global optimality conditions for nonlinear programming problems with bounds via quadratic...

This article was downloaded by: [University of Windsor]On: 10 November 2014, At: 17:49Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Optimization: A Journal ofMathematical Programming andOperations ResearchPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gopt20

Global optimality conditions fornonlinear programming problems withbounds via quadratic underestimatorsV. Jeyakumar a & N.Q. Huy ba Department of Applied Mathematics , University of New SouthWales , Sydney, 2052, Australiab Department of Mathematics , Hanoi Pedagogical University ,Vinh Phuc, VietnamPublished online: 03 Aug 2009.

To cite this article: V. Jeyakumar & N.Q. Huy (2010) Global optimality conditions fornonlinear programming problems with bounds via quadratic underestimators, Optimization:A Journal of Mathematical Programming and Operations Research, 59:2, 161-173, DOI:10.1080/02331930801951199

To link to this article: http://dx.doi.org/10.1080/02331930801951199

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

http://www.tandfonline.com/loi/gopt20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/02331930801951199

http://dx.doi.org/10.1080/02331930801951199

Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

OptimizationVol. 59, No. 2, February 2010, 161–173

Global optimality conditions for nonlinear programming problems with

bounds via quadratic underestimators

V. Jeyakumara* and N.Q. Huyb

aDepartment of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia;bDepartment of Mathematics, Hanoi Pedagogical University, Vinh Phuc, Vietnam

(Received 14 June 2006; final version received 22 November 2007)

In this article we develop conditions for feasible points to be global minimizers ofnonlinear programming problems with bounds on the variables. We obtainsufficient conditions for global optimality, first by constructing quadraticunderestimators of the objective function and then by deriving conditions forglobal minimizers of underestimators. We also apply this method to optimizationproblems with discrete variables and obtain global optimality conditions fornonlinear programming problems with discrete variables. Numerical examples arediscussed to illustrate the optimality conditions.

Keywords: smooth nonlinear programming problems; global optimization;optimality conditions; box constraints; discrete constraints

Mathematics Subject Classifications 2000: 90C30; 90C46

1. Introduction

In recent years, a great deal of attention has been focused on developing global optimalityconditions for nonconvex quadratic programming problems [1,2–6,8,10–12]. A completecharacterization of global optimality has also been given for box-constrained weightedleast squares problems in [9]. More recently, sufficient optimality conditions have beenpresented for smooth (not necessarily quadratic) minimization problems with linear matrixinequality constraints and bounds on the variables in [7].

The purpose of this work is to develop conditions for a feasible point (or a stationarypoint), to be a global minimizer of a general smooth mathematical programming modelproblem,

ðNLPÞ minx2Rn

fðxÞ

s:t: giðxÞ � 0, i 2M ¼ f1, . . . ,mg

hjðxÞ ¼ 0, j 2 K ¼ f1, . . . , kg

x 2Yni¼1

½ui, vi�,

*Corresponding author. Email: [email protected]

ISSN 0233–1934 print/ISSN 1029–4945 online

� 2010 Taylor & Francis

DOI: 10.1080/02331930801951199

http://www.informaworld.com

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

where f, gi and hj are twice continuously differentiable functions on an open subset of Rn

containing D :¼Qn

i¼1½ui, vi�, ui, vi 2 R , i ¼ 1, 2, . . . , n. Model problems of the form (NLP)cover large classes of nonconvex continuous optimization problems. Identifying andlocating global minimizers of NLPs are critical issues of global optimization. However, dueto the presence of nonconvex functions, such NLPs generally posses many local minimizerswhich are not global minimizers, and are inherently difficult global optimization problems.

In this article, we derive sufficient conditions for global optimality of (NLP). Thesufficient conditions are also expressed in terms of Kuhn–Tucker conditions and certaineigenvalue conditions on the Hessians of the functions involved. We obtain theseconditions by way of constructing quadratic underestimators of the objective function of(NLP) and then deriving conditions for global minimizers of underestimators. We alsoapply this method to optimization problems with discrete variables, and derive optimalityconditions for nonlinear programming problems with discrete variables. We illustrate byvarious examples how a global minimizer can be identified using our sufficient conditionsand local optimality conditions.

The layout of the article is as follows. Section 2 develops nonconvex quadraticunderestimators for (NLP) and derives sufficient global optimality conditions for (NLP).Section 3 presents global optimality conditions for nonlinear programming problems withdiscrete variables. Section 4 concludes by summarizing the approach and the results and bypresenting a discussion on further work.

2. Sufficient global optimality conditions

In this section, we derive sufficient global optimality conditions for nonlinearprogramming problems with box constraints. We begin by presenting basic definitionsand notations that will be used throughout the article. The real line is denoted by R andthe n-dimensional Euclidean space is denoted by R

n. The identity matrix is denoted by I.We first consider minimizing a smooth function over a box

ðP1Þ minx2Rn

fðxÞ

s:t: x 2 D :¼Yni¼1

½ui, vi�,

where ui, vi 2 R , i ¼ 1, . . . , n and f : Rn! R is a twice continuously differentiable function

on an open set containing D. For each �x¼ ( �x1, . . . , �x2)T2D, the gradient and Hessian of

f are denoted by rf( �x) and r2f( �x) respectively.

Definition 2.1 The function h : Rn! R is a quadratic underestimator of the objective

function f at �x over D, if h is a quadratic function, and, for each x2D, f(x)� h(x) andf( �x)¼ h( �x).

LEMMA 2.1 Let �x 2 D :¼Qn

i¼1½ui, vi�, ui, vi 2 R , i ¼ 1, . . . , n and let f be a twice continu-ously differentiable function on an open set containing D. Suppose that there exists a diagonalmatrix G such that, for each x2D, (r2f(x)�G) is positive semidefinite. LetlðxÞ ¼ ð1=2ÞxTGxþ ðrfð �xÞ � G �xÞTx, x 2 R

n. Then, the function h : Rn! R defined by

hðxÞ ¼ lðxÞ � lð �xÞ þ fð �xÞ,

is a quadratic underestimator of f at �x over D.

162 V. Jeyakumar and N.Q. Huy

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

Proof Let ’(x) :¼ f(x)� l(x), x2D. Then r’( �x)¼ 0 and r2’(x)¼r2f(x)�G, for each

x2D. So, ’(x)� ’( �x)� 0 for all x2D. Therefore, f(x)� f( �x)� l(x)� l( �x) for all x2D.

Thus, h is a quadratic underestimator of f at �x over D. g

Denote the eigenvalues of r2f( �x) by �r(x), r¼ 1, 2, . . . , n. Let

� :¼ minx2D

minr2f1, 2,..., ng

�rðxÞ: ð2:1Þ

In the case where f is a quadratic function with the constant Hessian A, � denotes the least

eigenvalue of A.In passing observe that � is often used as a ‘convexifier’ for ‘convexifying’ a twice

continuously differentiable function by a quadratic term as it was shown in [13]. In the

following lemma we use � to obtain a quadratic underestimator for a twice continuously

differentiable function.

LEMMA 2.2 Let �x2D let f be a twice continuously differentiable function on an open set

containing D. Let �� and let lðxÞ ¼ ð1=2Þ�xTxþ ðrfð �xÞ � G �xÞTx, x 2 Rn. Then, the

function h : Rn! R defined by

hðxÞ ¼ lðxÞ � lð �xÞ þ fð �xÞ, ð2:2Þ

is a quadratic underestimator of f at �x over D.

Proof Let G¼ �I. Then, for each x2D, the eigenvalues of r2f(x)�G are

non-negative, and so, r2f(x)�G is positive semidefinite. The conclusion now follows

from Lemma 2.1. g

Note that the underestimators developed here differ from the one employed in [7],

where diagonally dominant matrices played a key role.For a real number a, aþ is defined by aþ¼maxf0,�a}. Let �x¼ ( �x1, . . . , �xn)

T2D.

Define

exi ¼ �1 if �xi ¼ ui

1 if �xi ¼ vi ð2:3Þ

ðrfð �xÞÞi if �xi 2 ðui, viÞ: ð2:4Þ

8><>:THEOREM 2.1 For (P1), let �x 2 D :¼

Qni¼1½ui, vi�. Let ��. If

1

2�þðvi � uiÞ þexiðrfð �xÞÞi � 0, 8i 2 f1, 2, . . . , ng ð2:5Þ

then �x is a global minimizer of (P1).

Proof Let lðxÞ ¼ ð1=2Þ�xTxþ ðrfð �xÞ � G �xÞTx, x2D, where G¼ �I. The conclusion will

follow from Lemma 2.2, if we show that �x is a global minimizer of l( � ) over D. The point �x

is a global minimizer of l( � ) over D if and only if, for each x2D,

lðxÞ � lð �xÞ ¼1

2

Xni¼1

�x2i þXni¼1

ðrfð �xÞ � � �xÞixi �1

2

Xni¼1

� �x2i þXni¼1

ðrfð �xÞ � � �xÞi �xi

" #

¼Xni¼1

�

2ðxi � �xiÞ

2þXni¼1

rfð �xÞiðxi � �xiÞ

� 0:

Optimization 163

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

Thus, �x is a global minimizer of l( � ) if, and only if, for each xi2 [ui, vi], i¼ 1, . . . , n,

�

2ðxi � �xiÞ

2þ rfð �xÞiðxi � �xiÞ � 0: ð2:6Þ

We now establish simple conditions which are equivalent to Equation (2.6) by

considering the following three cases:

Case 1 �xi¼ ui. Then (2.6) holds if and only if,

�

2ðxi � �xiÞ þ rfð �xÞi � 0: ð2:7Þ

Subcase 1.1 �� 0. Then (2.7) holds if and only if, rf( �x)i� 0. Indeed, if rf( �x)i50,

by taking xi sufficiently close to �xi, we obtain that ð�=2Þðxi � �xiÞ þ rfð �xÞi 5 0. This

contradicts (2.7). The converse implication is immediate.

Subcase 1.2 �50. Then we show that (2.7) holds if and only if,

�

2ðvi � uiÞ þ rfð �xÞi � 0: ð2:8Þ

Clearly, (2.8) implies (2.7). To see the converse implication, suppose that (2.8) is false.

Then,

�

2ðvi � uiÞ þ rfð �xÞi < 0:

Choosing xi sufficiently close to vi, we have ð�=2Þðxi � �xiÞ þ rfð �xÞi 5 0, which is

impossible.

Case 2 �xi¼ vi. Then (2.6) holds if and only if,

�

2ðxi � �xiÞ þ rfð �xÞi � 0: ð2:9Þ

If �� 0 then, clearly, (2.9) holds if and only if, rf( �x)i� 0. If �50, then we see that (2.9)

holds if and only if,

�

2ðui � viÞ þ rfð �xÞi � 0: ð2:10Þ

Indeed, if (2.10) is false then, whenever xi is sufficiently close to ui, we have ð�=2Þ�ðxi � �xiÞ þ rfð �xÞi 4 0. This contradicts (2.9). The converse implication is immediate.

Case 3 �xi2 (ui, vi). If (2.6) holds then rf( �x)i¼ 0, and hence �� 0. To see this, we assume,

without loss of generality, that rf( �x)i40. Then, whenever xi is sufficiently close to �xiand xi5 �xi, we have ð�=2Þðxi � �xiÞ

2þ rfð �xÞiðxi � �xiÞ < 0, which contradicts (2.6). On the

other hand, it follows easily that for each i¼ 1, . . . , n, (2.6) holds whenever rf( �x)i¼ 0

and �� 0.

Using these three cases, we obtain that if (2.5) holds then (2.6) holds for each

xi2 [ui, vi], i¼ 1, 2, . . . , n, and hence �x is a global minimizer of (P1). g

In the special case where the function f is quadratic, i.e. f(x)¼ xTAxþ aTx,A is

a symmetric n� n matrix and a 2 Rn, we have the following result.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

COROLLARY 2.1 For (P1), let f(x)¼ xTAxþ aTx and let �x 2 D :¼Qn

i¼1½ui, vi�. Let � be the

least eigenvalue of A. If

1

2�þðvi � uiÞ þexiðaþ A �xÞi � 0, 8i 2 f1, 2, . . . , ng


Proof The conclusion easily follows from Theorem 2.1 by taking f(x)¼ xTAxþ aTx and

by noting that rf(x)¼Axþ a.

Recall that a point �x2D is a stationary point of (P1) if rf( �x)T (x� �x)� 0, 8 x2D,

which is equivalent to the condition that exiðrfð �xÞÞi � 0, 8i ¼ 1, 2, . . . , n. We now deduce

sufficient optimality conditions for a stationary point of (P1) to be a global minimizer.

COROLLARY 2.2 For (P1), let �x 2 D :¼Qn

i¼1½ui, vi� be a stationary point of (P1). Let ��.If

1

2�ðvi � uiÞ �exiðrfð �xÞÞi � 0, 8i 2 f1, 2, . . . , ng: ð2:11Þ


Proof Condition (2.11) implies that

1

2ð��Þðvi � uiÞ þexiðrfð �xÞÞi � 0, 8i 2 f1, 2, . . . , ng:

The stationary point condition gives us that

exiðrfð �xÞÞi � 0, 8i ¼ 1, 2, . . . , n:

The above two conditions together ensure that (2.5) holds. Hence the conclusion follows

from Theorem 2.1. g

Now, consider the smooth nonconvex programming problems with box constraints,

discussed in Section 1.

ðNLPÞ minx2Rn

fðxÞ

s:t: giðxÞ � 0, i 2M ¼ f1, . . . ,mg

hjðxÞ ¼ 0, j 2 K ¼ f1, . . . , kg

x 2Yni¼1

½ui, vi�:

Let

� :¼ fx 2 Rnj giðxÞ � 0, hjðxÞ ¼ 0, i 2M, j 2 Kg:

For each i2M and j2K, define

�i ¼ minx2D

minr2f1, 2,..., ng

�irðxÞ, ð2:12Þ

�j ¼ minx2D

minr2f1, 2,..., ng

�jrðxÞ, ð2:13Þ

Optimization 165

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

Gi ¼ �iI, Hj ¼ �jI ð2:14Þ

Q ¼ GþXi2M

�iGi þXj2K

�jHj, ð2:15Þ

where �ir(x) and �jr(x) are the eigenvalues of r2gi(x) and r

2hj(x), respectively. Let �x 2 Rn,

� 2 Rm and � 2 R

k. Define

bxi ¼�1 if �xi ¼ ui

1 if �xi ¼ vi ð2:16Þ

ðrfð �xÞ þPi2M

�irgið �xÞ þPj2K

�jrhjð �xÞÞi if �xi 2 ðui, viÞ; ð2:17Þ

8>><>>:� ¼ �þ

Xi2M

�i�i þXj2K

�j�j: ð2:18Þ

THEOREM 2.2 For (NLP), let �x2�\D. Suppose that there exist � 2 Rmþ and � 2 R

k such

thatP

i2M �igið �xÞ ¼ 0. Let �� . If

1

2�þðvi � uiÞ þbxiðrfð �xÞ þX

i2M

�irgið �xÞ þXj2K

�jrhjð �xÞÞi � 0, 8i 2 f1, 2, . . . , ng ð2:19Þ

then �x is a global minimizer of (NLP).

Proof Let

LðxÞ :¼ fðxÞ þXi2M

�igiðxÞ þXj2K

�jhjðxÞ,

pðxÞ :¼1

2xTQxþ ðrfð �xÞ þ

Xi2M


�jrhjð �xÞ �Q �xÞTx,

and

’ðxÞ :¼ LðxÞ � pðxÞ, x 2 � \D:

As it was shown in Lemma 2.2, it is easy to check that the function, q, defined by

qðxÞ ¼ pðxÞ � pð �xÞ þ Lð �xÞ,

is an underestimator of L at �x over �\D, since r2’(x) is positive semidefinite for each

x2D and r’( �x)¼ 0. Consequently,

fðxÞ � fðxÞ þXi2M

�igiðxÞ þXj2K

�jhjðxÞ

¼ fðxÞ þXi2M

�igiðxÞ þXj2K

�jhjðxÞ � fð �xÞ þXi2M

�igið �xÞ þXj2K

�jhjð �xÞ

!þ fð �xÞ

¼ LðxÞ � Lð �xÞ þ fð �xÞ

� qðxÞ � qð �xÞ þ fð �xÞ

¼ pðxÞ � pð �xÞ þ fð �xÞ, 8x 2 � \D:


Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

Thus, p( � )� p(x)þ f( �x) is an underestimator of f at �x over �\D. The conclu-

sion will follow if we show that �x is a global minimizer of p( � ) over D. Note that

for each x2D,

pðxÞ � pð �xÞ ¼1

2

Xni¼1

�x2i þXni¼1

rfð �xÞ þXi2M


�jrhjð �xÞ � � �x

!i

xi

�1

2

Xni¼1

� �x2i þXni¼1

rfð �xÞ þXi2M


�jrhjð �xÞ � � �x

!i

�xi

" #

¼Xni¼1

�

2ðxi � �xiÞ

2þXni¼1

rfð �xÞ þXi2M


�jrhjð �xÞ

!i

ðxi � �xiÞ

�Xni¼1

�

2ðxi � �xiÞ

2þXni¼1

rfð �xÞ þXi2M


�jrhjð �xÞ

!i

ðxi � �xiÞ:

So, �x is a global minimizer of p( � ) over D if, for each x2D,

Xni¼1

�

2ðxi � �xiÞ

2þXni¼1

rfð �xÞ þXi2M


�jrhjð �xÞ

!i

ðxi � �xiÞ � 0,

which means that, for each i¼ 1, . . . , n, xi2 [ui, vi],

�

2ðxi � �xiÞ

2þ rfð �xÞ þ

Xi2M


�jrhjð �xÞ

!i

ðxi � �xiÞ � 0: ð2:20Þ

The equivalence between (2.20) and (2.19) follows by considering the three simple cases as

in the proof of Theorem 2.1, where �xi¼ ui, �xi¼ vi and �xi2 (ui, vi). g

Example 2.1 Consider the following nonlinear programming problem

ðE2Þ minx2R3

fðxÞ ¼ x31 þ x32 þ x23 � 30x3

s:t: g1ðxÞ ¼ x31 þ x33 þ 16 � 0

h1ðxÞ ¼ x1 þ x2 þ x3 þ 4 ¼ 0

x 2 D :¼ ½�2, 0� � ½�2, 0� � ½�2, 0�:

Let �x¼ (�2,�2, 0)2�\D. Then

rfð �xÞ ¼ ð12, 12,�30ÞT, rgð �xÞ ¼ ð12, 0, 0ÞT, rhð �xÞ ¼ ð1, 1, 1ÞT,

� :¼ minx2D

mini2f1, 2, 3g

�iðr2fðxÞÞ ¼ �12,

�1 :¼ minx2D

mini2f1, 2, 3g

�iðr2g1ðxÞÞ ¼ �12,

�1 :¼ minx2D

mini2f1, 2, 3g

�iðr2h1ðxÞÞ ¼ 0:

Taking �1¼ 0 and �1¼ 1, we have �1g1( �x)¼ 0, �¼�12. It is easy to check that (2.19) holds

for (E2) with �¼ � ¼�12. Clearly, �x¼ (�2,�2, 0) is a global minimizer of (E2).

Optimization 167

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

Recall that if �x2�\D is a local minimizer of (NLP) then the following Kuhn–Tuckerconditions hold under suitable constraint qualifications:

rfð �xÞ þXi2M


�jrhjð �xÞ

!T

ðx� �xÞ � 0, 8x 2 D: ð2:21Þ

for some � 2 Rmþ and � 2 R

k such thatP

i2M �igið �xÞ ¼ 0. This condition (2.21) can also beexpressed as

bxi rfð �xÞ þXi2M


�jrhjð �xÞ

!i

� 0, 8i 2 f1, 2, . . . , ng

for some � 2 Rmþ and � 2 R

k such thatP

i2M �igið �xÞ ¼ 0. We now deduce sufficientoptimality conditions for (NLP) in terms of Kuhn-Tucker optimality conditions.

COROLLARY 2.3 For (NLP), let �x2�\D. Suppose that Kuhn-Tucker conditions that thereexist � 2 R

mþ and � 2 R

k such thatP

i2M �igið �xÞ ¼ 0 and bxiðrfð �xÞ þPi2M �irgið �xÞ þPj2K �jrhjð �xÞÞi � 0, 8i 2 f1, 2, . . . , ng hold at �x. If

1

2�ðvi � uiÞ �bxi rfð �xÞ þX

i2M


�jrhjð �xÞ

!i

� 0, 8i 2 f1, 2, . . . , ng ð2:22Þ

then �x is a global minimizer of (NLP).

Proof As in the proof of Corollary 2.2, the Kuhn–Tucker conditions at �x togetherwith (2.22) ensure that (2.19) holds at �x. Hence the conclusion follows fromTheorem 2.2. g

The following example shows how Corollary 2.3 can be used to identify a globalminimizer of a multi-extremal nonlinear programming problem using local optimalityconditions.


ðE3Þ minx2R2

fðxÞ ¼ ðx1 � 1Þ3 � 5x22 þ x2

s:t: g1ðxÞ ¼ x21 þ x22 � 9 � 0

h1ðxÞ ¼ x1 þ 3x2 þ 4 ¼ 0

x 2 D :¼ ½�1, 1� � ½�1, 1�:

It is easy to check that the Kuhn–Tucker conditions are satisfied at the localminimizers (�1,�1) and (�1, 1). Let �x¼ (�1,�1). Then

rfð �xÞ ¼ ð12, 9ÞT, rgð �xÞ ¼ ð�2,�2ÞT, rhð �xÞ ¼ ð1, 3ÞT,

� :¼ minx2D

mini2f1, 2g

�iðr2fðxÞÞ ¼ �12,

�1 :¼ minx2D

mini2f1, 2g

�iðr2g1ðxÞÞ ¼ 2,

�1 :¼ minx2D

mini2f1, 2g


Then the Kuhn–Tucker conditions hold with �1¼ 0 and �1¼ 1. Moreover �¼�12.Now clearly (2.22) holds at �x and it is in fact a global minimizer of (E3).


Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

Consider the following quadratic programming problems with boundson the variables,

ðQP2Þ minx2Rn

1

2xTAxþ aTx

s:t:1

2xTBixþ bTi xþ si � 0, i 2M ¼ f1, . . . ,mg

1

2xTCjxþ cTj xþ tj ¼ 0, j 2 K ¼ f1, . . . , kg

x 2Yni¼1

½ui, vi�,

ð2:23Þ

where A, B and C are symmetric n� n matrices, a, bi, cj 2 Rn and si, tj 2 R , for each i2M

and j2K. For each i2M and j2K, the smallest eigenvalues of A,Bi and Cj are denoted by

�,�i and �j, respestively. Let

G :¼ �I, Gi :¼ �iI, Hj :¼ �jI, ð2:24Þ

� :¼ �þXi2M

�i�i þXj2K

�j�j: ð2:25Þ

As an application of Theorem 2.2 we obtain sufficient conditions for global optimality of

(QP2).

COROLLARY 2.4 For (QP2), let �x2�\D. If there exist � 2 Rmþ and � 2 R

k such thatPi2M �iðð1=2Þ �x

TBi �xþ bTi �xþ siÞ ¼ 0 and for each i2 f1, 2, . . . , n},

1

2�þðvi � uiÞ þbxi Aþ

Xi2M

�iBi þXj2K

�jCj

!�xþ aþ

Xi2M

�ibi þXj2K

�jcj

!i

� 0, ð2:26Þ

then �x is a global minimizer of (QP2).

Proof The conclusion follows by applying Theorem 2.2 to (QP2), where fðxÞ ¼

ð1=2ÞxTAxþ aTx, giðxÞ ¼ ð1=2ÞxTBixþ bTi xþ si and hjðxÞ ¼ ð1=2Þx

TCjxþ cTj xþ tj. g

3. Applications to NLPs with discrete variables

In this section, we will apply the technique, described in Section 2, to smooth

nonlinear programming problems with discrete constraints. We first consider the following

problem,

ðBP1Þ minx2Rn

fðxÞ

s:t: x 2 C :¼Yni¼1

fui, vig,

where ui, vi 2 R , i¼ 1, . . . , n and f : Rn! R is a twice continuously differentiable function

on an open set containing C. Let �x¼ ( �x1, . . . , �xn)T2C, and define

�xi ¼�1 if �xi ¼ ui

1 if �xi ¼ vi:

�ð3:0Þ

Optimization 169

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

THEOREM 3.1 For (BP1), let �x 2 C :¼Qn

i¼1fui, vig. Let ��. If

�xiðrfð �xÞÞi �1

2�ðvi � uiÞ � 0, 8i 2 f1, 2, . . . , ng ð3:1Þ

then �x is a global minimizer of (BP1).

Proof Let lðxÞ ¼ ð1=2Þ�xTxþ ðrfð �xÞ � G �xÞTx, x2C. The conclusion will follow from

Lemma 2.2, if we show that �x is a global minimizer of l( � ) over C, that is,

lðxÞ � lð �xÞ ¼Xni¼1

�

2ðxi � �xiÞ

2þXni¼1

ðrfð �xÞÞiðxi � �xiÞ � 0, 8x 2 C:

Equivalently,

�

2ðxi � �xiÞ þ ðrfð �xÞÞi

h iðxi � �xiÞ � 0, 8xi 2 fui, vig: ð3:2Þ

Consider the following two cases:

Case 1 �x¼ ui. Then (3.2) holds if and only if,

�

2ðvi � uiÞ þ ðrfð �xÞÞi � 0:

Case 2. �x¼ vi. Then (3.2) holds if and only if,

�

2ðui � viÞ þ ðrfð �xÞÞi � 0:

Combining these cases, we see that �x is a global minimizer of l( � ) over C, if and only if for

each i¼ 1, . . . , n, �xiðrfð �xÞÞi � ð1=2Þ�ðvi � uiÞ � 0. Thus �x is a global minimizer of (BP1)

whenever (3.1) holds. g

Let us now consider the following nonlinear programming problem with discrete

variables,

ðBP2Þ minx2Rn

fðxÞ

s:t: giðxÞ � 0, i 2M

hjðxÞ ¼ 0, j 2 K

x 2 C ¼Yni¼1

fui, vig,

where f, gi and hj are twice continuously differentiable functions on an open set containing

C in Rn. Let x̌i and � be defined by (3.1) and (2.17), respectively.

THEOREM 3.2 For (BP2), let �x2�\C. Suppose that there exist � 2 Rmþ and � 2 R

k such

thatP

i2M �igið �xÞ ¼ 0. Let �� . If, for each i2 f1, 2, . . . , n},

�xi rfð �xÞ þXi2M


�jrhjð �xÞ

!i

�1

2�ðvi � uiÞ � 0 ð3:3Þ

then �x is a global minimizer of (BP2).


Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

Proof The method of proof is the same as in Theorems 2.1 and 3.1, and so

is omitted. g


ðE2Þ minx2R3

fðxÞ ¼1

2x21 þ

1

2x22 þ x33 � x1 � x2 þ 1

s:t: g1ðxÞ ¼ �x21 � x22 þ x23 þ 2 � 0

h1ðxÞ ¼ x21 þ x22 þ x23 � 2 ¼ 0

x 2 C :¼ f0, 1g � f0, 1g � f0, 1g:

Let �x¼ (1, 1, 0)2�\C and D :¼ [0, 1]� [0, 1]� [0, 1]. Then,

rfð �xÞ ¼ ð0, 0, 0ÞT, rgð �xÞ ¼ ð�2,� 2, 0ÞT, rhð �xÞ ¼ ð2, 2, 0ÞT,

� :¼ minx2D

mini2f1, 2, 3g

�iðr2fðxÞÞ ¼ 0,

�1 :¼ minx2D

mini2f1, 2, 3g

�iðr2g1ðxÞÞ ¼ �2,

�1 :¼ minx2D

mini2f1, 2, 3g


Taking �1¼ 1 and �1¼ 1, we have �1g1( �x)¼ 0 and �¼ 0. It is easy to verify that (3.3) holds

for (E2) with �¼ � ¼ 0, and �x¼ (1,1,0) is a global minimizer of (E2).Consider the following bivalent quadratic minimization problem with quadratic

constraints,

ðBQP2Þ minx2Rn

1

2xTAxþ aTx

s:t:1

2xTBixþ bTi xþ si � 0, i 2M ¼ f1, . . . ,mg

1

2xTCjxþ cTj xþ cj ¼ 0, j 2 K ¼ f1, . . . , kg

x 2Yni¼1

f�1, 1g:

COROLLARY 3.1 For (BQP2), let �x2�\C. If there exist � 2 Rmþ and � 2 R

k such thatPi2M �iðð1=2Þ �x

TBi �xþ bTi �xþ siÞ ¼ 0 and for each i2 f1, 2, . . . , n},

�xi AþXi2M

�iBi þXj2K

�jCj

!�xþ aþ

Xi2M

�ibi þXj2K

�jcj

!i

�� 0, ð3:4Þ

then �x is a global minimizer of (BQP2).

Proof The method of proof is the same as in Theorem 2.1 and so is omitted. g

4. Conclusions

In this article, we presented sufficient conditions for global optimality of the nonlinear

programming model problem (NLP) with bounds on the variables. We derived these

Optimization 171

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

conditions by way of constructing quadratic underestimators of the objective functionof (NLP) and then finding conditions for global minimizers of the underestimators.We illustrated by various examples, how a global minimizer can be identified using oursufficient conditions and local optimality conditions. We finally applied our methods tooptimization problems with discrete variables and derived global optimality conditions fornonlinear programming problems with discrete variables.

We expressed our sufficient optimality conditions for (NLP) in terms of eigenvalueconditions (such as (2.1), (2.12) and (2.13)) and Kuhn–Tucker conditions. The eigenvalueconditions involve minimizing eigenvalue functions subject to bounds on the variables,and provide the least eigenvalues for quadratic programming problems. However,verifying the eigenvalue conditions for general nonlinear programming problems (NLP)with non-quadratic functions presents an interesting problem for further research onthis topic.

Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions whichhave contributed to the final preparation of the article. The authors are also thankful toDr S. Srisatkunarajah, University of New South Wales, for his comments and suggestionsfor improvements to the final version. The work was partially supported by the AustralianResearch Council Discovery Project Grant. The work of N. Q. Huy was completed while hewas at UNSW.

References

[1] A. Beck and M. Teboulle, Global optimality conditions for quadratic optimization problems with

binary constraints, SIAM J. Optim. 11 (2000), pp. 179–188.

[2] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms

and Engineering Applications, SIAM-MPS, Philadelphia, 2000.

[3] C.A. Floudas and P.M. Pardalos, Optimization in Computational Chemistry and Molecular

Biology: Local and Global Approaches, Kluwer Academic Publishers, New York, 2000.

[4] C.A. Floudas and V. Visweswaran, Quadratic optimization, inHandbook of Global Optimization,

R. Horst and P.M. Pardalos, eds., Kluwer Academic Publishers, The Netherlands, 1995,

pp. 217–269.[5] J.B. Hiriart-Urruty, Global optimality conditions in maximizing a convex quadratic function under

convex quadratic constraints, J. Global Optim. 21 (2001), pp. 445–455.[6] J.B. Hiriart-Urruty, Conditions for global optimality 2, J. Global Optim. 13 (1998), pp. 349–367.[7] N.Q. Huy, V. Jeyakumar, and G.M. Lee, Sufficient global optimality conditions for multi-

extremal smooth minimization problems with bounds and linear matrix inequality constraints,

ANZIAM J. 47 (2006), pp. 439–450.[8] V. Jeyakumar and Z.Y. Wu, Conditions for global optimaliy of quadratic minimization problems

with LMI and bound constraints, Asia-Pac. J. Oper. Res. 24 (2007), pp. 149–160.[9] V. Jeyakumar, A.M. Rubinov, and Z.Y. Wu, Sufficient global optimality conditions for

non-convex quadratic minimization problems with box constraints, J. Global Optim. 36 (2006),

pp. 471–481.

[10] V. Jeyakumar, A.M. Rubinov, and Z.Y. Wu, Non-convex quadratic minimization problems with

quadratic constraints: Global optimality conditions, Math. Program., Ser. A. 110 (2007),

pp. 521–541.


Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

[11] M.C. P|nar, Sufficient global optimality conditions for bivalent quadratic optimization, J. Optim.Theory Appl. 122 (2004), pp. 433–440.

[12] Z.-Y. Wu, V. Jeyakumar, and A.M. Rubinov, Sufficient conditions for global optimality ofbivalent nonconvex quadratic programs with inequality constraints, J. Optim. Theor. Appl. 133

(2007), pp. 123–130.[13] S. Zlobec, Characterization of convexifiable functions, Optimization 55 (2006), pp. 251–261.

Optimization 173

Dow

nloa

ded

by [

Uni

vers

ity o

f W

inds

or]

at 1

7:49

10

Nov

embe

r 20

14

global optimality conditions for nonlinear programming problems with bounds via quadratic...

Documents