkk

�

�1 �

1pkxkk

�k Nxk �

pkxkk �

�1 �

1pkxkk

�k Nxk ! 1;

which is a contradiction. us (ii) holds.Next we show that (ii)H)(iii). Indeed, the violation of (iii) yields the existence of a constant

M > 0 and a sequence fxkg such that kxkk ! 1 as k ! 1 and f .xk/ < M for all k. But thisobviously contradicts the boundedness of the level set fx 2 Rn

j f .x/ < M g.Finally, we justify (iii)H)(i) leaving the proof of (iii)”(iv) as an exercise. Suppose that

(iii) is satisfied and forM > , find ˛ such that f .x/ � M whenever kxk > ˛. en

LM WD˚x 2 Rn

ˇf .x/ < M g � IB.0I˛/;

which shows that this set is bounded and thus verifies (i). �

4.2 OPTIMALITYCONDITIONSIn this section we first derive necessary and sufficient optimality conditions for optimal solutionsto the following set-constrained convex optimization problem:

minimize f .x/ subject to x 2 ˝; (4.3)

where ˝ is a nonempty, convex set and f W Rn! R is a convex function.

Definition 4.12 We say that Nx 2 ˝ is a to (4.3) if f . Nx/ < 1 and thereis > 0 such that

f .x/ � f . Nx/ for all x 2 IB. NxI / \˝: (4.4)If f .x/ � f . Nx/ for all x 2 ˝, then Nx 2 ˝ is a to (4.3).

e next result shows, in particular, that there is no difference between local and globalsolutions for problems of convex optimization.

Proposition 4.13 e following are equivalent in the convex setting of (4.3):(i) Nx is a local optimal solution to the constrained problem (4.3).(ii) Nx is a local optimal solution to the unconstrained problem

minimize g.x/ WD f .x/C ı.xI˝/ on Rn : (4.5)


(iii) Nx is a global optimal solution to the unconstrained problem (4.5).(iv) Nx is a global optimal solution to the constrained problem (4.3).

Proof. To verify (i)H)(ii), select > 0 such that (4.4) is satisfied. Since g.x/ D f .x/ for x 2

˝ \ IB. NxI / and since g.x/ D 1 for x 2 IB. NxI / and x … ˝, we have

g.x/ � g. Nx/ for all x 2 IB. NxI /;

which means (ii). Implication (ii)H)(iii) is a consequence of Proposition 2.33. e verificationof (iii)H)(iv) is similar to that of (i)H)(ii), and (vi)H)(i) is obvious. �

In what follows we use the term “optimal solutions” for problems of convex optimization,without mentioning global or local. Observe that the objective function g of the unconstrainedproblem (4.5) associated with the constrained one (4.3) is always extended-real-valued when˝ ¤

Rn. is reflects the “infinite penalty” for the constraint violation.e next theorem presents basic necessary and sufficient optimality conditions for optimal

solutions to the constrained problem (4.3) with a general (convex) cost function.

eorem 4.14 Consider the convex problem (4.3) with f W Rn! R such that f . Nx/ < 1 for

some point Nx 2 ˝ under the qualification condition

@1f . Nx/ \�

�N. NxI˝/�

D f0g; (4.6)

which automatically holds if f is continuous at Nx. en the following are equivalent:(i) Nx is an optimal solution to (4.3).(ii) 0 2 @f . Nx/CN. NxI˝/, i.e., @f . Nx/ \ Œ�N. NxI˝/� ¤ ;.(iii) f 0. NxI d/ � 0 for all d 2 T . NxI˝/.

Proof. Observe first that @1f . Nx/ D f0g and thus (4.6) holds if f is continuous at Nx. is followsfrom eorem 2.29. To verify now (i)H)(ii), we use the unconstrained form (4.5) of problem(4.3) due to Proposition 4.13. en Proposition 2.35 provides the optimal solution characteriza-tion 0 2 @g. Nx/. Applying the subdifferential sum rule from eorem 2.44 under the qualificationcondition (4.6) gives us the optimality condition in (ii).

To show that (ii)H)(iii), we have by the tangent-normal duality (3.4) the equivalence

d 2 T . NxI˝/ ”�hv; d i � 0 for all v 2 N. NxI˝/

�:

Fix an element w 2 @f . Nx/ with �w 2 N. NxI˝/. en hw; d i � 0 for all d 2 T . NxI˝/ and wearrive at f 0. NxI d/ � hw; d i � 0 by eorem 2.84, achieving (iii) in this way.

It remains to justify implication (iii)H)(i). It follows from Lemma 2.83 that

f . Nx C d/ � f . Nx/ � f 0. NxI d/ � 0 for all d 2 T . NxI˝/:


By definition of the tangent cone, we have T . NxI˝/ D cl�cone.˝ � Nx/

�, which tells us that d WD

w � Nx 2 T . NxI˝/ if w 2 ˝. is yields

f .w/ � f . Nx/ D f .w � Nx C Nx/ � f . Nx/ D f . Nx C d/ � f . Nx/ � 0;

which verifies (i) and completes the proof of the theorem. �

Corollary 4.15 Assume that in the framework of eorem 4.14 the cost function f W Rn! R is

differentiable at Nx 2 domf . en the following are equivalent:(i) Nx is an optimal solution to problem (4.3).(ii) �rf . Nx/ 2 N. NxI˝/, which reduces to rf . Nx/ D 0 if Nx 2 int˝.

Proof. Since the differentiability of the convex function f at Nx implies its continuity and so Lip-schitz continuity around this point by eorem 2.29. Since @f . Nx/ D frf . Nx/g in this case byeorem 3.3, we deduce this corollary directly from eorem 4.14. �

Next we consider the following convex optimization problem containing functional in-equality constraints given by finite convex functions, along with the set/geometric constraint as in(4.3). For simplicity, suppose that f is also finite on Rn. Problem .P / is:

minimize f .x/

subject to gi .x/ � 0 for i D 1; : : : ; m;

x 2 ˝:

Define the Lagrange function for .P / involving only the cost and functional constraints by

L.x; �/ WD f .x/C

mXiD1

�igi .x/

with �i 2 R and the active index set given by I.x/ WD fi 2 f1; : : : ; mg j gi .x/ D 0g.

eorem 4.16 Let Nx be an optimal solution to .P /. en there are multipliers �0 � 0 and� D .�1; : : : ; �m/ 2 Rm, not equal to zero simultaneously, such that �i � 0 as i D 0; : : : ; m,

0 2 �0@f . Nx/C

mXiD1

�i@gi . Nx/CN. NxI˝/;

and �igi . Nx/ D 0 for all i D 1; : : : ; m.


Proof. Consider the finite convex function

'.x/ WD max˚f .x/ � f . Nx/; gi .x/

ˇi D 1; : : : ; m

and observe that Nx solves the following problem with no functional constraints:

minimize '.x/ subject to x 2 ˝:

Since ' is finite and hence locally Lipschitz continuous, we get from eorem 4.14 that 0 2

@'. Nx/CN. NxI˝/. e subdifferential maximum rule from Proposition 2.54 yields

0 2 co�@f . Nx/ [ Œ

[i2I. Nx/

@gi . Nx/��

CN. NxI˝/:

is gives us �0 � 0 and �i � 0 for i 2 I. Nx/ such that �0 CP

i2I. Nx/ �i D 1 and

0 2 �0@f . Nx/CX

i2I. Nx/

�i@gi . Nx/CN. NxI˝/: (4.7)

Letting �i WD 0 for i … I. Nx/, we get �igi . Nx/ D 0 for all i D 1; : : : ; m with .�0; �/ ¤ 0. �

Many applications including numerical algorithms to solve problem .P / require effectiveconstraint qualification conditions that ensure �0 ¤ 0 in eorem 4.16. e following one is clas-sical in convex optimization.

Definition 4.17 S’ .CQ/ holds in .P / if there exists u 2 ˝

such that gi .u/ < 0 for all i D 1; : : : ; m.

e next theorem provides necessary and sufficient conditions for convex optimization inthe qualified, normal, or Karush-Kuhn-Tucker .KKT/ form.

eorem 4.18 Suppose that Slater’s CQ holds in .P /. en Nx is an optimal solution to .P / ifand only if there exist nonnegative Lagrange multipliers .�1; : : : ; �m/ 2 Rm such that

0 2 @f . Nx/C

mXiD1

�i@gi . Nx/CN. NxI˝/ (4.8)


Proof. To verify the necessity part, we only need to show that �0 ¤ 0 in (4.7). Suppose on thecontrary that �0 D 0 and find .�1; : : : ; �m/ ¤ 0, vi 2 @gi . Nx/, and v 2 N. NxI˝/ satisfying

0 D

mXiD1

�ivi C v:


is readily implies that

0 D

mXiD1

�i hvi ; x � Nxi C hv; x � Nxi �

mXiD1

�i

�gi .x/ � gi . Nx/

�D

mXiD1

�igi .x/ for all x 2 ˝;

which contradicts since the Slater condition impliesPm

iD1 �igi .u/ < 0.To check the sufficiency of (4.8), take v0 2 @f . Nx/, vi 2 @gi . Nx/, and v 2 N. NxI˝/ such that

0 D v0 CPm

iD1 �ivi C v with �igi . Nx/ D 0 and �i � 0 for all i D 1; : : : ; m. en the optimalityof Nx in .P / follows directly from the definitions of the subdifferential and normal cone. Indeed,for any x 2 ˝ with gi .x/ � 0 for i D 1; : : : ; m one has

0 D h�ivi C v; x � Nxi D

mXiD1

�i hvi ; x � Nxi C hv; x � Nxi

�

mXiD1

�i Œgi .x/ � gi . Nx/�C f .x/ � f . Nx/

D

mXiD1

�igi .x/C f .x/ � f . Nx/ � f .x/ � f . Nx/;

which completes the proof. �

Finally in this section, we consider the convex optimization problem .Q/ with inequalityand linear equality constraints

minimize f .x/

subject to gi .x/ � 0 for i D 1; : : : ; m;

Ax D b;

where A is an p � n matrix and b 2 Rp.

Corollary 4.19 Suppose that Slater’s CQ holds for problem .Q/ with the set˝ WD fx 2 Rnj Ax �

b D 0g in Definition 4.17. en Nx is an optimal solution to .Q/ if and only if there exist nonnegativemultipliers �1; : : : ; �m such that

0 2 @f . Nx/C

mXiD1

�i@gi . Nx/C imA�


Proof. e result follows from eorem 4.16 and Proposition 2.12, where the normal cone to theset ˝ is calculated via the image of the adjoint operator A�. �


e concluding result presents the classical version of the Lagrange multiplier rule for con-vex problems with differentiable data.

Corollary 4.20 Let f and gi for i D 1; : : : ; m be differentiable at Nx and let frgi . Nx/ j i 2 I. Nx/g belinearly independent. en Nx is an optimal solution to problem .P / with ˝ D Rn if and only if thereare nonnegative multipliers �1; : : : ; �m such that

0 2 rf . Nx/C

mXiD1

�irgi . Nx/ D Lx. Nx; �/


Proof. In this case N. NxI˝/ D f0g and it follows from (4.7) that �0 D 0 contradicts the linearindependence of the gradient elements frgi . Nx/ j i 2 I. Nx/g. �

4.3 SUBGRADIENTMETHODS INCONVEXOPTIMIZATION

is section is devoted to algorithmic aspects of convex optimization and presents two versions ofthe classical subgradient method: one for unconstrained and the other for constrained optimiza-tion problems. First we study the unconstrained problem:

minimize f .x/ subject to x 2 Rn; (4.9)

where f W Rn! R is a convex function. Let f˛kg as k 2 N be a sequence of positive numbers. e

subgradient algorithm generated by f˛kg is defined as follows. Given a starting point x1 2 Rn,consider the iterative procedure:

xkC1 WD xk � ˛kvk with some vk 2 @f .xk/; k 2 N : (4.10)

In this section we assume that problem (4.9) admits an optimal solution and that f is convex andLipschitz continuous on Rn. Our main goal is to find conditions that ensure the convergence ofthe algorithm. We split the proof of the main result into several propositions, which are of theirown interest. e first one gives us an important subgradient property of convex Lipschitzianfunctions.

Proposition 4.21 Let ` � 0 be a Lipschitz constant of f on Rn. For any x 2 Rn, we have thesubgradient estimate

kvk � ` whenever v 2 @f .x/:

4.3. SUBGRADIENTMETHODS INCONVEXOPTIMIZATION 141

Proof. Fix x 2 Rn and take any subgradient v 2 @f .x/. It follows from the definition that

hv; u � xi � f .u/ � f .x/ for all u 2 Rn :

Combining this with the Lipschitz property of f gives us

hv; u � xi � ` ku � xk ; u 2 Rn;

which yields in turn that

hv; y C x � xi � ` ky C x � xk ; i.e. hv; yi � ` kyk ; y 2 Rn :

Using y WD v, we arrive at the conclusion kvk � `. �

Proposition 4.22 Let ` � 0 be a Lipschitz constant of f on Rn. For the sequence fxkg generated byalgorithm (4.10), we have

kxkC1 � xk2

� kxk � xk2

� 2˛k

�f .xk/ � f .x/

�C ˛2

k`2 for all x 2 Rn; k 2 N : (4.11)

Proof. Fix x 2 Rn, k 2 N and get from (4.10) that

kxkC1 � xk2

D kxk � ˛kvk � xk2

D kxk � x � ˛kvkk2

D kxk � xk2

� 2˛khvk; xk � xi C ˛2k kvkk

2 :

Since vk 2 @f .xk/ and kvkk � ` by Proposition 4.21, it follows that

hvk; x � xki � f .x/ � f .xk/:

is gives us (4.11) and completes the proof of the proposition. �

e next proposition provides an estimate of closeness of iterations to the optimal value of(4.9) for an arbitrary sequence f˛kg in the algorithm (4.10). Denote

V WD min˚f .x/

ˇx 2 Rn

and Vk WD min

˚f .x1/; : : : ; f .xk/

:

Proposition 4.23 Let ` � 0 be a Lipschitz constant of f on Rn and let S be the set of optimalsolutions to (4.9). For all k 2 N, we have

0 � Vk � V �

d.x1IS/2 C `2kP

iD1

˛2i

2kP

iD1

˛i

: (4.12)


Proof. Substituting any Nx 2 S into estimate (4.11) gives us

kxkC1 � Nxk2

� kxk � Nxk2

� 2˛k

�f .xk/ � f . Nx/

�C ˛2

k`2:

Since Vk � f .xi / for all i D 1; : : : ; k and V D f . Nx/, we get

kxkC1 � Nxk2

� kx1 � Nxk2

� 2

kXiD1

˛i

�f .xi / � f . Nx/

�C `2

kXiD1

˛2i

� kx1 � Nxk2

� 2

kXiD1

˛i .Vk � V /C `2

kXiD1

˛2i :

It follows from kxkC1 � Nxk2

� 0 that

2

kXiD1

˛i

�Vk � V

�� kx1 � Nxk

2C `2

kXiD1

˛2i :

Since Vk � V for every k, we deduce the estimate

0 � Vk � V �kx1 � Nxk

2C `2

PkiD1 ˛

2i

2Pk

iD1 ˛i

and hence arrive at (4.12) since Nx 2 S was chosen arbitrarily. �

e corollary below shows the number of steps in order to achieve an appropriate errorestimate for the optimal value in the case of constant step sizes.

Corollary 4.24 Suppose that ˛k D � for all k 2 N in the setting of Proposition 4.23. en thereexists a positive constant C such that

0 � Vk � V < `2� whenever k > C=�2:

Proof. From (4.12) we readily get

0 � Vk � V �aC k`2�2

2k�D

a

2k�C`2�

2with a WD d.x1IS/2:

Since a=.2k�/C `2�=2 < `2� as k > a=.`2�2/, the conclusion holds with C WD a=`2. �

As a direct consequence of Proposition 4.23, we get our first convergence result.

Proposition 4.25 In the setting of Proposition 4.23 suppose in addition that1X

kD1

˛k D 1 and limk!1

˛k D 0: (4.13)

en we have the convergence Vk ! V as k ! 1.


Proof. Given any � > 0, choose K 2 N such that ˛i` < � for all i � K. For any k > K, using(4.12) ensures the estimates

0 � Vk � V �

d.x1IS/2 C `2KP

iD1

˛2i

2kP

iD1

˛i

C

`2kP

iDK

˛2i

2kP

iD1

˛i

�

d.x1IS/2 C `2KP

iD1

˛2i

2kP

iD1

˛i

C

�kP

iDK

˛i

2kP

iD1

˛i

�

d.x1IS/2 C `2KP

iD1

˛2i

2kP

iD1

˛i

C �=2:

e assumptionP1

kD1 ˛k D 1 yields limk!1

PkiD1 ˛i D 1, and hence

0 � lim infk!1

.Vk � V / � lim supk!1

.Vk � V / � �=2:

Since � > 0 is chosen arbitrarily, one has limk!1.Vk � V / D 0 or Vk ! V as k ! 1. �

e next result gives us important information about the value convergence of the subgra-dient algorithm (4.10).

Proposition 4.26 In the setting of Proposition 4.25 we have

lim infk!1

f .xk/ D V :

Proof. Since V is the optimal value in (4.9), we get f .xk/ � V for any k 2 N. is yields

lim infk!1

f .xk/ � V :

To verify the opposite inequality, suppose on the contrary that lim infk!1 f .xk/ > V and thenfind a small number � > 0 such that

lim infk!1

f .xk/ � 2� > V D minx2Rn

f .x/ D f . Nx/;

where Nx is an optimal solution to problem (4.9). us

lim infk!1

f .xk/ � � > f . Nx/C �:


In the case where lim infk!1 f .xk/ < 1, by the definition of lim inf, there is k0 2 N such thatfor every k � k0 we have that

f .xk/ > lim infk!1

f .xk/ � �;

and hence f .xk/ � f . Nx/ > � for every k � k0. Note that this conclusion also holds iflim infk!1 f .xk/ D 1. It follows from (4.11) with x D Nx that

kxkC1 � Nxk2

� kxk � Nxk2

� 2˛k

�f .xk/ � f . Nx/

�C ˛2

k`2 for every k � k0: (4.14)

Let k1 be such that ˛k`2 < � for all k � k1 and let Nk1 WD maxfk1; k0g. For any k > Nk1, it easily

follows from estimate (4.14) that

kxkC1 � Nxk2

� kxk � Nxk2

� ˛k� � : : : �

x Nk1� Nx

2

� �

kXj D Nk1

j ;

which contradicts the assumptionP1

kD1 ˛k D 1 in (4.13) and so completes the proof. �

Now we are ready to justify the convergence of the subgradient algorithm (4.10).

eorem 4.27 Suppose that the generating sequence f˛kg in (4.10) satisfies1X

kD1

˛k D 1 and1X

kD1

˛2k < 1: (4.15)

en the sequence fVkg converges to the optimal value V and the sequence of iterations fxkg in(4.10) converges to some optimal solution Nx of (4.9).

Proof. Since (4.15) implies (4.13), the sequence fVkg converges to V by Proposition 4.25. Itremains to prove that fxkg converges to some optimal solution Nx of problem (4.9). Let ` � 0 bea Lipschitz constant of f on Rn. Since the set S of optimal solutions to (4.9) is assumed to benonempty, we pick anyex 2 S and substitute x D ex in estimate (4.11) of Proposition 4.22. It givesus

kxkC1 � exk2

� kxk � exk2

� 2˛k

�f .xk/ � V

�C ˛2

k`2 for all k 2 N : (4.16)

Since f .xk/ � V � 0, inequality (4.16) yields

kxkC1 � exk2

� kxk � exk2

C ˛2k`

2:

en we get by induction that

kxkC1 � exk2

� kx1 � exk2

C `2

kXiD1

˛2i : (4.17)


e right-hand side of (4.17) is finite by our assumption thatP1

kD1 ˛2k< 1, and hence

supk2N

kxkC1 � exk < 1:

is implies that the sequence fxkg is bounded. Furthermore, Proposition 4.26 tells us that

lim infk!1

f .xk/ D V :

Let fxkjg be a subsequence of fxkg such that

limj !1

f .xkj/ D V : (4.18)

en fxkjg is also bounded and contains a limiting point Nx. Without loss of generality assume

that xkj! Nx as j ! 1. We directly deduce from (4.18) and the continuity of f that Nx is an

optimal solution to (4.9). Fix any j 2 N and any k � kj . us we can rewrite (4.16) with ex D Nx

and see that

kxkC1 � Nxk2

� kxk � Nxk2

C ˛2k`

2� : : : �

xkj� Nx

2C `2

kXiDkj

˛2i :

Taking first the limit as k ! 1 and then the limit as j ! 1 in the resulting inequality abovegives us the estimate

lim supk!1

kxkC1 � Nxk2

� limj !1

xkj� Nx

2C `2 lim

j !1

1XiDkj

˛2i :

Since xkj! Nx and

P1

kD1 ˛2k< 1, this implies that

lim supk!1

kxkC1 � Nxk2

D 0;

and thus justifies the claimed convergence of xk ! Nx as k ! 1. �

Finally in this section, we present a version of the subgradient algorithm for convex prob-lems of constrained optimization given by

minimize f .x/ subject to x 2 ˝; (4.19)

where f W Rn! R is a convex function that is Lipschitz continuous on Rn with some constant

` � 0, while—in contrast to (4.9)—we have now the constraint on x given by a nonempty, closed,convex set˝ � Rn.e following counterpart of the subgradient algorithm (4.10) for constrainedproblems is known as the projected subgradient method.


Given a sequence of positive numbers f˛kg and given a starting point x1 2 ˝, the sequenceof iterations fxkg is constructed by

xkC1 WD ˘.xk � ˛kvkI˝/ with some vk 2 @f .xk/; k 2 N; (4.20)

where ˘.xI˝/ stands for the (unique) Euclidean projection of x onto ˝.e next proposition, which is a counterpart of Proposition 4.22 for algorithm (4.20), plays

a major role in justifying convergence results for the projected subgradient method.

Proposition 4.28 In the general setting of (4.20) we have the estimate

kxkC1 � xk2

� kxk � xk2

� 2˛k

�f .xk/ � f .x/

�C ˛2

k`2 for all x 2 ˝; k 2 N :

Proof. Define zkC1 WD xk � ˛kvk and fix any x 2 ˝. It follows from Proposition 4.22 that

kzkC1 � xk2

� kxk � xk2

� 2˛k

�f .xk/ � f .x/

�C ˛2

k`2 for all k 2 N : (4.21)

Now by projecting zkC1 onto ˝ and using Proposition 1.79, we get

kxkC1 � xk D k˘.zkC1I˝/ �˘.xI˝/k � kzkC1 � xk :

Combining this with estimate (4.21) gives us

kxkC1 � xk2

� kxk � xk2

� 2˛k

�f .xk/ � f .x/

�C ˛2

k`2 for all k 2 N;

which thus completes the proof of the proposition. �

Based on this proposition and proceeding in the same way as for the subgradient algorithm(4.10) above, we can derive similar convergence results for the projected subgradient algorithm(4.20) under the assumptions (4.13) on the generating sequence f˛kg.

4.4 THEFERMAT-TORRICELLI PROBLEMIn 1643 Pierre de Fermat proposed to Evangelista Torricelli the following optimization problem:Given three points in the plane, find another point such that the sum of its distances to the givenpoints is minimal. is problem was solved by Torricelli and was named the Fermat-Torricelliproblem. A more general version of the Fermat-Torricelli problem asks for a point that minimizesthe sum of the distances to a finite number of given points in Rn. is is one of the main problemsin location science, which has been studied by many researchers. e first numerical algorithm forsolving the general Fermat-Torricelli problem was introduced by Endre Weiszfeld in 1937. As-sumptions that guarantee the convergence of the Weiszfeld algorithm were given by Harold Kuhnin 1972. Kuhn also pointed out an example in which the Weiszfeld algorithm fails to converge.Several new algorithms have been introduced recently to improve the Weiszfeld algorithm. e

4.4. THE FERMAT-TORRICELLI PROBLEM 147

goal of this section is to revisit the Fermat-Torricelli problem from both theoretical and numericalviewpoints by using techniques of convex analysis and optimization.

Given points ai 2 Rn as i D 1; : : : ; m, define the (nonsmooth) convex function

'.x/ WD

mXiD1

kx � aik: (4.22)

en the mathematical description of the Fermat-Torricelli problem is

minimize '.x/ over all x 2 Rn: (4.23)

Proposition 4.29 e Fermat-Torricelli problem (4.23) has at least one solution.

Proof. e conclusion follows directly from eorem 4.10. �

Proposition 4.30 Suppose that ai as i D 1; : : : ; m are not collinear, i.e., they do not belong to thesame line. en the function ' in (4.22) is strictly convex, and hence the Fermat-Torricelli problem(4.23) has a unique solution.

Proof. Since each function 'i .x/ WD kx � aik as i D 1; : : : ; m is obviously convex, their sum' D

PmiD1 'i is convex as well, i.e., for any x; y 2 Rn and � 2 .0; 1/ we have

'��x C .1 � �/y

�� '.x/C .1 � �/'.y/: (4.24)

Supposing by contradiction that ' is not strictly convex, find Nx; Ny 2 Rn with Nx ¤ Ny and � 2 .0; 1/

such that (4.24) holds as equality. It follows that

'i

�� Nx C .1 � �/ Ny

�D �'i . Nx/C .1 � �/'i . Ny/ for all i D 1; : : : ; m;

which ensures therefore that

k�. Nx � ai /C .1 � �/. Ny � ai /k D k�. Nx � ai /k C k.1 � �/. Ny � ai /k; i D 1; : : : ; m:

If Nx ¤ ai and Ny ¤ ai , then there exists ti > 0 such that ti�. Nx � ai / D .1 � �/. Ny � ai /, and hence

Nx � ai D i . Ny � ai / with i WD1 � �

ti�:

Since Nx ¤ Ny, we have i ¤ 1. us

ai D1

1 � i

Nx � i

1 � i

Ny 2 L. Nx; Ny/;

where L. Nx; Ny/ signifies the line connecting Nx and Ny. Both cases where Nx D ai and Ny D ai give usai 2 L. Nx; Ny/. Hence ai 2 L. Nx; Ny/ for i D 1; : : : ; m, a contradiction. �


We proceed further with solving the Fermat-Torricelli problem for three points in Rn. Letus first present two elementary lemmas.

Lemma 4.31 Let a1 and a2 be any two unit vectors in Rn. en we have �a1 � a2 2 IB if andonly if ha1; a2i � �

1

2.

Proof. If �a1 � a2 2 IB , then a1 C a2 2 IB and

ka1 C a2k � 1:

By squaring both sides, we get the inequality

ka1 C a2k2

� 1;

which subsequently implies that

ka1k2

C 2ha1; a2i C ka2k2

D 2C 2ha1; a2i � 1:

erefore, ha1; a2i � �1

2. e proof of the converse implication is similar. �

Lemma4.32 Let a1; a2; a3 2 Rn with ka1k D ka2k D ka3k D 1. en a1 C a2 C a3 D 0 if andonly if we have the equalities

ha1; a2i D ha2; a3i D ha3; a1i D �1

2: (4.25)

Proof. It follows from the condition a1 C a2 C a3 D 0 that

ka1k2

C ha1; a2i C ha1; a3i D 0; (4.26)

ha2; a1i C ka2k2

C ha2; a3i D 0; (4.27)

ha3; a1i C ha3; a2i C ka3k2

D 0: (4.28)

Subtracting (4.28) from (4.26) gives us ha1; a2i D ha2; a3i and similarly ha2; a3i D ha3; a1i. Sub-stituting these equalities into (4.26), (4.27), and (4.28) ensures the validity of (4.25).

Conversely, assume that (4.25) holds. en we have

ka1 C a2 C a3k2

D

3XiD1

kaik2

C

3Xi;j D1;i¤j

hai ; aj i D 0;

which readily yields the equality a1 C a2 C a3 D 0 and thus completes the proof. �


Now we use subdifferentiation of convex functions to solve the Fermat-Torricelli problemfor three points in Rn. Given two nonzero vectors u; v 2 Rn, define

cos.u; v/ WDhu; vi

kuk � kvk

and also consider, when Nx ¤ ai , the three vectors

vi DNx � ai

k Nx � aik; i D 1; 2; 3:

Geometrically, each vector vi is a unit one pointing in the direction from the vertex ai to Nx.Observe that the classical (three-point) Fermat-Torricelli problem always has a unique solutioneven if the given points belong to the same line. Indeed, in the latter case the middle point solvesthe problem.

e next theorem fully describes optimal solutions of the Fermat-Torricelli problem forthree points in two distinct cases.

eorem 4.33 Consider the Fermat-Torricelli problem (4.23) for the three points a1; a2; a3 2

Rn. e following assertions hold:(i) Let Nx … fa1; a2; a3g. en Nx is the solution of the problem if and only if

cos.v1; v2/ D cos.v2; v3/ D cos.v3; v1/ D �1=2:

(ii) Let Nx 2 fa1; a2; a3g, say Nx D a1. en Nx is the solution of the problem if and only if

cos.v2; v3/ � �1=2:

Proof. In case (i) function (4.22) is differentiable at Nx. en Nx solves (4.23) if and only if

r'. Nx/ D v1 C v2 C v3 D 0:

By Lemma 4.32 this holds if and only if

hvi ; vj i D cos.vi ; vj / D �1=2 for any i; j 2˚1; 2; 3

with i ¤ j:

To verify (ii), employ the subdifferential Fermat rule (2.35) and sum rule from Corollary 2.45,which tell us that Nx D a1 solves the Fermat-Torricelli problem if and only if

0 2 @'.a1/ D v2 C v3 C IB:

Since v2 and v3 are unit vectors, it follows from Lemma 4.31 that

hv2; v3i D cos.v2; v3/ � �1=2;



Figure 4.2: Construction of the Torricelli point.

Example 4.34 is example illustrates geometrically the solution construction for the Fermat-Torricelli problem in the plane. Consider (4.23) with the three given points A, B , and C onR2 as shown in Figure 4.2. If one of the angles of the triangle ABC is greater than or equal to120ı, then the corresponding vertex is the solution to (4.23) by eorem 4.33(ii). Consider thecase where none of the angles of the triangle is greater than or equal to 120ı. Construct the twoequilateral triangles ABD and ACE and let S be the intersection ofDC and BE as in the figure.Two quadrilateralsADBC andABCE are convex, and hence S lies inside the triangleABC . It isclear that two trianglesDAC andBAE are congruent. A rotation of 60ı aboutAmaps the triangleDAC to the triangleBAE.e rotationmapsCD toBE, so †DSB D 60ı. Let T be the image ofS though this rotation.en T belongs toBE. It follows that †AST D †ASE D 60ı.Moreover,†DSA D 60ı, and hence †BSA D 120ı. It is now clear that †ASC D 120ı and †BSC D 120ı.By eorem 4.33(i) the point S is the solution of the problem under consideration.

Next we present the Weiszfeld algorithm [32] to solve numerically the Fermat-Torricelliproblem in the general case of (4.23) form points in Rn that are not collinear. is is our standingassumption in the rest of this section. To proceed, observe that the gradient of the function ' from(4.22) is calculated by

r'.x/ D

mXiD1

x � ai

kx � aikfor x …

˚a1; : : : ; am

:

Solving the gradient equation r'.x/ D 0 gives us

x D

PmiD1

ai

kx � aikPmiD1

1

kx � aik

DW f .x/: (4.29)


We define f .x/ on the whole space Rn by letting f .x/ WD x for x 2 fa1; : : : ; amg.Weiszfeld algorithm consists of choosing an arbitrary starting point x1 2 Rn, taking f is

from (4.29), and defining recurrently

xkC1 WD f .xk/ for k 2 N; (4.30)

where f .x/ is called the algorithm mapping. In the rest of this section we present the preciseconditions and the proof of convergence for Weiszfeld algorithm in the way suggested by Kuhn[10] who corrected some errors from [32]. In contrast to [10], we employ below subdifferentialrules of convex analysis, which allow us to simplify the proof of convergence.

e next proposition ensures decreasing the function value of ' after each iteration, i.e.,the descent property of the algorithm.

Proposition 4.35 If f .x/ ¤ x, then '.f .x// < '.x/.

Proof. e assumption of f .x/ ¤ x tells us that x is not a vertex from fa1; : : : ; amg. Observealso that the point f .x/ is a unique minimizer of the strictly convex function

g.z/ WD

mXiD1

kz � aik2

kx � aik

and that g.f .x// < g.x/ D '.x/ since f .x/ ¤ x. We have furthermore that

g�f .x/

�D

mXiD1

kf .x/ � aik2

kx � aik

D

mXiD1

.kx � aik C kf .x/ � aik � kx � aik/2

kx � aik

D '.x/C 2�'.f .x// � '.x/

�C

mXiD1

.kf .x/ � aik � kx � aik/2

kx � aik:

is clearly implies the inequality

2'�f .x/

�C

mXiD1

.kf .x/ � aik � kx � aik/2

kx � aik< 2'.x/


e following two propositions show how the algorithm mapping f behaves near a vertexthat solves the Fermat-Torricelli problem. First we present a necessary and sufficient conditionensuring the optimality of a vertex in (4.23). Define

Rj WD

mXiD1;i¤j

ai � aj

kai � aj k; j D 1; : : : ; m:


Proposition 4.36 e vertex aj solves problem (4.23) if and only if kRj k � 1.

Proof. Employing the Fermat rule from Proposition 2.35 and the subdifferential sum rule fromCorollary 2.45 shows that the vertex aj solves problem (4.23) if and only if

0 2 @'.aj / D �Rj C IB.0; 1/;

which is clearly equivalent to the condition kRj k � 1. �

Proposition 4.36 from convex analysis allows us to essentially simplify the proof of thefollowing result established by Kuhn. We use the notation:

f s.x/ WD f�f s�1.x/

�for s 2 N with f 0.x/ WD x; x 2 Rn :

Proposition 4.37 Suppose that the vertex aj does not solve (4.23). en there is ı > 0 such that thecondition 0 < kx � aj k � ı implies the existence of s 2 N with

kf s.x/ � aj k > ı and kf s�1.x/ � aj k � ı: (4.31)

Proof. For any x … fa1; : : : ; amg, we have from (4.29) that

f .x/ � aj D

PmiD1;i¤j

ai � aj

kx � aikPmiD1

1

kx � aik

and so

limx!aj

f .x/ � aj

kx � aj kD lim

x!aj

PmiD1;i¤j

ai � aj

kx � aik

1CPm

iD1;i¤j

kx � aj k

kx � aik

D Rj ; j D 1; : : : ; m:

It follows from Proposition 4.36 that

limx!aj

kf .x/ � aj k

kx � aj kD kRj k > 1; j D 1; : : : ; m; (4.32)

which allows us to select � > 0; ı > 0 such that

kf .x/ � aj k

kx � aj k> 1C � whenever 0 < kx � aj k � ı:


us for every s 2 N the validity of 0 < kf q�1.x/ � aj k � ı with any q D 1; : : : ; s yields

kf s.x/ � aj k � .1C �/kf s�1.x/ � aj k � : : : � .1C �/skx � aj k:

Since .1C �/skx � aj k ! 1 as s ! 1, it gives us (4.31) and completes the proof. �

We finally present the main result on the convergence of the Weiszfeld algorithm.

eorem 4.38 Let fxkg be the sequence generated by Weiszfeld algorithm (4.30) and let xk …

fa1; : : : ; amg for k 2 N. en fxkg converges to the unique solution Nx of (4.23).

Proof. If xk D xkC1 for some k D k0, then fxkg is a constant sequence for k � k0, and henceit converges to xk0

. Since f .xk0/ D xk0

and xk0is not a vertex, xk0

solves the problem. uswe can assume that xkC1 ¤ xk for every k. It follows from Proposition 4.35 that the sequencef'.xk/g of nonnegative numbers is decreasing, so it converges. is gives us

limk!1

�'.xk/ � '.xkC1/

�D 0: (4.33)

By the construction of fxkg, we have that each xk belongs to the compact set co fa1; : : : ; amg.is allows us to select from fxkg a subsequence fxk�

g converging to some point Nz as � ! 1.We have to show that Nz D Nx, i.e., it is the unique solution to (4.23). Indeed, we get from (4.33)that

lim�!1

�'.xk�

/ � '�f .xk�

/��

D 0;

which implies by the continuity of the functions ' and f that '. Nz/ D '.f . Nz//, i.e., f . Nz/ D Nz andthus Nz solves (4.23) if it is not a vertex.

It remains to consider the case where Nz is a vertex, say a1. Suppose by contradiction that Nz D

a1 ¤ Nx and select ı > 0 sufficiently small so that (4.31) holds for some s 2 N and that IB.a1I ı/

does not contain Nx and ai for i D 2; : : : ; m. Since xk�! Nz D a1, we can assume without loss of

generality that this sequence is entirely contained in IB.a1I ı/. For x D xk1select l1 so that xl1

2

IB.a1I ı/ and f .xl1/ … IB.a1I ı/. Choosing now k� > l1 and applying Proposition 4.37 give us

l2 > l1 with xl22 IB.a1I ı/ and f .xl2

/ … IB.a1I ı/. Repeating this procedure, find a subsequencefxl�

g such that xl�2 IB.a1I ı/ while f .xl�

/ is not in this ball. Extracting another subsequenceif needed, we can assume that fxl�

g converges to some point Nq satisfying f . Nq/ D Nq. If Nq is not avertex, then it solves (4.23) by the above, which is a contradiction because the solution Nx is notin IB.a1I ı/. us Nq is a vertex, which must be a1 since the other vertexes are also not in the ball.en we have

lim�!1

kf .xl�/ � a1k

kxl�� a1k

D 1;

which contradicts (4.32) in Proposition 4.37 and completes the proof of the theorem. �


4.5 AGENERALIZEDFERMAT-TORRICELLI PROBLEMere are several generalized versions of the Fermat-Torricelli problem known in the literatureunder different names; see, e.g., our papers [15, 16, 17] and the references therein. Note thatthe constrained version of the problem below is also called a “generalized Heron problem” afterHeron from Alexandria who formulated its simplest version in the plane in the 1st century AD.Following [15, 16], we consider here a generalized Fermat-Torricelli problem defined via theminimal time functions (3.9) for finitely many closed, convex target sets in Rn with a constraintset of the same structure. In [6, 11, 29] and their references, the reader can find some particularversions of this problem and related material on location problems. We also refer the reader to[15, 17] to nonconvex problems of this type.

Let the target sets ˝i for i D 1; : : : ; m and the constraint set ˝0 be nonempty, closed, con-vex subsets of Rn. Consider the minimal time functions TF .xI˝i / D T F

˝i.x/ from (3.9) for ˝i ,

i D 1; : : : ; m, with the constant dynamics defined by a closed, bounded, convex set F � Rn thatcontains the origin as an interior point. en the generalized Fermat-Torricelli problem is formu-lated as follows:

minimize H.x/ WD

mXiD1

TF .xI˝i / subject to x 2 ˝0: (4.34)

Our first goal in this section is to establish the existence and uniqueness of solutions to the opti-mization problem (4.34). We split the proof into several propositions of their own interest. Letus start with the following property of the Minkowski gauge from (3.11).

Proposition 4.39 Assume that F is a closed, bounded, convex subset of Rn such that 0 2 intF . en�F .x/ D 1 if and only if x 2 bdF .

Proof. Suppose that �F .x/ D 1. en there exists a sequence of real numbers ftkg that convergesto t D 1 and such that x 2 tkF for every k. Since F is closed, it yields x 2 F . To complete theproof of the implication, we show that x … intF . By contradiction, assume that x 2 intF andchoose t > 0 so small that x C tx 2 F . en x 2 .1C t/�1F . is tells us �F .x/ � .1C t/�1 <

1, a contradiction.Now let x 2 bdF . SinceF is closed, we have x 2 F , and hence �F .x/ � 1. Suppose on the

contrary that WD �F .x/ < 1. en there exists a sequence of real numbers ftkg that convergesto with x 2 tkF for every k. en x 2 F D F C .1 � /0 � F . By Proposition 3.32, theMinkowski gauge �F is continuous, so the set fu 2 Rn

j �F .u/ < 1g is an open subset of Fcontaining x. us x 2 intF , which completes the proof. �

Recall further that a setQ � Rn is said to be strictly convex if for any x; y 2 Q with x ¤ y

and for any t 2 .0; 1/ we have tx C .1 � t/y 2 intQ.

4.5. AGENERALIZEDFERMAT-TORRICELLI PROBLEM 155

Proposition 4.40 Assume that F is a closed, bounded, strictly convex subset of Rn and such that0 2 intF . en

�F .x C y/ D �F .x/C �F .y/ with x; y ¤ 0 (4.35)

if and only if x D �y for some � > 0.

Proof. Denote ˛ WD �F .x/, ˇ WD �F .y/ and observe that ˛; ˇ > 0 since F is bounded. If thelinearity property (4.35) holds, then

�F

�x C y

˛ C ˇ

�D 1;


�F

�x˛

˛

˛ C ˇCy

ˇ

ˇ

˛ C ˇ

�D 1:

is allows us to conclude thatx

˛

˛

˛ C ˇCy

ˇ

ˇ

˛ C ˇ2 bdF:

Since x˛

2 F , yˇ

2 F , and ˛

˛ C ˇ2 .0; 1/, it follows from the strict convexity of F that

x

˛Dy

ˇ; i.e., x D �y with � D

�F .x/

�F .y/:

e opposite implication in the proposition is obvious. �

Proposition 4.41 Let F be as in Proposition 4.40 and let˝ be a nonempty, closed, convex subset ofRn. For any x 2 Rn, the set˘F .xI˝/ is a singleton.

Proof. We only need to consider the case where x … ˝. It is clear that ˘F .xI˝/ ¤ ;. Supposeby contradiction that there are u1; u2 2 ˘F .xI˝/ with u1 ¤ u2. en

TF .xI˝/ D �F .u1 � x/ D �F .u2 � x/ D r > 0;

which implies that u1 � x

r2 bdF and u2 � x

r2 bdF . Since F is strictly convex, we have

1

2

u1 � x

rC1

2

u2 � x

rD1

r

�u1 C u2

2� x

�2 intF;

and so we get �F .u � x/ < r D TF .xI˝/with u WDu1 C u2

22 ˝. is contradiction completes

the proof of the proposition. �


In what follows we identify the generalized projection ˘F .xI˝/ with its unique elementwhen both sets F and ˝ are strictly convex.

Proposition 4.42 Let F be a closed, bounded, convex subset of Rn such that 0 2 intF and let ˝ bea nonempty, closed set˝ � Rn with Nx … ˝. If N! 2 ˘F . NxI˝/, then N! 2 bd˝.

Proof. Suppose that N! 2 int˝. en there is � > 0 such that IB. N!I �/ � ˝. Define

z WD N! C �Nx � N!

k Nx � N!k2 IB. N!I �/:

and observe from the construction of the Minkowski gauge (3.11) that

�F .z � Nx/ D �F

�N! C �

Nx � N!

k Nx � N!k� Nx

�D

�1 �

�

k Nx � N!k

��F . N! � Nx/ < �F . N! � Nx/ D TF . NxI˝/:

is shows that N! must be a boundary point of ˝. �

Figure 4.3: ree sets that satisfy assumptions of Proposition 4.43.

Proposition 4.43 In the setting of problem (4.34) suppose that the sets ˝i for i D 1; : : : ; m arestrictly convex. If for any x; y 2 ˝0 with x ¤ y there is i0 2 f1; : : : ; mg such that

L.x; y/ \˝i0 D ;

for the line L.x; y/ connecting x and y, then the function

H.x/ D

mXiD1

TF .xI˝i /

defined in (4.34) is strictly convex on the constraint set˝0.


Proof. Suppose by contradiction that there are x; y 2 ˝0, x ¤ y, and t 2 .0; 1/ with

H�tx C .1 � t /y

�D tH.x/C .1 � t/H.y/:

Using the convexity of each function TF .xI˝i / for i D 1; : : : ; m, we have

TF

�tx C .1 � t /yI˝i

�D tTF .xI˝i /C .1 � t/TF .yI˝i /; i D 1; : : : ; m: (4.36)

Suppose further that L.x; y/ \˝i0 D ; for some i0 2 f1; : : : ; mg and define

u WD ˘F .xI˝i0/ and v WD ˘F .yI˝i0/:

en it follows from (4.36), eorem 3.33, and the convexity of the Minkowski gauge that

t�F .u � x/C .1 � t/�F .v � y/ D tTF .xI˝i0/C .1 � t/TF .yI˝i0/

D TF

�tx C .1 � t/yI˝i0

�� F

�tuC .1 � t /v � .tx C .1 � t/y/

�� t�F .u � x/C .1 � t /�F .v � y/;

which shows that tuC .1 � t /v D ˘F .tx C .1 � t/yI˝i0/. us u D v, since otherwise we havetuC .1 � t/v 2 int˝i0 , a contradiction. is gives us u D v D ˘F .tx C .1 � t /yI˝i0/. Apply-ing (4.36) again tells us that

�F

�u � .tx C .1 � t/y

�D �F

�t.u � x/

�C �F

�.1 � t /.u � y/

�;

so u � x; u � y ¤ 0 due to x; y … ˝i0 . By Proposition 4.40, we find � > 0 such that t.u � x/ D

�.1 � t /.u � y/, which yields

u � x D .u � y/ with D�.1 � t /

t¤ 1:

is justifies the inclusion

u D1

1 � x �

1 � y 2 L.x; y/;

which is a contradiction that completes the proof of the proposition. �

Now we are ready to establish the existence and uniqueness theorem for (4.34).

eorem 4.44 In the setting of Proposition 4.43 suppose further that at least one of the setsamong ˝i for i D 0; : : : ; m is bounded. en the generalized Fermat-Torricelli problem (4.34)has a unique solution.


Proof. By the strict convexity of H from Proposition 4.43, it suffices to show that problem (4.34)admits an optimal solution. If ˝0 is bounded, then it is compact and the conclusion followsfrom the (Weierstrass) eorem 4.10 since the function H in (4.34) Lipschitz continuous byCorollary 3.34. Consider now the case where one of the sets˝i for i D 1; : : : ; m is bounded; sayit is ˝1. In this case we get that for any ˛ 2 R the set

L˛ D˚x 2 ˝0

ˇH.x/ � ˛

�

˚x 2 ˝0

ˇTF .xI˝1/ � ˛

� ˝0 \ .˝1 � ˛F /

is compact. is ensures the existence of a minimizer for (4.34) by eorem 4.10. �

Next we completely solve by means of convex analysis and optimization the following sim-plified version of problem (4.34):

minimize H.x/ D

3XiD1

d.xI˝i /; x 2 R2; (4.37)

where the target sets ˝i WD IB.ci I ri / for i D 1; 2; 3 are the closed balls of R2 whose centers ci aresupposed to be distinct from each other to avoid triviality.

We first study properties of this problem that enable us to derive verifiable conditions interms of the initial data .ci ; ri / ensuring the uniqueness of solutions to (4.37).

Proposition 4.45 e solution set S of (4.37) is a nonempty, compact, convex set in R2.

Proof. e existence of solutions (4.37) follows from the proof of eorem 4.44. It is also easy tosee that S is closed and bounded. e convexity of S can be deduced from the convexity of thedistance functions d.xI˝i /. �

To proceed, for any u 2 R2 define the index subset

I.u/ WD˚i 2 f1; 2; 3g

ˇu 2 ˝i

; (4.38)

which is widely used in what follows.

Proposition 4.46 Suppose that Nx does not belong to any of the sets ˝i , i D 1; 2; 3, i.e., I. Nx/ D ;.en Nx is an optimal solution to (4.37) if and only if Nx solves the classical Fermat-Torricelli problem(4.23) generated by the centers of the balls ai WD ci for i D 1; 2; 3 in (4.22).

Proof. Assume that Nx is a solution to (4.37) with Nx … ˝i for i D 1; 2; 3. Choose ı > 0 such thatIB. NxI ı/ \˝i D ; as i D 1; 2; 3 and get following for every x 2 IB. NxI ı/:

3XiD1

k Nx � cik �

3XiD1

ri D infx2R2

H.x/ D H. Nx/ � H.x/ D

3XiD1

kx � cik �

3XiD1

ri :


is shows that Nx is a local minimum of the problem

minimize F.x/ WD

3XiD1

kx � cik; x 2 R2; (4.39)

so it is a global absolute minimum of this problem due to the convexity of F . us Nx solves theclassical Fermat-Torricelli problem (4.23) generated by c1; c2; c3. e converse implication is alsostraightforward. �

e next result gives a condition for the uniqueness of solutions to (4.37) in terms of theindex subset defined in (4.38).

Proposition 4.47 Let Nx be a solution to (4.37) such that I. Nx/ D ; in (4.38). en Nx is the uniquesolution to (4.37), i.e., S D f Nxg.

Proof. Suppose by contradiction that there is u 2 S such that u ¤ Nx. Since I. Nx/ D ;, we haveNx … ˝i for i D 1; 2; 3. en Proposition 4.46 tells us that Nx solves the classical Fermat-Torricelliproblem generated by c1; c2; c3. It follows from the convexity of S in Proposition 4.45 thatŒ Nx; u� � S . us we find v ¤ Nx such that v 2 S and v … ˝i for i D 1; 2; 3. It shows that v alsosolves the classical Fermat-Torricelli problem (4.39). is contradicts the uniqueness result for(4.39) and hence justifies the claim of the proposition. �

e following proposition verifies a visible geometric property of ellipsoids.

Proposition 4.48 Given ˛ > 0 and a; b 2 R2 with a ¤ b, consider the set

E WD˚x 2 R2

ˇkx � ak C kx � bk D ˛

and suppose that ka � bk < ˛. For x; y 2 E with x ¤ y, we have x C y

2� a

C

x C y

2� b

< ˛;which ensures, in particular, that x C y

2… E.

Proof. It follows from the triangle inequality that x C y

2� a

C

x C y

2� b

D1

2

�kx � aC y � ak C kx � b C y � bk

��1

2

�kx � ak C ky � ak C kx � bk C ky � bk

�D ˛:


Suppose by contradiction that x C y

2� a

C

x C y

2� b

D ˛; (4.40)

which means that x C y

22 E. Since our norm is (always) Euclidean, it gives

x � a D ˇ.y � a/ and x � b D .y � b/

for some numbers ˇ; 2 .0;1/ n f1g since x ¤ y. is implies in turn that

a D1

1 � ˇx �

ˇ

1 � ˇy 2 L.x; y/ and b D

1

1 � x �

1 � y 2 L.x; y/:

Since ˛ > ka � bk, we see that x; y 2 L.a; b/ n Œa; b�. To check now that x C y

22 Œa; b�, impose

without loss of generality the following order of points: x, x C y

2, a; b, and y. en x C y

2� a

C

x C y

2� b

< kx � ak C kx � bk D ˛;

which contradicts (4.40). Since x C y

22 Œa; b�, it yields x C y

2� a

C

x C y

2� b

D ka � bk < ˛;

which is a contradiction. us x C y

2… E and the proof is complete. �

e next result ensures the uniqueness of solutions to (4.37) via the index set (4.38) differ-ently from Proposition 4.47.

Proposition 4.49 Let S be the solution set for problem (4.37) and let Nx 2 S satisfy the conditions:I. Nx/ D fig and Nx … Œcj ; ck�, where i; j; k are distinct indices in f1; 2; 3g.enS is a singleton, namelyS D f Nxg.

Proof. Suppose for definiteness that I. Nx/ D f1g. en we have Nx 2 ˝1, Nx … ˝2, Nx … ˝3, andNx … Œc2; c3�. Denote

˛ WD k Nx � c2k C k Nx � c3k D infx2R2

H.x/C r2 C r3

and observe that ˛ > kc2 � c3k. Consider the ellipse

E WD˚x 2 R2

ˇkx � c2k C kx � c3k D ˛


for which we get Nx 2 E \˝1. Arguing by contradiction, suppose that S is not a singleton, i.e.,there is u 2 S with u ¤ Nx. e convexity of S implies that Œ Nx; u� � S . We can choose v 2 Œ Nx; u�

sufficiently close to but different from Nx so that Œ Nx; v� \˝2 D ; and Œ Nx; v� \˝3 D ;. Let us firstshow that v … ˝1. Assuming the contrary gives us d.vI˝1/ D 0, and hence

H.v/ D infx2R2

H.x/ D d.vI˝2/C d.vI˝3/ D kv � c2k � r2 C kv � c3k � r3;

which implies that v 2 E. It follows from the convexity of S and ˝1 that Nx C v

22 S \˝1. We

have furthermore that Nx C v

2… ˝2 and Nx C v

2… ˝3, which tell us that Nx C v

22 E. But this

cannot happen due to Proposition 4.48. erefore, v 2 S and I.v/ D ;, which contradict theresult from Proposition 4.47 and thus justify that S is a singleton. �

Figure 4.4: A Fermat-Torricelli problem with jI. Nx/j D 2.

Now we verify the expected boundary property of optimal solutions to (4.37).

Proposition 4.50 Let Nx 2 S and let I. Nx/ D fi; j g � f1; 2; 3g. en Nx 2 bd .˝i \ j /.

Proof. Assume for definiteness I. Nx/ D f1; 2g, i.e., Nx 2 ˝1 \˝2 and Nx … ˝3. Arguing by con-tradiction, suppose that Nx 2 int .˝1 \˝2/, and therefore there exists ı > 0 such that IB. NxI ı/ �

˝1 \˝2. Denotep WD ˘. NxI˝3/, WD k Nx � pk > 0 and take q 2 Œ Nx; p� \ IB. Nx; ı/ satisfying thecondition kq � pk < k Nx � pk D d. NxI˝3/. en we get

H.q/ D

3XiD1

d.qI˝i / D d.qI˝3/ � kq � pk < k Nx � pk D d. NxI˝3/ D

3XiD1

d. NxI˝i / D H. Nx/;

which contradicts the fact that Nx 2 S and thus completes the proof. �

e following result gives us verifiable necessary and sufficient conditions for the solutionuniqueness in (4.37).


Figure 4.5: A generalized Fermat-Torricelli problem with infinite solutions.

eorem 4.51 Let \3iD1˝i D ;. en problem (4.37) has more than one solutions if and only

if there is a set Œci ; cj � \˝k for distinct indices i; j; k 2 f1; 2; 3g that contains a point u 2 R2

belonging to the interior of ˝k and such that the index set I.u/ is a singleton.

Proof. Let u 2 Œc1; c2� \˝3, u 2 int˝3 satisfy the conditions in the “if ” part of the theo-rem. en u … ˝1, u … ˝2 and we can choose v ¤ u with v 2 Œc1; c2], v … ˝1, v … ˝2, andv 2 int˝3. is shows that the set I.v/ is a singleton. Let us verify that in this case u is a solu-tion to (4.37). To proceed, we observe first that

H.u/ D d.uI˝1/C d.uI˝2/C d.uI˝3/ D kc1 � c2k � r1 � r2:

Choose ı > 0 such that IB.uI ı/ \˝1 D ;, IB.uI ı/ \˝2 D ;, and IB.uI ı/ � ˝3. For everyx 2 IB.uI ı/, we get

H.x/ D d.xI˝2/C d.xI˝3/ D kx � c1k C kx � c2k � r1 � r2 � kc1 � c2k � r1 � r2 D H.u/;

which implies that u is a local (and hence global) minimizer of H. Similar arguments show thatv 2 S , so the solution set S is not a singleton.

To justify the “only if ” part, suppose that S is not a singleton and pick two distinct elementsNx; Ny 2 S , which yields Œ Nx; Ny� � S by Proposition 4.45. If there is a solution to (4.37) not belongingto˝i for every i 2 f1; 2; 3g, then S reduces to a singleton due to Proposition 4.47, a contradiction.us we can assume by the above that ˝1 contains infinitely many solutions. en its interiorint˝1 also contains infinitely many solutions by the obvious strict convexity of the ball ˝1. Ifthere is such a solution u with I.u/ D f1g, then u 2 int˝1 and u … ˝2 as well as u … ˝3. isimplies that u 2 Œc2; c3�, since the opposite ensures the solution uniqueness by Proposition 4.49.Consider the case where the set I.u/ contains two elements for every solution belonging to int˝1.en there are infinitely many solutions belonging to the intersection of these two sets, which isstrictly convex in this case. Hence there is a solution to (4.37) that belongs to the interior of this


intersection, which contradicts Proposition 4.50 and thus completes the proof of the theorem.�

Let us now consider yet another particular case of problem (4.34), where the constant dy-namics F is given by the closed unit ball. It is formulated as

minimize D.x/ WD

mXiD1

d.xI˝i / subject to x 2 ˝0: (4.41)

For x 2 Rn, define the index subsets of f1; : : : ; mg by

I.x/ WD˚i 2 f1; : : : ; mg

ˇx 2 ˝i

and J.x/ WD

˚i 2 f1; : : : ; mg

ˇx … ˝i

: (4.42)

It is obvious that I.x/ [ J.x/ D f1; : : : ; mg and I.x/ \ J.x/ D ; for all x 2 Rn. e next theo-rem fully characterizes optimal solutions to (4.41) via projections to ˝i .

eorem 4.52 e following assertions hold for problem (4.41):(i) Let Nx 2 ˝0 be a solution to (4.41). For ai . Nx/ � Ai . Nx/ as i 2 J. Nx/, we have

�X

i2J. Nx/

ai . Nx/ 2X

i2I. Nx/

Ai . Nx/CN. NxI˝0/; (4.43)

where each set Ai . Nx/, i D 1; : : : ; m, is calculated by

Ai . Nx/ D

8<:

Nx �˘. NxI˝i /

d. NxI˝i /for Nx … ˝i ;

N. NxI˝i / \ IB for Nx 2 ˝i :

(4.44)

(ii) Conversely, if Nx 2 ˝0 satisfies (4.43), then Nx is a solution to (4.41).

Proof. To verify (i), fix an optimal solution Nx to problem (4.41) and equivalently describe it as anoptimal solution to the unconstrained optimization problem:

minimize D.x/C ı.xI˝/; x 2 Rn : (4.45)

Applying the subdifferential Fermat rule to (4.45), we characterize Nx by

0 2 @� nX

iD1

d.�I˝i /C ı.�I˝/�. Nx/: (4.46)


Since all of the functions d.�I˝i / for i D 1; : : : ; n are convex and continuous, we employ thesubdifferential sum rule of eorem 2.15 to (4.46) and arrive at

0 2X

i2J. Nx/

@d. NxI˝i /CX

i2I. Nx/

d. NxI˝i /CN. NxI˝0/

DX

i2J. Nx/

Ai . Nx/CX

i2I. Nx/

Ai . Nx/CN. NxI˝0/

DX

i2J. Nx/

ai . Nx/CX

i2I. Nx/

Ai . Nx/CN. NxI˝0/;

which justifies the claimed inclusion

�X

i2J. Nx/

ai . Nx/ 2X

i2I. Nx/

Ai . Nx/CN. NxI˝0/:

e converse assertion (ii) is verified similarly by taking into account the “if and only if ” propertyof the proof of (i). �

Let us specifically examine problem (4.41) in the case where the intersections of the con-straint set ˝0 with the target sets ˝i are empty.

Corollary 4.53 Consider problem (4.41) with˝0 \˝i D ; for i D 1; : : : ; m. Given Nx 2 ˝0, de-fine the elements

ai . Nx/ WDNx �˘. NxI˝i /

d. NxI˝i /¤ 0; i D 1; : : : ; m: (4.47)

en Nx 2 ˝0 is an optimal solution to (4.41) if and only if

�

mXiD1

ai . Nx/ 2 N. NxI˝0/: (4.48)

Proof. In this case we have that Nx … ˝i for all i D 1; : : : ; m, which means that I. Nx/ D ; andJ. Nx/ D f1; : : : ; mg. us our statement is a direct consequence of eorem 4.52. �

For more detailed solution of (4.41), consider its special case of m D 3.

eorem4.54 Letm D 3 in the framework of eorem 4.52, where the target sets˝1;˝2, and˝3 are pairwise disjoint, and where ˝0 D Rn. e following hold for the optimal solution Nx to(4.41) with the sets ai . Nx/ defined by (4.47):(i) If Nx belongs to one of the sets ˝i , say ˝1, then we have˝

a2; a3i � �1=2 and � a2 � a3 2 N. NxI˝1/:


(ii) If Nx does not belong to any of the three sets ˝1, ˝2, and ˝3, then

hai ; aj i D �1=2 for i ¤ j as i; j 2˚1; 2; 3

:

Conversely, if Nx satisfies either (i) or (ii), then it is an optimal solution to (4.41).

Proof. To verify (i), we have from eorem 4.52 that Nx 2 ˝1 solves (4.41) if and only if

0 2 @d. NxI˝1/C @d. NxI˝2/C @d. NxI˝3/

D @d. NxI˝1/C a2 C a3 D N. NxI˝1/ \ IB C a2 C a3:

us �a2 � a3 2 N. NxI˝1/ \ IB , which is equivalent to the validity of �a2 � a3 2 IB and �a2 �

a3 2 N. NxI˝1/. By Lemma 4.31, this is also equivalent to˝a2; a3i � �1=2 and � a2 � a3 2 N. NxI˝1/;

which gives (i). To justify (ii), we similarly conclude that in the case where Nx … ˝i for i D 1; 2; 3,this element solves (4.41) if and only if

0 2 @d. NxI˝1/C @d. NxI˝2/C @d. NxI˝3/ D a1 C a2 C a3:

en Lemma 4.32 tells us that the latter is equivalent to

hai ; aj i D �1=2 for i ¤ j as i; j 2˚1; 2; 3

:

Conversely, the validity of either (i) or (ii) and the convexity of ˝i ensure that

0 2 @d. NxI˝1/C @d. NxI˝2/C @d. NxI˝3/ D @D. Nx/;

and thus Nx is an optimal solution to the problem under consideration. �

Figure 4.6: Generalized Fermat-Torricelli problem for three disks.

Example 4.55 Let the sets ˝1, ˝2, and ˝3 in problem (4.37) be closed balls in R2 of radiusr D 1 centered at the points .0; 2/, .�2; 0/, and .2; 0/, respectively.We can easily see that the point


.0; 1/ 2 ˝1 satisfies all the conditions in eorem 4.54(i), and hence it is an optimal solution (infact the unique one) to this problem.

More generally, consider problem (4.37) in R2 generated by three pairwise disjoint disksdenoted by ˝i , i D 1; 2; 3. Let c1, c2, and c3 be the centers of the disks. Assume first that eitherthe line segment Œc2; c3� 2 R2 intersects ˝1, or Œc1; c3� intersects ˝2, or Œc1; c2� intersects ˝3. Itis not hard to check that any point of the intersections (say, of the sets ˝1 and Œc2; c3� for defi-niteness) is an optimal solution to the problem under consideration, since it satisfies the necessaryand sufficient optimality conditions of eorem 4.54(i). Indeed, if Nx is such a point, then a2 anda3 are unit vectors with ha2; a3i D �1 and �a2 � a3 D 0 2 N. NxI˝1/. If the above intersectionassumptions are violated, we define the three points q1; q2, and q3 as follows. Let u and v be in-tersection points of Œc1; c2� and Œc1; c3� with the boundary of the disk centered at c1, respectively.en we see that there is a unique point q1 on the minor curve generated by u and v such thatthe measures of angle c1q1c2 and c1q1c3 are equal. e points q2 and q3 are defined similarly.eorem 4.54 yields that whenever the angle c2q1c3, or c1q2c3, or c2q3c1 equals or exceeds 120ı

(say, the angle c2q1c3 does), then the point Nx WD q1 is an optimal solution to the problem underconsideration. Indeed, in this case a2 and a3 from (4.47) are unit vectors with ha2; a3i � �1=2

and �a2 � a3 2 N. NxI˝1/ because the vector �a2 � a3 is orthogonal to ˝1.If none of these angles equals or exceeds 120ı, there is a point q not belonging to ˝i

as i D 1; 2; 3 such that the angles c1qc2 D c2qc3 D c3qc1 are of 120ı and that q is an optimalsolution to the problem under consideration. Observe that in this case the point q is also a uniqueoptimal solution to the classical Fermat-Torricelli problem determined by the points c1; c2, andc3; see Figure 4.6.

Finally in this section, we discuss the application of the subgradient algorithm from Sec-tion 4.3 to the numerical solution of the generalized Fermat-Torricelli problem (4.34).

eorem4.56 Consider the generalized Fermat-Torricelli problem (4.34) and assume that S ¤

; for its solution set. Picking a sequence f˛kg of positive numbers and a starting point x1 2 ˝0,form the iterative sequence

xkC1 WD ˘�xk � ˛k

mXiD1

vikI˝0

�; k D 1; 2; : : : ; (4.49)

with an arbitrary choice of subgradient elements

vik 2 �@�F .!ik � xk/ \N.!ikI˝i / for some !ik 2 ˘F .xkI˝i / if xk … ˝i (4.50)

and vik WD 0 otherwise, where ˘F .xI˝/ stands for the generalized projection operator (3.21),and where �F is the Minkowski gauge (3.11). Define the value sequence

Vk WD min˚H.xj /

ˇj D 1; : : : ; k

:


(i) If the sequence f˛kg satisfies conditions (4.13), then fVkg converges to the optimal value V ofproblem (4.34). Furthermore, we have the estimate

0 � Vk � V �d.x1IS/2 C `2

P1

kD1 ˛2k

2Pk

iD1 ˛k

;

where ` � 0 is a Lipschitz constant of cost function H.�/ on Rn.(ii) If the sequence f˛kg satisfies conditions (4.15), then fVkg converges to the optimal value Vand fxkg in (4.49) converges to an optimal solution Nx 2 S of problem (4.34).

Proof. is is a direct application of the projected subgradient algorithm described in Proposi-tion 4.28 to the constrained optimization problem (4.34) by taking into account the results forthe subgradient method in convex optimization presented in Section 4.3. �

Let us specify the above algorithm for the case of the unit ball F D IB in (4.34).

Corollary 4.57 Let F D IB � Rn in (4.34), let f˛kg be a sequence of positive numbers, and letx1 2 ˝0 be a starting point. Consider algorithm (4.49) with @�F .�/ calculated by

@�F .!ki � xk/ D

8<:˘.xkI˝i / � xk

d.xkI˝i /if xk … ˝i ;

0; if xk 2 ˝i :

en all the conclusions of eorem 4.56 hold for this specification.

Proof. Immediately follows from eorem 4.56 with �F .x/ D kxk. �

Next we illustrate applications of the subgradient algorithm of eorem 4.56 to solvingthe generalized Fermat-Torricelli problem (4.34) formulated via the minimal time function withnon-ball dynamics. Note that to implement the subgradient algorithm (4.49), it is sufficient tocalculate just one subgradient from the set on the right-hand side of (4.50) for each target set. Inthe following example we consider for definiteness the dynamics F given by the square Œ�1; 1� �Œ�1; 1� in the plane. In this case the correspondingMinkowski gauge (3.11) is given by the formula

�F .x1; x2/ D max˚jx1j; jx2j

; .x1; x2/ 2 R2 : (4.51)

Example 4.58 Consider the unconstrained generalized Fermat-Torricelli problem (4.34) withthe dynamics F D Œ�1; 1� � Œ�1; 1� and the square targets˝i of right position (with sides parallelto the axes) centered at .ai ; bi / with radii ri (half of the side) as i D 1; : : : ; m. Given a sequence


Figure 4.7: Minimal time function with square dynamic

of positive numbers f˛kg and a starting point x1, construct the iterations xk D .x1k; x2k/ byalgorithm (4.49). By eorem 3.39, subgradients vik can be chosen by

vik D

8ˆˆ<ˆˆ:

.1; 0/ if jx2k � bi j � x1k � ai and x1k > ai C ri ;

.�1; 0/ if jx2k � bi j � ai � x1k and x1k < ai � ri ;

.0; 1/ if jx1k � ai j � x2k � bi and x2k > bi C ri ;

.0;�1/ if jx1k � ai j � bi � x2k and x2k < bi � ri ;

.0; 0/ otherwise.

4.6 AGENERALIZED SYLVESTERPROBLEMIn [31] James Joseph Sylvester proposed the smallest enclosing circle problem: Given a finite numberof points in the plane, find the smallest circle that encloses all of the points. After more than acentury, location problems of this type remain active as they are mathematically beautiful andhave meaningful real-world applications.

is section is devoted to the study by means of convex analysis of the following generalizedversion of the Sylvester problem formulated in [18]; see also [19, 20] for further developments.Let F � Rn be a closed, bounded, convex set containing the origin as an interior point. Definethe generalized ball centered at x 2 Rn with radius r > 0 by

GF .xI r/ WD x C rF:

4.6. AGENERALIZED SYLVESTERPROBLEM 169

It is obvious that for F D IB the generalized ball GF .xI r/ reduces to the closed ball of radiusr centered at x. Given now finitely many nonempty, closed, convex sets ˝i for i D 0; : : : ; m,the generalized Sylvester problem is formulated as follows: Find a point x 2 ˝0 and the smallestnumber r � 0 such that

GF .xI r/ \˝i ¤ ; for all i D 1; : : : ; m:

Observe that when ˝0 D R2 and all the sets ˝i for i D 1; : : : ; m are singletons in R2, the gen-eralized Sylvester problem becomes the classical Sylvester enclosing circle problem. e sets ˝i

for i D 1; : : : ; m are often called the target sets.To study the generalized Sylvester problem, we introduce the cost function

T .x/ WD max˚TF .xI˝i /

ˇi D 1; : : : ; m

via the minimal time function (3.9) and consider the following optimization problem:

minimize T .x/ subject to x 2 ˝0: (4.52)

We can show that the generalized ball GF .xI r/ with Nx 2 ˝0 is an optimal solution of the gener-alized Sylvester problem if and only if Nx is an optimal solution of the optimization problem (4.52)with r D T . Nx/.

Our first result gives sufficient conditions for the existence of solutions to (4.52).

Proposition 4.59 e generalized Sylvester problem (4.52) admits an optimal solution if the set˝0 \

� TmiD1.˝i � ˛F /

�is bounded for all ˛ � 0. is holds, in particular, if there exists an index

i0 2 f0; : : : ; mg such that the set˝i0 is compact.

Proof. For any ˛ � 0, consider the level set

L˛ WD˚x 2 ˝0

ˇT .x/ � ˛

and then show that in fact we have

L˛ D ˝0 \ Œ

m\iD1

.˝i � ˛F /�: (4.53)

Indeed, it holds for any nonempty, closed, convex set Q that TF .xIQ/ � ˛ if and only if .x C

˛F / \Q ¤ ;, which is equivalent to x 2 Q � ˛F . If x 2 L˛, then x 2 ˝0 and TF .xI˝i / � ˛

for i D 1; : : : ; m. us x 2 ˝i � ˛F for i D 1; : : : ; m, which verifies the inclusion “�” in (4.53).e proof of the opposite inclusion is also straightforward. It is clear furthermore that the bound-edness of the set ˝0 \

� TmiD1.˝i � ˛F /

�for all ˛ � 0 ensures that the level sets for the cost

function in (4.52) are bounded. us it follows from eorem 4.11 that problem (4.52) has anoptimal solution. e last statement is obvious. �


Now we continue with the study of the uniqueness of optimal solutions to (4.52) splittingthe proof into several propositions.

Proposition 4.60 Consider a convex function g.x/ � 0 on a nonempty, convex set ˝ � Rn. enthe function defined by h.x/ WD .g.x//2 is strictly convex on˝ if and only if g is not constant on anyline segment Œa; b� � ˝ with a ¤ b.

Proof. Suppose that h is strictly convex on ˝ and suppose on the contrary that g is constant ona line segment Œa; b� with a ¤ b, then h is also constant on this line segment, so it cannot bestrictly convex. Conversely, suppose g is not constant on any segment Œa; b� � ˝ with a ¤ b.Observe that h is a convex function on˝ since it is a composition of a quadratic function, whichis a nondecreasing convex function on Œ0;1/, and a convex function whose range is a subsetof Œ0;1/. Let us show that it is strictly convex on ˝. Suppose by contradiction that there aret 2 .0; 1/ and x; y 2 ˝, x ¤ y, with

h.z/ D th.x/C .1 � t /h.y/ with z WD tx C .1 � t/y:

en .g.z//2 D t.g.x//2 C .1 � t /.g.y//2. Since g.z/ � tg.x/C .1 � t/g.y/, one has�g.z/

�2� t2

�g.x/

�2C .1 � t/2

�g.y/

�2C 2t.1 � t /g.x/g.y/;

which readily implies that

t�g.x/

�2C .1 � t/

�g.y/

�2� t2

�g.x/

�2C .1 � t /2

�g.y/

�2C 2t.1 � t/g.x/g.y/:

us .g.x/ � g.y//2 � 0, so g.x/ D g.y/. is shows that h.x/ D h.z/ D h.y/ for the pointz 2 .x; y/ defined above. We arrive at a contradiction by showing that h is constant on the linesegment Œz; y�. Indeed, fix any u 2 .z; y/ and observe that

h.u/ � �h.z/C .1 � �/h.y/ D h.z/ for some � 2 .0; 1/:

On the other hand, since z lies between x and u, we have

h.z/ � �h.x/C .1 � �/h.u/ � �h.z/C .1 � �/h.z/ D h.z/ for some � 2 .0; 1/:

It gives us h.z/ D �h.x/C .1 � �/h.u/ D �h.z/C .1 � �/h.u/, and hence h.u/ D h.z/. iscontradicts the assumption that g.u/ D

ph.u/ is not constant on any line segment Œa; b� with

a ¤ b and thus completes the proof of the proposition. �

Proposition 4.61 Let both setsF and˝ inRn be nonempty, closed, strictly convex, whereF containsthe origin as an interior point. en the convex function TF .�I˝/ is not constant on any line segmentŒa; b� � Rn such that a ¤ b and Œa; b� \˝ D ;.


Proof. To prove the statement of the proposition, suppose on the contrary that there is Œa; b� �

Rn with a ¤ b such that Œa; b� \˝ D ; and TF .xI˝/ D r for all x 2 Œa; b�. Choosing u 2

˘F .aI˝/ and v 2 ˘F .bI˝/, we get �F .u � a/ D �F .v � b/ D r . Let us verify that u ¤ v. In-deed, the opposite gives us �F .u � a/ D �F .u � b/ D r , which implies by r > 0 that

�F

�u � a

r

�D �F

�u � b

r

�D 1;

and thus u � a

r2 bdF and u � b

r2 bdF . Since F is strictly convex, it yields

1

2

�u � a

r

�C1

2

�u � b

r

�D1

r

�u �

aC b

2

�2 intF:

Hence we arrive at a contradiction by having

TF

�aC b

2I˝

�� F

�u �

aC b

2

�< r:

To finish the proof of the proposition, fix t 2 .0; 1/ and get

r D TF

�taC .1 � t/bI˝

�� F

�tuC .1 � t/v � .taC .1 � t /b/

�� t�F .u � a/C .1 � t/�F .v � b/ D r;

which implies that tuC .1 � t/v 2 ˘F .taC .1 � t /bI˝/. us tuC .1 � t /v 2 bd˝, whichcontradicts the strict convexity of ˝ and completes the proof. �

Figure 4.8: Minimal time function for strictly convex sets.

Proposition 4.62 Let˝ � Rn be a nonempty, closed, convex set and let hi W˝ ! R be nonnegative,continuous, convex functions for i D 1; : : : ; m. Define

�.x/ WD max˚h1.x/; : : : ; hm.x/


and suppose that �.x/ D r > 0 on Œa; b� for some line segment Œa; b� � ˝ with a ¤ b. en thereexists a line segment Œ˛; ˇ� � Œa; b� with ˛ ¤ ˇ and an index i0 2 f1; : : : ; mg such that we havehi0.x/ D r for all x 2 Œ˛; ˇ�.

Proof. ere is nothing to prove for m D 1. When m D 2, consider the function

�.x/ WD max˚h1.x/; h2.x/

:

en the conclusion is obvious if h1.x/ D r for all x 2 Œa; b�. Otherwise we find x0 2 Œa; b� withh1.x0/ < r and a subinterval Œ˛; ˇ� � Œa; b�, ˛ ¤ ˇ, such that

h1.x/ < r for all x 2 Œ˛; ˇ�:

us h2.x/ D r on this subinterval. To proceed further by induction, we assume that the conclu-sion holds for a positive integer m � 2 and show that the conclusion holds for mC 1 functions.Denote

�.x/ WD max˚h1.x/; : : : ; hm.x/; hmC1.x/

:

en we have �.x/ D maxfh1.x/; k1.x/g with k1.x/ WD maxfh2.x/; : : : ; hmC1.x/g, and the con-clusion follows from the case of two functions and the induction hypothesis. �

Now we are ready to study the uniqueness of an optimal solution to the generalized Sylvesterproblem (4.52).

eorem 4.63 In the setting of problem (4.52) suppose further that the sets F and ˝i fori D 1; : : : ; m are strictly convex. en problem (4.52) has at most one optimal solution if andonly if \m

iD0˝i contains at most one element.

Proof. Define K.x/ WD .T .x//2 and consider the optimization problem

minimize K.x/ subject to x 2 ˝0: (4.54)

It is obvious that Nx 2 ˝0 is an optimal solution to (4.52) if and only if it is an optimal solutionto problem (4.54). We are going to prove that if \m

iD0˝i contains at most one element, then K isstrictly convex on˝0, and hence problem (4.52) has atmost one optimal solution. Indeed, supposethat K is not strictly convex on˝0. By Proposition 4.60, we find a line segment Œa; b� � ˝0 witha ¤ b and a number r � 0 so that

T .x/ D max˚TF .xI˝i /

ˇi D 1; : : : ; m

D r for all x 2 Œa; b�:

It is clear that r > 0, since otherwise Œa; b� �Tm

iD0˝i , which contradicts the assumption thatthis set contains at most one element. Proposition 4.62 tells us that there exist a line segmentŒ˛; ˇ� � Œa; b� with ˛ ¤ ˇ and an index i0 2 f1; : : : ; mg such that

TF .xI˝i0/ D r for all x 2 Œ˛; ˇ�:


Since r > 0, we have x … ˝i0 for all x 2 Œ˛; ˇ�, which contradicts Proposition 4.61.Conversely, let problem (4.52) have at most one solution. Suppose now that the set \m

iD0˝i

contains more than one element. It is clear thatTm

iD0˝i is the solution set of (4.52) with the op-timal value equal to zero. us this problem has more than one solution, which is a contradictionthat completes the proof of the theorem. �

e following corollary gives a sufficient condition for the uniqueness of optimal solutionsto the generalized Sylvester problem (4.52).

Corollary 4.64 In the setting of (4.52) suppose that the sets F and ˝i for i D 1; : : : ; m are strictlyconvex. If

TmiD0˝i contains at most one element and if one of the sets among ˝i for i D 0; : : : ; m is

bounded, then problem (4.52) has a unique optimal solution.

Proof. SinceTm

iD0˝i contains at most one element, problem (4.52) has at most one optimalsolution by eorem 4.63. us it has exactly one solution because the solution set is nonemptyby Proposition 4.59. �

Next we consider a special unconstrained case of (4.52) given by

minimize M.x/ WD max˚d.xI˝i /

ˇi D 1; : : : ; m

subject to x 2 Rn; (4.55)

for nonempty, closed, convex sets ˝i under the assumption that \niD1˝i D ;. Problem (4.55)

corresponds to (4.52) with F D IB � Rn. Denote the set of active indices for (4.55) by

I.x/ WD˚i 2 f1; : : : ; mg

ˇM.x/ D d.xI˝i /

; x 2 Rn :

Further, for each x … ˝i , define the singleton

ai .x/ WDx �˘.xI˝i /

d.xI˝i /(4.56)

where the projection ˘.xI˝i / is unique. We have M.x/ > 0 as i 2 I.x/ due to the assumptionTmiD1˝i D ;. is ensures that x … ˝i automatically for any i 2 I.x/.

Let us now present necessary and sufficient optimality conditions for (4.55) based on the gen-eral results of convex analysis and optimization developed above.

eorem 4.65 e element Nx 2 Rn is an optimal solution to problem (4.55) under the assump-tions made if and only if we have the inclusion

Nx 2 co˚˘. NxI˝i /

ˇi 2 I. Nx/

(4.57)

In particular, for the case where ˝i D faig, i D 1; : : : ; m, problem (4.55) has a unique solutionand Nx is the problem generated by ai if and only if

Nx 2 co˚ai

ˇi 2 I. Nx/

: (4.58)


Proof. It follows from Proposition 2.35 and Proposition 2.54 that the condition

0 2 co[

i2I. Nx/

@d. NxI˝i /

is necessary and sufficient for optimality of Nx in (4.55). Employing now the distance functionsubdifferentiation from eorem 2.39 gives us the minimum characterization

0 2 co˚ai . Nx/

ˇi 2 I. Nx/

; (4.59)

via the singletons ai . Nx/ calculated in (4.56). It is easy to observe that (4.59) holds if and only ifthere are �i � 0 for i 2 I. Nx/ such that

Pi2I. Nx/ �i D 1 and

0 DX

i2I. Nx/

�i

Nx � N!i

M. Nx/with !i WD ˘. NxI˝i /:

is equation is clearly equivalent to

0 DX

i2I. Nx/

�i . Nx � N!i /; or Nx DX

i2I. Nx/

�i N!i ;

which in turn is equivalent to inclusion (4.57). Finally, for ˝i D faig as i D 1; : : : ; m, the latterinclusion obviously reduces to condition (4.58). In this case the problem has a unique optimalsolution by Corollary 4.64. �

To formulate the following result, we say that the ball IB. NxI r/ touches the set ˝i if the in-tersection set ˝i \ IB. NxI r/ is a singleton.

Corollary 4.66 Consider problem (4.55) under the assumptions above.en any smallest intersectingball touches at least two target sets among˝i , i D 1; : : : ; m.

Proof. Let us first verify that there are at least two indices in I. Nx/ provided that Nx is a solution to(4.55). Suppose the contrary, i.e., I. Nx/ D fi0g. en it follows from (4.57) that

Nx 2 ˘. NxI˝i0/ and d. NxI˝i0/ D M. Nx/ > 0;

a contradiction. Next we show that IB. NxI r/ with r WD M. Nx/ touches˝i when i 2 I. Nx/. Indeed,if on the contrary there are u; v 2 ˝i such that

u; v 2 IB. NxI r/ \˝i with u ¤ v;

en, by taking into account that d. NxI˝i / D r , we get

ku � Nxk � r D d. NxI˝i / and kv � Nxk � r D d. NxI˝i /;

so u; v 2 ˘. NxI˝i /. is contradicts the fact that the projector ˘. NxI˝i / is a singleton and thuscompletes the proof of the corollary. �


Now we illustrate the results obtained by solving the following two examples.

Example 4.67 Let Nx solve the smallest enclosing ball problem generated by ai 2 R2, i D 1; 2; 3.Corollary 4.66 tells us that there are either two or three active indices. If I. Nx/ consists oftwo indices, say I. Nx/ D f2; 3g, then Nx 2 co fa2; a3g and k Nx � a2k D k Nx � a3k > k Nx � a1k byeorem 4.65. In this case we have Nx D

a2 C a3

2and ha2 � a1; a3 � a1i < 0. If conversely

ha2 � a1; a3 � a1i < 0, then the angle of the triangle formed by a1, a2, and a3 at the vertexa1 is obtuse; we include here the case where all the three points ai are on a straight line. It iseasy to see that I

�a2 C a3

2

�D f2; 3g and that the point Nx D

a2 C a3

2satisfies the assumption

of eorem 4.65 thus solving the problem. Observe that in this case the solution of (4.55) is themidpoint of the side opposite to the obtuse vertex.

If none of the angles of the triangle formed by a1; a2; a3 is obtuse, then the active index setI. Nx/ consists of three elements. In this case Nx is the unique point satisfying

Nx 2 co˚a1; a2; a3

and k Nx � a1k D k Nx � a2k D k Nx � a3k:

In the other words, Nx is the center of the circumscribing circle of the triangle.

Figure 4.9: Smallest intersecting ball problem for three disks.

Example 4.68 Let ˝i D IB.ci ; ri / with ri > 0 and i D 1; 2; 3 be disjoint disks in R2, and let Nx

be the unique solution of the corresponding problem (4.55). Consider first the case where one ofthe line segments connecting two of the centers intersects the other disks, e.g., the line segmentconnecting c2 and c3 intersects˝1. Denote u2 WD c2c3 \ bd˝2 and u3 WD c2c3 \ bd˝3, and letNx be the midpoint of u2u3. en I. Nx/ D f2; 3g and we can apply eorem 4.65 to see that Nx isthe solution of the problem.


It remains to consider the case where any line segment connecting two centers of the disksdoes not intersect the remaining disk. In this case denote

u1 WD c1c2 \ bd˝1; v1 WD c1c3 \ bd˝1;

u2 WD c2c3 \ bd˝2; v2 WD c2c1 \ bd˝2;

u3 WD c3c1 \ bd˝3; v3 WD c3c2 \ bd˝3;

a1 WD c1m1 \ bd˝1; a2 D c2m2 \ bd˝2; a3 WD c3m3 \ bd˝3;

where m1 is the midpoint of u2v3, m2 is the midpoint of u3v1, and m3 is the midpoint of u1v2.Suppose that one of the angles 2u2a1v3, 2u3a2v1, 2u1a3v2 is greater than or equal to 90ı; e.g.,2u2a1v3 is greater than 90ı. en I.m1/ D f2; 3g, and thus Nx D m1 is the unique solution of thisproblem by eorem 4.65. If now we suppose that all of the aforementioned angles are less thanor equal to 90ı, then I. Nx/ D 3 and the smallest disk we are looking for is the unique disk thattouches three other disks. e construction of this disk is the celebrated problem of Apollonius.

Finally in this section, we present and illustrate the subgradient algorithm to solve numeri-cally the generalized Sylvester problem (4.52).

eorem 4.69 In the setting of problem (4.52) suppose that the solution set S of the problemis nonempty. Fix x1 2 ˝0 and define the sequences of iterates by

xkC1 WD ˘�xk � ˛kvkI˝0

�; k 2 N;

where f˛kg is a sequence of positive numbers, and where

vk 2

�f0g if xk 2 ˝i ;�

� @�F .!k � xk/�

\N.!kI˝i / if xk … ˝i ;

where !k 2 ˘F .xkI˝i / for some i 2 I.xk/. Define the value sequence

Vk WD min˚T .xj /

ˇj D 1; : : : ; k

:

(i) If the sequence f˛kg satisfies conditions (4.13), then fVkg converges to the optimal value V ofthe problem. Furthermore, we have the estimate

0 � Vk � V �d.x1IS/2 C `2

P1

kD1 ˛2k

2Pk

iD1 ˛k

;

where ` � 0 is a Lipschitz constant of cost function T .�/ on Rn.(ii) If the sequence f˛kg satisfies conditions (4.15), then fVkg converges to the optimal value Vand fxkg in (4.49) converges to an optimal solution Nx 2 S of the problem.

Proof. Follows from the results of Section 4.3 applied to problem (4.52). �


e next examples illustrate the application of the subgradient algorithm of eorem 4.69in particular situations.

Example 4.70 Consider problem (4.55) in R2 with m target sets given by the squares

˝i D S.!i I ri / WD�!1i � ri ; !1i C ri

��

�!2i � ri ; !2i C ri

�; i D 1; : : : ; m:

Denote the vertices of the i th square by v1i WD .!1i C ri ; !2i C ri /; v2i WD .!1i � ri ; !2i C

ri /; v3i WD .!1i � ri ; !2i � ri /; v4i WD .!1i C ri ; !2i � ri /, and let xk WD .x1k; x2k/. Fix an in-dex i 2 I.xk/ and then choose the elements vk from eorem 4.69 by

vk D

8ˆˆ<ˆˆ:

xk � v1i

kxk � v1ikif x1k � !1i > ri and x2k � !2i > ri ;

xk � v2i

kxk � v2ikif x1k � !1i < �ri and x2k � !2i > ri ;

xk � v3i

kxk � v3ikif x1k � !1i < �ri and x2k � !2i < �ri ;

xk � v4i

kxk � v4ikif x1k � !1i > ri and x2k � !2i < �ri ;

.0; 1/ if jx1k � !1i j � ri and x2k � !2i > ri ;

.0;�1/ if jx1k � !1i j � ri and x2k � !2i < �ri ;

.1; 0/ if x1k � !1i > ri and jx2k � !2i j � ri ;

.�1; 0/ if x1k � !1i < �ri and jx2k � !2i j � ri :

Figure 4.10: Euclidean projection to a square.

Now we consider the case of a non-Euclidean norm in the minimal time function.


Example 4.71 Consider problem (4.52) with F WD˚.x1; x2/ 2 R2

ˇjx1j C jx2j � 1

and a

nonempty, closed, convex target set ˝. e generalized ball GF . NxI t/ centered at Nx D . Nx1; Nx2/

with radius t > 0 has the following diamond shape:

GF . NxI t/ D˚.x1; x2/ 2 R2

ˇjx1 � Nx1j C jx2 � Nx2j � t

:

e distance from Nx to ˝ and the corresponding projection are given by

TF . NxI˝/ D min˚t � 0

ˇGF . NxI t / \˝ ¤ ;

; (4.60)

˘F . NxI˝/ D GF . NxI t / \˝ with t D TF . NxI˝/:

Let˝i be the squares S.!i I ri /, i D 1; : : : ; m, given in Example 4.70. en problem (4.55) can beinterpreted as follows: Find a set of the diamond shape in R2 that intersects all m given squares.Using the same notation as in Example 4.70, we can see that the elements vk in eorem 4.69are chosen by

vk D

8ˆ<ˆˆ:

.1; 1/ if x1k � !1i > ri and x2k � !2i > ri ;

.�1; 1/ if x1k � !1i < �ri and x2k � !2i > ri ;

.�1;�1/ if x1k � !1i < �ri and x2k � !2i < �ri ;

.1;�1/ if x1k � !1i > ri and x2k � !2i < �ri ;

.0; 1/ if jx1k � !1i j � ri and x2k � !2i > ri ;

.0;�1/ if jx1k � !1i j � ri and x2k � !2i < �ri ;

.1; 0/ if x1k � !1i > ri and jx2k � !2i j � ri ;

.�1; 0/ if x1k � !1i < �ri and jx2k � !2i j � ri :


Exercise 4.1 Given a lower semicontinuous function f W Rn! R, show that its multiplication

. f / 7! f .x/ by any scalar ˛ > 0 is lower semicontinuous as well.

Exercise 4.2 Given a function f W Rn! R, show that its lower semicontinuity is equivalent

to openess, for every � 2 R, of the set

U� WD˚x 2 Rn

ˇf .x/ > �

:

Exercise 4.3 Give an example of two lower semicontinuous real-valued functions whose prod-uct is not lower semicontinuous.


Figure 4.11: Minimal time function with diamond-shape dynamic.

Exercise 4.4 Let f W Rn! R. Prove the following assertions:

(i) f is upper semicontinuous (u.s.c.) at Nx in the sense of Definition 2.89 if and only if �f islower semicontinuous at this point.(ii) f is u.s.c. at Nx if and only if lim supk!1

f .xk/ � f . Nx/ for every xk ! Nx.(iii) f is u.s.c. at Nx if and only if we have lim supx! Nx f .x/ � f . Nx/.(iv) f is u.s.c. on Rn if and only if the upper level set fx 2 Rn

j f .x/ � �g is closed for any realnumber �.(v) f is u.s.c. on Rn if and only if its hypergraphical set (or simply ) hypo f WD

f.x; �/ 2 Rn� R j � � f .x/g is closed.

Exercise 4.5 Show that a function f W Rn! R is continuous at Nx if and only if it is both lower

semicontinuous and upper semicontinuous at this point.

Exercise 4.6 Prove the equivalence (iii)”(iv) in eorem 4.11.

Exercise 4.7 Consider the real-valued function

g.x/ WD

mXiD1

kx � aik2; x 2 Rn;

where ai 2 Rn for i D 1; : : : ; m. Show that g is a convex function and has an absolute minimumat the mean point Nx D

a1 C a2 C : : :C am

m.


Exercise 4.8 Consider the problem:

minimize kxk2 subject to Ax D b;

whereA is anp � nmatrix with rank A D p, and where b 2 Rp. Find the unique optimal solutionof this problem.

Exercise 4.9 Give a direct proof of Corollary 4.15 without using eorem 4.14.

Exercise 4.10 Given ai 2 Rn for i D 1; : : : ; m, write MATLAB programs to implement thesubgradient algorithm for minimizing the function f given by:(i) f .x/ WD

PmiD1 kx � aik; x 2 Rn.

(ii) f .x/ WDPm

iD1 kx � aik1; x 2 Rn.(iii) f .x/ WD

PmiD1 kx � aik1; x 2 Rn.

Exercise 4.11 Write a MATLAB program to implement the projected subgradient algorithmfor solving the problem:

minimize kxk1 subject to Ax D b;

where A is an p � n matrix with rank A D p, and where b 2 Rp.

Exercise4.12 Consider the optimization problem (4.9) and the iterative procedure (4.10) underthe standing assumptions imposed in Section 4.3. Suppose that the optimal value V is known andthat the sequence f˛kg is given by

˛k WD

8<:0 if 0 2 @f .xk/;

f .xk/ � V

kvkk2otherwise;

(4.61)

where vk 2 @f .xk/. Prove that:(i) f .xk/ ! V as k ! 1.(ii) 0 � Vk � V �

`d.x1IS/pk

for k 2 N.

Exercise 4.13 Let ˝i for i D 1; : : : ; m be nonempty, closed, convex subsets of Rn having anonempty intersection. Use the iterative procedure (4.10) for the function

f .x/ WD max˚d.xI˝i /

ˇi D 1; : : : ; m

with the sequence f˛kg given in (4.61) to derive an algorithm for finding Nx 2

TmiD1˝i .

Exercise 4.14 Derive the result of Corollary 4.20 from condition (ii) of eorem 4.14.


Exercise 4.15 Formulate and prove a counterpart ofeorem 4.27 for the projected subgradientalgorithm (4.20) to solve the constrained optimization problem (4.19).

Exercise 4.16 Prove the converse implication in Lemma 4.31.

Exercise 4.17 Write a MATLAB program to implement the Weiszfeld algorithm.

Exercise 4.18 Let ai 2 R for i D 1; : : : ; m. Solve the problem

minimizemX

iD1

jx � ai j; x 2 R :

Exercise 4.19 Give a simplified proof of eorem 4.51.

Exercise 4.20 Prove assertion (ii) of eorem 4.52.

Exercise 4.21 Give a complete proof of eorem 4.56.

Exercise 4.22 Let ˝i WD Œai ; bi � 2 R for i D 1; : : : ; m be m disjoint intervals. Solve the fol-lowing problem:

minimize f .x/ WD

mXiD1

d.xI˝i /; x 2 R :

Exercise 4.23 Write a MATLAB program to implement the subgradient algorithm for solvingthe problem:

minimize f .x/; x 2 Rn;

where f .x/ WDPm

iD1 d.xI˝i / with ˝i WD IB.ci I ri / for i D 1; : : : ; m.

Exercise 4.24 Complete the solution of the problem in Example 4.58. Write a MATLABprogram to implement the algorithm.

Exercise 4.25 Give a simplified proof of eorem 4.63 in the case where F is the closed unitball of Rn and the target sets are closed Euclidean balls of Rn.

Exercise 4.26 Give a simplified proof of eorem 4.63 in the case where F is the closed unitball of Rn.

Exercise 4.27 Write a MATLAB program to implement the subgradient algorithm for solvingthe problem:

minimize f .x/; x 2 Rn;


where f .x/ WD maxfd.xI˝i / j i D 1; : : : ; mg with ˝i WD IB.ci I ri / for i D 1; : : : ; m. Note thatif Nx is an optimal solution to this problem with the optimal value r , then IB. NxI r/ is the smallestball that intersects ˝i for i D 1; : : : ; m.

Exercise 4.28 Give a complete proof of eorem 4.69.



183

Solutions andHints forSelected Exercises

is final part of the book contains hints for selected exercises. Rather detailed solutionsare given for several problems.

EXERCISES FORCHAPTER 1

Exercise 1.1. Suppose that ˝1 is bounded. Fix a sequence fxkg � ˝1 �˝2 converging to Nx.For every k 2 N, xk is represented as xk D yk � zk with yk 2 ˝1, zk 2 ˝2. Since ˝1 is closedand bounded, fykg has a subsequence fyk`

g converging to Ny 2 ˝1. en fzk`g converges to

Ny � Nx 2 ˝2. us Nx 2 ˝1 �˝2 and ˝ is closed. Let ˝1 WD f.x; y/ 2 R2j y � 1=x; x > 0g

and ˝2 WD f.x; y/ 2 R2j y � �1=x; x > 0g. en ˝1 and ˝2 are nonempty, closed, convex

sets while ˝1 �˝2 D f.x; y/ 2 R2j y > 0

is not closed.

Exercise 1.4. (i) Use Exercise 1.3(ii).(ii) We only consider the case where m D 2 since the general case is similar. From thedefinition of cones, both sets ˝1 and ˝2 contain 0, and so ˝1 [˝2 � ˝1 C˝2. enco f˝1 [˝2g � ˝1 C˝2. To verify the opposite inclusion, fix any x D w1 C w2 2 ˝1 C˝2

with w1 2 ˝1 and w2 2 ˝2 and then write x D Œ.2w1 C 2w2/=2� 2 co f˝1 [˝2g.

Exercise 1.7. Compute the first or second derivative of each function. en apply eithereorem 1.45 or Corollary 1.46.

Exercise 1.8. (i) Use f .x; y/ WD xy; .x; y/ 2 R2. (ii) Take f1.x/ WD x and f2.x/ WD �x.

Exercise 1.9. Let f1.x/ WD x and f2.x/ WD x2, x 2 R.

Exercise 1.12. Use Proposition 1.39. Note that the function f .x/ WD xq is convex on Œ0;1/.

Exercise 1.16. Define F W Rn!! Rp by F.x/ WD K, x 2 Rn. en gphF is a convex subset of

Rn� Rp. e conclusion follows from Proposition 1.50.

184 SOLUTIONSANDHINTS FOREXERCISES

Exercise 1.18. (i) gphF D C . (ii) Consider the function

f .x/ WD

(�

p1 � x2 if jxj � 1;

1 otherwise:

(iii) Apply Proposition 1.50.

Exercise 1.20. (i) Use the definition. (ii) Take f .x/ WDp

jxj.

Exercise 1.22. Suppose that �v CPm

iD1 �ivi D 0 with �CPm

iD1 �i D 0. If � ¤ 0, then

�Pm

iD1

�i

�D 1, and so

v D �

mXiD1

�i

�vi 2 aff

˚v1; : : : ; vm

;

which is a contradiction. us � D 0, and hence �i D 0 for i D 1; : : : ; m because the elementsv1; : : : ; vm are affinely independent.

Exercise 1.23. Since�m � ˝ � aff˝, we have aff�m � aff˝. To prove the opposite inclusion,it remains to show that ˝ � aff�m. By contradiction, suppose that there are v 2 ˝ such thatv … aff�m. It follows from Exercise 1.22 that the vectors v; v1; : : : ; vm are affinely independentand all of them belong to ˝. is contradicts the condition dim˝ D m. e second equalitycan also be proved similarly.

Exercise 1.24. (i) We obviously have aff˝ � aff˝. Observe that the set aff˝ is closed andcontains ˝ since aff˝ D w0 C L, where w0 2 ˝ and L is the linear subspace parallel to aff˝,which is a closed set. us ˝ � aff˝ and the conclusion follows.(ii) Choose ı > 0 so that IB. NxI ı/ \ aff˝ � ˝. Since Nx C t. Nx � x/ D .1C t/ Nx C .�t/x is anaffine combination of some elements in aff˝ D aff˝, we have Nx C t . Nx � x/ 2 aff˝. usu WD Nx C t. Nx � x/ 2 ˝ when t is small.(iii) It follows from (i) that ri˝ � ri˝. Fix any x 2 ri˝ and Nx 2 ri˝. By (ii) we havez WD x C t.x � Nx/ 2 ˝ when t > 0 is small, and so x D z=.1C t /C t Nx=.1C t / 2 .z; Nx/ � ri˝.

Exercise 1.26. If ˝1 D ˝2, then ri˝1 D ri˝2, and so ri˝1 D ri˝2 by Exercise 1.24(iii).

Exercise 1.27. (i) Fix y 2 B.ri˝/ and find x 2 ri˝ such that y D B.x/. en pick a vectorNy 2 riB.˝/ � B.˝/ and take Nx 2 ˝ with Ny D B. Nx/. If x D Nx, then y D Ny 2 riB.˝/. Considerthe case where x ¤ Nx. Following the solution of Exercise 1.24(iii), findex 2 ˝ such that x 2 .ex; Nx/

and let ey WD B.ex/ 2 B.˝/. us y D B.x/ 2 .B.ex/; B. Nx// D .ey; Ny/, and so y 2 riB.˝/. Tocomplete the proof, it remains to show thatB.˝/ D B.ri˝/ and then use Exercise 1.26 to obtain

SOLUTIONSANDHINTS FOREXERCISES 185

the inclusionriB.˝/ D riB.ri˝/ � B.ri˝/:

Since B.ri˝/ � B.˝/, the continuity of B and Proposition 1.73 imply that

B.˝/ � B.˝/ D B.ri˝/ � B.ri˝/:

us we have B.˝/ � B.ri˝/ and complete the proof.(ii) Consider B.x; y/ WD x � y defined on Rn

� Rn.

Exercise 1.28. (i) Let us show that

aff .epif / D aff .domf / � R : (4.62)

Fix any .x; / 2 aff .epif /. en there exist �i 2 R and .xi ; i / 2 epif for i D 1; : : : ; m withPmiD1 �i D 1 such that

.x; / D

mXiD1

�i .xi ; i /:

Since xi 2 domf for i D 1; : : : ; m, we have x DPm

iD1 �ixi 2 aff .domf /, which yields

aff .epif / � aff .domf / � R :

To verify the opposite inclusion, fix any .x; / 2 aff .domf / � R and find xi 2 domf and �i 2

R for i D 1; : : : ; m withPm

iD1 �i D 1 and x DPm

iD1 �ixi . Define further ˛i WD f .xi / 2 R and˛ WD

PmiD1 �i˛i for which .xi ; ˛i / 2 epif and .xi ; ˛i C 1/ 2 epif . us

mXiD1

�i .xi ; ˛i / D .x; ˛/ 2 aff .epif /;

mXiD1

�i .xi ; ˛i C 1/ D .x; ˛ C 1/ 2 aff .epif /:

Letting � WD ˛ � C 1 gives us

.x; / D �.x; ˛/C .1 � �/.x; ˛ C 1/ 2 aff .epif /;

which justifies the opposite inclusion in (4.62).(ii) See the proof of [9, Proposition 1.1.9].

Exercise 1.30. (i) ˘.xI˝/ D x Cb � ha; xi

kak2a.

(ii) ˘.xI˝/ D x C AT .AAT /�1.b � Ax/.(iii) ˘.xI˝/ D maxfx; 0g (componentwise maximum).


EXERCISES FORCHAPTER 2Exercise 2.1. (i) Apply Proposition 2.1 and the fact that tx 2 ˝ for all t � 0.(ii) Use (i) and observe that �x 2 ˝ whenever x 2 ˝.

Exercise 2.2. Observe that the sets ri˝i for i D 1; 2 are nonempty and convex. By eorem 2.5,there is 0 ¤ v 2 Rn such that

hv; xi � hv; yi for all x 2 ri˝1; y 2 ri˝2:

Fix x0 2 ri˝1 and y0 2 ri˝2. It follows from eorem 1.72(ii) that for any x 2 ˝1, y 2 ˝2,and t 2 .0; 1/ we have that x C t.x0 � x/ 2 ri˝1 and y C t.y0 � y/ 2 ri˝2. en apply theinequality above for these elements and let t ! 0C.

Exercise 2.3. We split the solution into four steps.Step 1: If L is a subspace of Rn that contains a nonempty, convex set˝ with Nx … ˝ and Nx 2 L, thenthere is 0 ¤ v 2 L such that

sup˚hv; xi

ˇx 2 ˝

< hv; Nxi: (4.63)

Proof. By Proposition 2.1, find w 2 Rn such that

sup˚hw; xi

ˇx 2 ˝

< hw; Nxi: (4.64)

Represent Rn as L˚ L?, where L? WD fu 2 Rnj hu; xi D 0 for all x 2 Lg. en w D uC v

with u 2 L? and v 2 L. Substituting w into (4.64) and taking into account that hu; xi D 0 forall x 2 ˝ and that hu; Nxi D 0, we get (4.63). Note that (4.63) yields v ¤ 0.Step 2: Let ˝ � Rn be a nonempty, convex set such that 0 … ri˝. en the sets ˝ and f0g can beseparated properly.Proof. In the case where 0 … ˝, we can separate ˝ and f0g strictly, so the conclusion is obvi-ous. Suppose now that 0 2 ˝ n ri˝. Let L WD aff˝ D span˝, where the latter holds due to0 2 aff˝, which follows from the fact that ˝ � L since L is closed. Repeating the proof ofLemma 2.4, find a sequence fxkg � L with xk ! 0 and xk … ˝ for every k. It follows fromStep 1 that there is fvkg � L with vk ¤ 0 and

sup˚hvk; xi

ˇx 2 ˝

< 0:

Dividing both sides by kvkk, we suppose without loss of generality that kvkk D 1 and that vk !

v 2 L with kvk D 1. ensup

˚hv; xi

ˇx 2 ˝

� 0:

To finish this proof, it remains to show that there exists x 2 ˝ with hv; xi < 0. By contradiction,suppose that hv; xi � 0 for all x 2 ˝. en hv; xi D 0 on ˝. Since v 2 L D aff˝, we can rep-resent v as v D

PmiD1 �iwi , where

PmiD1 �i D 1 and wi 2 ˝ for i D 1; : : : ; m. is implies that


kvk2 D hv; vi DPm

iD1 �i hv;wi i D 0, which is a contradiction.Step 3: Let ˝ � Rn be a nonempty, convex set. If the sets ˝ and f0g can be separated properly, thenwe have 0 … ri˝.Proof. By definition, find 0 ¤ v 2 Rn such that

sup˚hv; xi

ˇx 2 ˝

� 0

and then take Nx 2 ˝ with hv; Nxi < 0. Arguing by contradiction, suppose that 0 2 ri˝. enExercise 1.24(ii) gives us t > 0 such that 0C t.0 � Nx/ D �t Nx 2 ˝. us hv;�t Nxi � 0, whichyields hv; Nxi � 0, a contradiction.Step 4: Justifying the statement of Exercise 2.3.Proof. Define ˝ WD ˝1 �˝2. By Exercise 1.27(ii) we have ri˝1 \ ri˝2 D ; if and only if

0 … ri .˝1 �˝2/ D ri˝1 � ri˝2:

en the conclusion follows from Step 2 and Step 3.

Exercise 2.4. Observe that ˝1 \ Œ˝2 � .0; �/� D ; for any � > 0.

Exercise 2.5. Employing eorem 2.5 and arguments similar to those in Exercise 2.2, we canshow that there is 0 ¤ v 2 Rn such that

hv; xi � hv; yi for all x 2 ˝1; y 2 ˝2:

By Proposition 2.13, choose 0 ¤ v 2 N. NxI˝1/ \ Œ�N. NxI˝2/� and, using the normal conedefinition, arrive at the conclusion with v1 WD v, v2 WD �v, ˛1 WD hv1; Nxi, and ˛2 WD hv2; Nxi:

Exercise 2.6. Fix Nx � 0 and v 2 N. NxI˝/. en

hv; x � Nxi � 0 for all x 2 ˝:

Using this inequality for 0 2 ˝ and 2 Nx 2 ˝, we get hv; Nxi D 0. Since ˝ D .�1; 0� � : : : �

.�1; 0�, it follows from Proposition 2.11 that v � 0. e opposite inclusion is a direct conse-quence of the definition.

Exercise 2.8. Suppose that this holds for any k convex sets with k � m and m � 2. Consider thesets ˝i for i D 1; : : : ; mC 1 satisfying the qualification condition

h mC1XiD1

vi D 0; vi 2 N. NxI˝i /i

H)�vi D 0 for all i D 1; : : : ; mC 1

�:


Since 0 2 N. NxI˝mC1/, this condition also holds for the first m sets. us

N�

NxI

m\iD1

˝i

�D

mXiD1

N. NxI˝i /:

en apply the induction hypothesis to �1 WD \miD1˝i and �2 WD ˝mC1.

Exercise 2.10. Since v 2 @f . Nx/, we have

v.x � Nx/ � f .x/ � f . Nx/ for all x 2 R :

en pick any x1 < Nx and use the inequality above.

Exercise 2.12. Define the function

g.x/ WD f .x/C ıf Nxg.x/ D

(f . Nx/ if x D Nx;

1 otherwise

via the indicator function of f Nxg. en epig D f Nxg � Œf . Nx/;1/, and henceN.. Nx; g. Nx//I epig/ D

Rn�.�1; 0�. It follows from Proposition 2.31 that @g. Nx/ D Rn and @ıf Nxg. Nx/ D Rn. Employing

the subdifferential sum rule from Corollary 2.45 gives us

RnD @g. Nx/ D @f . Nx/C Rn;

which ensures that @f . Nx/ ¤ ;.

Exercise 2.13. We proceed step by step as in the proof of eorem 2.15. It follows from Exer-cise 1.28 that ri�1 D ri˝1 � .0;1/ and

ri�2 D˚.x; �/

ˇx 2 ri˝1; � < hv; x � Nxi

:

It is easy to check, arguing by contradiction, that ri�1 \ ri�2 D ;. Applying the separation resultfrom Exercise 2.3, we find 0 ¤ .w; / 2 Rn

� R such that

hw; xi C �1 � hw; yi C �2 for all .x; �1/ 2 �1; .y; �2/ 2 �2:

Moreover, there are .ex;e�1/ 2 �1 and .ey;e�2/ 2 �2 satisfying

hw;exi C e�1 < hw;eyi C e�2 :

As in the proof of eorem 2.15, it remains to verify that < 0. By contradiction, assume that D 0 and then get

hw; xi � hw; yi for all x 2 ˝1; y 2 ˝2;

hw;exi < hw;eyi with ex 2 ˝1; ey 2 ˝2:


us the sets ˝1 and ˝2 are properly separated, and hence it follows from Exercise 2.3 thatri˝1 \ ri˝2 D ;. is is a contradiction, which shows that < 0. To complete the proof, weonly need to repeat the proof of Case 2 in eorem 2.15.

Exercise 2.14. Use Exercise 2.13 and proceed as in the proof of eorem 2.44.

Exercise 2.16. Use the same procedure as in the proof of eorem 2.51 and the construction ofthe singular subdifferential to obtain

@1.f ı B/. Nx/ D A��@1f . Ny/

�D

˚A�v

ˇv 2 @1f . Ny/

: (4.65)

Alternatively, observe the equality dom .f ı B/ D B�1.dom f /. en apply Proposition 2.23and Corollary 2.53 to complete the proof.

Exercise 2.17. Use the subdifferential formula from eorem 2.61 and then the representationof the composition from Proposition 1.54.

Exercise 2.19. For any v 2 Rn, we have �˝1.v/ D kvk1 and �˝2

.v/ D kvk1.

Exercise 2.21. Use first the representations

.ı˝/�.v/ D sup

˚hv; xi � ı.xI˝/

ˇx 2 Rn

D sup

˚hv; xi

ˇx 2 ˝

D �˝.v/; v 2 Rn :

Applying then the Fenchel conjugate one more time gives us

.�˝/�.x/ D sup

˚hx; vi � �˝.v/

ˇv 2 Rn

D ıco˝.x/:

Exercise 2.23. Since f W Rn! R is convex and positively homogeneous, we have @f .0/ ¤ ; and

f .0/ D 0. is implies in the case where v 2 ˝ D @f .0/ that hv; xi � f .x/ for all x 2 Rn, andso the equality holds when x D 0. us we have


ˇx 2 Rn

D 0:

If v … ˝ D f .0/, there is Nx 2 Rn satisfying hv; Nxi > f . Nx/. It gives us by homogeneity that


ˇx 2 Rn

� sup

˚hv; t Nxi � f .t Nx/

ˇt > 0

D 1:

Exercise 2.26. f 0.�1I d/ D �1 and f 0.1I d/ D 1.

Exercise 2.27. Write

Nx C t2d Dt2

t1. Nx C t1d/C .1 �

t2

t1/ Nx for fixed t1 < t2 < 0:


Exercise 2.28. (i) For d 2 Rn, consider the function ' defined in Lemma 2.80. en show thatthe inequalities � < 0 < t imply that '.�/ � '.t/.(ii) It follows from (i) that '.t/ � '.1/ for all t 2 .0; 1/. Similarly, '.�1/ � '.t/ for allt 2 .�1; 0/. is easily yields the conclusion.

Exercise 2.31. Fix Nx 2 Rn, define P. NxI˝/ WD fw 2 ˝ j k Nx � wk D �˝. Nx/g, and consider thefollowing function

fw.x/ WD kx � wk for w 2 ˝ and x 2 Rn :

en we get from eorem 2.93 that

@�˝. Nx/ D co˚@fw. Nx/

ˇw 2 P. NxI˝/

:

In the case where ˝ is not a singleton we have @fw. Nx/ DNx � w

�˝. Nx/, and so

@�˝. Nx/ DNx � coP. NxI˝/

�˝. Nx/:

e case where ˝ is a singleton is trivial.

Exercise 2.33. (i) Fix any v 2 Rn with kvk � 1 and any w 2 ˝. en

hx; vi � �˝.v/ � hx; vi � hw; vi D hx � w; vi � kx � wk � kvk � kx � wk:

It follows therefore thatsup

kvk�1

˚hx; vi � �˝.v/

� d.xI˝/:

To verify the opposite inequality, observe first that it holds trivially if x 2 ˝. In the case wherex … ˝, let w WD ˘.xI˝/ and u WD x � w; thus kuk D d.xI˝/. Following the proof of Propo-sition 2.1, we obtain the estimate

hu; zi � hu; xi � kuk2 for all z 2 ˝:

Divide both sides by kuk and let v WD u=kuk. en �˝.v/ � hv; xi � d.xI˝/, and so

d.xI˝/ � supkvk�1

˚hx; vi � �˝.v/

:

EXERCISES FORCHAPTER 3Exercise 3.1. Suppose that f is Gâteaux differentiable at Nx with f 0

G. Nx/ D v. en

hf 0G. Nx/; vi D lim

t!0

f . Nx C td / � f . Nx/

tD lim

t!0C

f . Nx C td / � f . Nx/

tfor all d 2 Rn :


is implies by the definition of the directional derivative that f 0. NxI d/ D hv; d i for all d 2 Rn.Conversely, suppose that f 0. NxI d/ D hv; d i for all d 2 Rn. en

f 0. NxI d/ D hv; d i D limt!0C

f . Nx C td / � f . Nx/

tfor all d 2 Rn :

Moreover, f 0�. NxI d/ D �f 0. NxI �d/ D �hv;�d i D hv; d i for all d 2 Rn. us

f 0�. NxI d/ D hv; d i D lim

t!0�

f . Nx C td / � f . Nx/

tfor all d 2 Rn :

is implies that f is Gâteaux differentiable at Nx with f 0G. Nx/ D v.

Exercise 3.2. (i) Since the function '.t/ WD t2 is nondecreasing on Œ0;1/, following the proofof Corollary 2.62, we can show that if g W Rn

! Œ0;1/ is a convex function, then @.g2/. Nx/ D

2g. Nx/@g. Nx/ for every Nx 2 Rn, where g2.x/ WD Œg.x/�2. It follows from eorem 2.39 that

@f . Nx/ D 2d. NxI˝/@d. NxI˝/ D

(f0g if Nx 2 ˝;˚2Œ Nx �˘. NxI˝/�

otherwise:

us in any case @f . Nx/ D˚2Œ Nx �˘. NxI˝/�

is a singleton, which implies the differentiability of

f at Nx by eorem 3.3 and rf . Nx/ D 2Œ Nx �˘. NxI˝/�.(ii) Use eorem 2.39 in the out-of-set case and eorem 3.3.Exercise 3.3. For every u 2 F , the function fu.x/ WD hAx; ui �

1

2kuk2 is convex, and hence f

is a convex function by Proposition 1.43. It follows from the equality hAx; ui D hx;A�ui that

rfu.x/ D A�u:

Fix x 2 Rn and observe that the function gx.u/ WD hAx; ui �1

2kuk2 has a unique maximizer on

F denoted by u.x/. By eorem 2.93 we have @f .x/ D fA�u.x/g, and thus f is differentiableat x by eorem 3.3. We refer the reader to [22] for more general results and their applicationsto optimization.

Exercise 3.5. Consider the set Y WD fd 2 Rnj f 0. NxI d/ D f 0

�. NxI d/g and verify that Y is asubspace containing the standard orthogonal basis fei j i D 1; : : : ; ng. us V D Rn, whichimplies the differentiability of f at Nx. Indeed, fix any v 2 @f . Nx/ and get by eorem 2.84(iii)that f 0. NxI d/ D hv; d i for all d 2 Rn. Hence @f . Nx/ is a singleton, which ensures that f isdifferentiable at Nx by eorem 3.3.

Exercise 3.6. Use Corollary 2.37 and follow the proof of eorem 2.40.


Exercise 3.7. An appropriate example is given by the function f .x/ WD x4 on R.

Exercise 3.10. Fix any sequence fxkg � co˝. By the Carathéodory theorem, find �k;i � 0 andwk;i 2 ˝ for i D 0; : : : ; n with

xk D

nXiD0

�k;iwk;i ;

nXiD0

�k;i D 1 for every k 2 N :

en use the boundedness of f�k;igk for i D 0; : : : ; n and the compactness of ˝ to extract asubsequence of fxkg converging to some element of co˝.

Exercise 3.16. By Corollary 2.18 we have N. NxI˝1 \˝2/ D N. NxI˝1/CN. NxI˝2/. Applyingeorem 3.16 tells us that

T . NxI˝1 \˝2/ D�N. NxI˝1 \˝2/

�ı

D�N. NxI˝1/CN. NxI˝2/

�ıD

�N. NxI˝1/

�ı\

�N. NxI˝2/

�ı

D T . NxI˝1/ \ T . NxI˝2/:

Exercise 3.18. Consider the set ˝ WD f.x; y/ 2 R2j y � x2g.

Exercise 3.19. Fix any a; b 2 Rn with a ¤ b. It follows from eorem 3.20 that there existsc 2 .a; b/ such that

f .b/ � f .a/ D hv; b � ai for some c 2 @f .c/:

en jf .b/ � f .a/j � kvk � kb � ak � `kb � ak.

Exercise 3.21. (i) F1 D f0g � RC. (ii) F1 D F . (iii) F1 D RC � RC.(iv) F1 D f.x; y/ j y � jxjg.

Exercise 3.23. (i) Apply the definition. (ii) A condition is A1 \ .�B1/ D f0g.

Exercise 3.24. Suppose that T F˝ .x/ D 0. en there exists a sequence of tk ! 0, tk � 0, with

.x C tkF / \˝ ¤ ; for every k 2 N. Choose fk 2 F and wk 2 ˝ such that x C tkfk D wk .Since F is bounded, the sequence fwkg converges to x, and so x 2 ˝ since ˝ is closed. econverse implication is trivial.

Exercise 3.25. Follow the proof of eorem 3.37 and eorem 3.38.

EXERCISES FORCHAPTER 4Exercise 4.2. e set U� D Lc

�is closed by Proposition 4.4.


Exercise 4.3. Consider the l.s.c. function f .x/ WD 0 for x ¤ 0 and f .0/ WD �1. en the productof f � f is not l.s.c.

Exercise 4.7. Solve the equation rg.x/ D 0, where rg.x/ DPm

iD1 2.x � ai /.

Exercise 4.8. Let˝ WD fx 2 Rnj Ax D bg. enN.xI˝/ D fAT� j � 2 Rp

g for x 2 ˝. Corol-lary 4.15 tells us that x 2 ˝ is an optimal solution to this problem if and only if there exists � 2 Rp

such that�2x D AT � and Ax D b:

Solving this system of equations yields x D AT .AAT /�1b.

Exercise 4.11. e MATLAB function sign x gives us a subgradient of the functionf .x/ WD kxk1 for all x 2 R. Considering the set ˝ WD fx 2 Rn

j Ax D b, we have

˘.xI˝/ D x C AT .AAT /�1.b � Ax/, which can be found by minimizing the quadraticfunction kx � uk2 over u 2 ˝; see Exercise 4.8.

Exercise 4.12. If 0 2 @f .xk0/ for some k0 2 N, then ˛k0

D 0, and hence xk D xk02 S and

f .xk/ D V for all k � k0. us we assume without loss of generality that 0 … @f .xk/ as k 2 N.It follows from the proof of Proposition 4.23 that

0 � kxkC1 � Nxk2

� kx1 � Nxk2

� 2

kXiD1

˛i Œf .xi / � V �C

kXiD1

˛2i kvik

2

for any Nx 2 S . Substituting here the formula for ˛k gives us

0 � kxkC1 � Nxk2

� kx1 � Nxk2

� 2

kXiD1

Œf .xi / � V �2

kvik2

C

kXiD1

Œf .xi / � V �2

kvik2

D kx1 � Nxk2

�

kXiD1

Œf .xi / � V �2

kvik2

and

kXiD1

Œf .xi / � V �2

kvik2

� kx1 � Nxk2:

Since kvik � ` for every i , we getkX

iD1

Œf .xi / � V �2

`2� kx1 � Nxk

2;

which implies in turn thatkX

iD1

Œf .xi / � V �2 � `2kx1 � Nxk

2: (4.66)


en the seriesP1

kD1Œf .xk/ � V �2 is convergent. Hence f .xk/ ! V as k ! 1, which justifies(i). From (4.66) and the fact that Vk � f .xj / for i 2 f1; : : : ; kg it follows that

k.Vk � V /2 � `2kx1 � Nxk

2:

us 0 � Vk � V �`kx1 � Nxk

pk

. Since Nx was taken arbitrarily in S , we arrive at (ii).

Exercise 4.17. Define the MATLAB function

f .x/ WD

PmiD1

ai

kx � aikPmiD1

1

kx � aik

; x … fa1; : : : ; amg

and f .x/ WD x for x 2 fa1; : : : ; amg. Use the FOR loop with a stopping criteria to findxkC1 WD f .xk/, where x0 is a starting point.

Exercise 4.18. Use the subdifferential Fermat rule. Consider the cases where m is odd and wherem is even. In the first case the problem has a unique optimal solution. In the second case theproblem has infinitely many solutions.

Exercise 4.22. For the interval˝ WD Œa; b� � R, the subdifferential formula for the distance func-tion 2.39 gives us

@d. NxI˝/ D

8ˆ<ˆ:

f0g if Nx 2 .a; b/;

Œ0; 1� if Nx D b;

f1g if Nx > b;

Œ�1; 0� if Nx D a;

f�1g if Nx < b:

Exercise 4.23. For ˝ WD IB.cI r/ � Rn, a subgradient of the function d.xI˝/ at Nx is

v. Nx/ WD

8<:0 if Nx 2 ˝;Nx � c

k Nx � ckif Nx … ˝:

en a subgradient of the function f can be found by the subdifferential sum rule of Corol-lary 2.46. Note that the qualification condition (2.29) holds automatically in this case.

Exercise 4.27. Use the subgradient formula in the hint of Exercise 4.23 together with Propo-sition 2.54. Note that we can easily find an element i 2 I. Nx/, and then any subgradient of thefunction fi .x/ WD d.xI˝i / at Nx belongs to @f . Nx/.

195

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Authors’ Biographies

BORISMORDUKHOVICHBorisMordukhovich is Distinguished University Professor of Mathematics at Wayne State Uni-versity holding also Chair Professorship of Mathematics and Statistics at the King Fahd Univer-sity of Petroleum and Minerals. He has more than 350 publications including several mono-graphs. Among his best known achievements are the introduction and development of powerfulconstructions of generalized differentiation and their applications to broad classes of problems invariational analysis, optimization, equilibrium, control, economics, engineering, and other fields.Mordukhovich is a SIAM Fellow, an AMS Fellow, and a recipient of many international awardsand honors including Doctor Honoris Causa degrees from six universities over the world. He isnamed a Highly Cited Researcher inMathematics by the Institute of Scientific Information (ISI).His research has been supported by continued grants from the National Science Foundations.Department of Mathematics, Wayne State University, Detroit, MI [email protected]. Homepage: math.wayne.edu/�boris

NGUYENMAUNAMNguyenMau Nam received his B.S. from Hue University, Vietnam, in 1998 and a Ph.D. fromWayne State University in 2007. He is currently Assistant Professor of Mathematics at PortlandState University. He has around 30 papers on convex and variational analysis, optimization, andtheir applications published in or accepted by high-level mathematical journals. His research issupported by a collaboration grant from the Simons Foundation.Fariborz Maseeh Department of Mathematics and Statistics,Portland State University, Portland, OR [email protected]. Homepage: web.pdx.edu/�mnn3

201

Index

affine combination, 22affine hull, 21affine mapping, 7affine set, 21affinely independent, 23asymptotic cone, 108

biconjugacy equality, 80Bolzano-Weierstrass theorem, 3boundary of set, 3Brouwer’s fixed point theorem, 120

Carathéodory theorem, 99closed ball, 2closure of set, 3coderivative, 70coercivity, 134compact set, 3complement of set, 2cone, 33, 97constrained optimization, 135convex combination, 5convex combinations, 99convex cone, 33convex cone generated by a set, 97convex conic hull, 97convex function, 11convex hull, 8, 99convex optimization, 135convex set, 5

derivative, 55

descent property, 151dimension of convex set, 23directional derivative, 82distance function, 28domain, 11domain of multifunction, 18

epigraph, 11, 131Euclidean projection, 146Euclidean projection to set, 29extended-real-valued function, 4

Farkas lemma, 105Farkas’ lemma, 99Fenchel conjugate, 77Fenchel-Moreau theorem, 80Fermat rule, 56Fermat-Torricelli problem, 146Fréchet differentiability, 55

GOateaux differentiability, 95generalized ball, 168generalized Fermat-Torricelli problem, 154generalized Sylvester problem, 169global/absolute minimum, 4graph of multifunction, 18

Helly’s theorem, 102horizon cone, 108, 117hypograph, 179

image, 44

202 INDEX

indicator function, 54infimal convolution, 20, 75infimum, 4interior of set, 2

Jensen inequality, 12

Karush-Kuhn-Tucker condition, 138

Lagrange function, 137level set, 34, 113, 131Lipschitz continuity, 114Lipschitz continuous functions, 49local minimum, 4locally Lipschitz functions, 49lower semicontinuous, 178lower semicontinuous function, 129

marginal function, 72maximum function, 14, 86mean value theorem, 17, 106minimal time function, 111Minkowski gauge, 111, 113, 156mixed strategies, 120monotone mappings, 125Moreau-Rockafellar theorem, 63

Nash equilibrium, 120noncooperative games, 120normal cone, 42, 117

open ball, 2optimal value function, 70optimal value/marginal function, 18

parallel subspace, 22polar of a set, 103positive semidefinite matrix, 12

positively homogeneous functions, 34projected subgradient method, 145

Radon’s theorem, 102relative interior, 25

set separation, 39, 40set separation, strict, 37set-valued mapping/multifunction, 18simplex, 24singular subdifferential, 49Slater constraint qualification, 138smallest enclosing circle problem, 168span, 23strictly convex, 147, 155strictly convex functions, 170strictly convex sets, 154strictly monotone mappings, 125strongly convex, 124strongly monotone mappings, 125subadditive functions, 34subdifferential, 53, 117subgradient, 53subgradient algorithm, 141subgradients, 53, 117subspace, 22support function, 117support functions, 75supremum, 4supremum function, 16

tangent cone, 104two-person games, 120

upper semicontinuous function, 87

Weierstrass existence theorem, 5Weiszfeld algorithm, 150, 181

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