Struct Multidisc OptimDOI 10.1007/s00158-007-0145-z

RESEARCH PAPER

Optimization under uncertainty with applicationsto design of truss structures

Giuseppe C. Calafiore · Fabrizio Dabbene

Received: 6 November 2006© Springer-Verlag 2007

Abstract Many real-world engineering design prob-lems are naturally cast in the form of optimizationprograms with uncertainty-contaminated data. In thiscontext, a reliable design must be able to cope insome way with the presence of uncertainty. In thispaper, we consider two standard philosophies forfinding optimal solutions for uncertain convex opti-mization problems. In the first approach, classical inthe stochastic optimization literature, the optimal de-sign should minimize the expected value of the ob-jective function with respect to uncertainty (averageapproach), while in the second one it should mini-mize the worst-case objective (worst-case or min–maxapproach). Both approaches are briefly reviewed inthis paper and are shown to lead to exact and nu-merically efficient solution schemes when the uncer-tainty enters the data in simple form. For generaluncertainty dependence however, these problems arenumerically hard. In this paper, we present two tech-niques based on uncertainty randomization that permitto solve efficiently some suitable probabilistic relax-ation of the indicated problems, with full generalitywith respect to the way in which the uncertainty en-ters the problem data. In the specific context of trusstopology design, uncertainty in the problem arises,

G. C. Calafiore (B)Dipartimento di Automatica e Informatica,Politecnico di Torino,Turin, Italye-mail: giuseppe.calafiore@polito.it

F. DabbeneIEIIT-CNR, Politecnico di Torino,Turin, Italye-mail: fabrizio.dabbene@polito.it

for instance, from imprecise knowledge of materialcharacteristics and/or loading configurations. In thispaper, we show how reliable structural design can beobtained using the proposed techniques based on theinterplay of convex optimization and randomization.

Keywords Uncertainty · Convex optimization ·Truss topology design · Randomized algorithms

1 Introduction

A standard convex optimization program is usuallyformulated as the problem of minimizing a convex costfunction f (x) over a convex set X , i.e.,

minx∈X

f (x).

In practice, however, the objective function f itselfmight be imprecisely known. We account for this sit-uation by assuming that function f depends not onlyon the decision vector x but also on a vector of un-certain parameters δ that are assumed to be randomwith known distribution over a compact set � ⊂ R

�.In this setting, the formulation of the optimizationproblem and the meaning of solution need be clarified.In fact, it is possible to devise different paradigms thatconsider the effect of the uncertainty from differentviewpoints. One may consider the problem from a min–max viewpoint and look for a solution that minimizesthe “worst-case” value (with respect to the uncertainty)of the objective function. Alternatively, one may beinterested in considering the effect of the uncertainty“on average,” and this corresponds to minimizing anuncertainty-averaged version of the cost function. In

G.C. Calafiore, F. Dabbene

this paper, we present these two standard philosophies,and we discuss recent techniques based either on ex-act or probabilistically approximate solutions. Exactsolutions are obtained when the uncertainty enters theobjective function in some simply structured form andare discussed in Section 2.1. In the general case, exactsolutions are numerically hard to compute. Hence, wepresent in Section 2.2 algorithms based on uncertaintyrandomization for efficient solution of probabilistic re-laxations of convex problems with generic uncertaintydependence. We refer the reader to Tempo et al. (2004)and Calafiore and Dabbene (2006) for an overview ofrandomized algorithms and their application to robust-ness problems and for pointers to the related literature.

Section 4 describes an application of the discussedtechniques to problems related to design of mechan-ical truss structures under uncertainty on the loadingpatterns and material characteristics.

2 Worst-case and average designs

We formally state the two classes of optimization prob-lems that are the object of our study.

Consider an objective function f (x, δ) : X × � → R,where X ⊆ R

n is a convex set and � ⊂ R� is a compact

set. Let P denote a probability measure on δ.We make the standing assumption that the objective

is convex in the decision variables for any fixed value ofthe uncertainty:

Assumption 1 The function f (x, δ) is convex in x forevery δ ∈ �.

2.1 Worst-case approach

In the worst-case optimization approach, one seeks asolution guaranteed for all possible values taken by theuncertainty δ ∈ �. This leads to the following min–maxoptimization problem

minx∈X

maxδ∈�

f (x, δ). (1)

When � is convex and f (x, δ) is concave in δ, prob-lem (1) is a saddle-point optimization problem that canbe solved, for instance, by means of cutting-plane tech-niques (see, e.g., Atkinson and Vaidya 1995; Goffin andVial 2002). Problem (1) can also be restated in an epi-graphic form as a robust, or semi-infinite, optimizationproblem:

mint,x∈X

t subject to

f (x, δ) ≤ t, ∀δ ∈ �. (2)

An important class of such problems involves symmet-ric linear matrix functions of x, with f being the largesteigenvalue function:

f (x, δ) = λmax (F(x, δ)) ,

F(x, δ) = F0(δ) +n∑

i=1

xi Fi(δ),

where Fi(δ) are symmetric matrices that depend ina possibly nonlinear way on the uncertainties δ. Theconstraint λmax (F(x, δ)) ≤ 0 is equivalent to requiringthat F(x, δ) is negative semidefinite, a condition that weindicate with the notation F(x, δ) � 0. For fixed δ, thisis a linear matrix inequality (LMI) condition on x. Forexamples of different problems that can be cast in LMIform and for additional details on the subject, the inter-ested reader is referred to Boyd et al. (1994) and Loboet al. (1998). Minimizing a linear objective subject to anLMI constraint is a convex problem [usually referred toas semidefinite optimization program (SDP)] that canbe solved very efficiently by means, for instance, of inte-rior point methods (see Todd 2001; Vandenberghe andBoyd 1996). In our context, however, we are interestedin problems that are subject to an infinite number ofLMI constraints, that is in problems of the form

minx∈Rn

cT x subject to

F(x, δ) � 0, ∀δ ∈ �. (3)

These problems are called robust SDPs in El Ghaouiet al. (1998) and are, in general, very hard to solvenumerically (see, e.g., Ben-Tal and Nemirovski 1998,2002). Efficiently computable and exact solutions areavailable only when the uncertainty enters the functionF in a simple form (such as affine; see, for instance,Ben-Tal and Nemirovski 1998, 2002; El Ghaoui et al.1998).

We next recall two key results that permit to recastexactly the semi-infinite optimization problem (3) inthe form of a standard semidefinite program.

Lemma 1 (Polytopic SDP) Let F(x, δ) � 0 be an LMIin x ∈ R

n, parameterized in δ ∈ Rp, which is written in

standard form as

F(x, δ).= F0(δ) +

n∑

i=1

xi Fi(δ) � 0,

where Fi(δ) = FTi (δ) are affine functions of δ. Let δ ∈ �,

where � is a polytope of vertices δ1, . . . , δv . Then

F(x, δ) � 0, ∀δ∈� ⇔ F(x, δi) � 0, i=1, . . . , v. (4)

Optimization under uncertainty

A proof of this lemma is given in the Appendix.When � is a polytope and δ enters F(x, δ) affinely, theprevious result permits to substitute the semi-infiniteconstraint F(x, δ) � 0, ∀� ∈ � with a finite number ofLMI constraints, thus transforming problem (3) into astandard and efficiently solvable SDP.

Along the same line, the next lemma provides a wayto recast a robust SDP into a standard one, when F isaffected additively by a norm-bounded uncertainty (seeEl Ghaoui et al. 1998).

Lemma 2 (Norm-bounded SDP) Let F(x, δ) have theform

F(x, δ) = F(x) + LδR(x) + RT(x)δTLT ,

where F(x) is an affine, symmetric matrix function ofx, L is a constant matrix, R(x) is affine in x, and δ ∈� with � = {δ ∈ R

p,m : ‖δ‖ ≤ 1}. Then the semi-infinitecondition on x

F(x) + LδR(x) + RT(x)δTLT � 0, ∀δ ∈ � (5)

is equivalent to the standard LMI condition on x and aslack variable τ

[F(x) + τLLT RT(x)

R(x) −τ I

]� 0. (6)

This lemma allows to replace the semi-infinite and pos-sibly uncountable set of constraints in (5)—one for eachpossible value of δ in �—with the single LMI constraint(6) expressed over an enlarged set of variables. For ad-ditional details, the reader is referred to El Ghaoui et al.(1998). Extensions of this result exist, dealing with situ-ations where the set � contains elements with block-diagonal structure, i.e., � = {δ = diag(δ1, . . . , δ�), δi ∈R

pi,mi : ‖δi‖ ≤ 1}. In this case, sufficient LMI conditionsfor the satisfaction of the semi-infinite constraint in(3) have been derived in El Ghaoui et al. (1998). InSection 4, we shall see an application of the aboveresults to problems arising in truss structure design.

2.2 Average approach

In the average approach, one aims at solving the sto-chastic program

minx∈X

Eδ[ f (x, δ)] (7)

where Eδ[ f (x, δ)] is the expectation of f (x, δ) withrespect to the probability measure P defined on δ ∈�. Problem (7) is a classical stochastic optimizationproblem, and it has been extensively treated in theliterature (see, for instance, Kushner and Yin 2003;Marti 2005; Ruszczynski and Shapiro 2003, and the

references therein). Determining an exact solution fora stochastic program is, in general, computationallyprohibitive (indeed, just evaluating the expectation fora fixed x amounts to computing a multi-dimensionalintegral, which might be numerically hard). There exist,however, simple situations where the expectation canbe computed explicitly; hence, the problem convertsto a standard convex program. For instance, a simplesituation arises when f (x, δ) is a quadratic function inδ, i.e.,

f (x, δ) = δT A(x)δ + b T(x)δ + c(x).

In this case, it is straightforward to see that the expecta-tion can be explicitly computed if the first two momentsof δ are given:

Eδ[ f (x, δ)] = tr(

A(x)Eδ[δδT ]) + b T(x)Eδ[δ] + c(x).

3 Sampling-based approximate solutions

In this section, we propose a randomized approach forapproximate solutions of both worst-case and averageproblems that cannot be solved efficiently via exactmethods.

The main idea in randomized relaxations is a simpleone: we collect a finite number N of random samples ofthe uncertainty

δ(1), δ(1), . . . , δ(N)

extracted independently according to P, and we con-struct a suitable “sampled” approximation of the prob-lems previously considered.

We next show how to construct these approxima-tions, and we highlight the theoretical properties of thesolutions obtained via the approximated problems.

3.1 Worst-case design

A sampled approximation of problem (1) can be natu-rally stated as

minx∈X

maxi=1,...,N

f (x, δ(i)).

This problem may also be rewritten in epigraph form as

minx∈X

t subject to

f (x, δ(i)) ≤ t, for i = 1, . . . , N. (8)

Notice that the possibly infinite number of constraintsof problem (2) is substituted in problem (8) by a finitenumber N of sampled scenarios of the uncertainty. Thisscenario approximation of the worst-case problem has

G.C. Calafiore, F. Dabbene

been first introduced in Calafiore and Campi (2005),where it has been shown that the solution obtainedfrom the scenario problem is actually approximatelyfeasible for the original worst-case problem, in a senseexplained next.

For any given t, we define the probability of violationof x as

PV(x, t) .= P{δ ∈ � : f (x, δ) − t > 0}.For example, if a uniform (with respect to Lebesguemeasure) probability distribution is assumed, thenPV(x, t) measures the volume of ‘bad’ parameters δ

such that the constraint f (x, δ) ≤ t is violated. Clearly,a solution x with small associated PV(x, t) is feasible formost of the problem instances; i.e., it is approximatelyfeasible for the robust problem.

Definition 1 (ε-level solution) Let ε ∈ (0, 1). We saythat x ∈ X is an ε-level robustly feasible (or, moresimply, an ε-level) solution if PV(x, t) ≤ ε.

The following theorem establishes the probabilisticproperties of the scenario solution.

Theorem 1 (Corollary 1 of Calafiore and Campi 2006)Assume that, for any extraction of δ(1),. . . , δ(N), thescenario problem (8) attains a unique optimal solutionx(N)

wc . Fix two real numbers ε ∈ (0, 1) (robustness level)and β ∈ (0, 1) (confidence parameter) and let1

N ≥ Nwc(ε, β).=

⌈2

εln

1

β+ 2(n+1) + 2(n+1)

εln

2

ε

⌉

then, with probability no smaller than (1 − β), x(N)wc is

ε-level robustly feasible.

Note that this bound is a simplification of a tighter (butmore involved) bound given in Theorem 1 of Calafioreand Campi (2006). We also remark that this result holdsfor any probability distribution P on δ; i.e., it is distrib-ution independent. In many specific problem instances,the distribution of uncertainties is known (for instance,Gaussian); hence, the random samples δ(1),. . . , δ(N) areextracted according to this distribution. If this is notthe case, a uniform distribution on � might be assumeddue to its worst-case properties (see, for instance,Barmish and Lagoa 1997); hence, δ(1),. . . , δ(N) must beextracted uniformly.

1The notation � denotes the smallest integer greater than orequal to the argument.

3.2 Average design

For notation ease, we define the function

φav(x).= Eδ[ f (x, δ)].

We denote by x∗av a minimizer of φav(x) and define the

achievable minimum as

φ∗av

.= minx∈X

φav(x) = φav(x∗av).

We state a further assumption on the function f ,namely, that the total variation of the function isbounded.

Assumption 2 Let f ∗(δ) .= minx∈X f (x, δ), and assumethat the total variation of f is bounded by a constantV > 0, i.e.,

f (x, δ) − f ∗(δ) ≤ V, ∀x ∈ X , ∀δ ∈ �.

This implies that the total variation of the expectedvalue is also bounded by V, i.e.,

φav(x) − φ∗av ≤ V, ∀x ∈ X .

Notice that we only assume that there exists a constantV such that the above holds, but we do not need toactually know its numerical value.

An approximate solution to (7) may be obtainedby constructing an empirical estimate of φav based onrandom samples:

φ(N)av (x)

.= 1

N

N∑

i=1

f (x, δ(i)).

The function φ(N)av is convex in x, being the sum of

convex functions.Now, we study the convergence of the empirical

minimum φ(N)av (xk) to the actual unknown minimum

φav(x∗). To this end, notice first that as x varies over X ,f (x, ·) spans a family F of measurable functions of δ,namely, F .= { f (x, δ) : x ∈ X } . A key step for assessingconvergence is to bound (in probability) the relativedeviation between the actual mean φav(x) = Eδ[ f (x, δ)]and the empirical mean φ

(N)av (x) for all f (·, δ) belonging

to the family F . In other words, for the given relativescale error ε ∈ (0, 1), we require that

Pr

{supx∈X

|φav(x) − φ(N)av (x)|

V> ε

}≤ α(N), (9)

with α(N) → 0 as N → ∞. Notice that the uniformityof the bound (9) with respect to x is crucial, as x isnot fixed and known in advance: The uniform “close-ness” of φ

(N)av (x) to φav(x) is the feature that allows us

to perform the minimization on φ(N)av (x) instead of on

Optimization under uncertainty

φav(x). Property (9) is usually referred to as the uniformconvergence of the empirical mean (UCEM) property.A fundamental result of the Learning Theory statesthat the UCEM property holds for a function class Fwhenever a particular measure of the complexity of theclass, called the P-dimension of F (P-dim (F)), is finite.The interested reader can refer to the monographs(Vapnik 1998; Vidyasagar 1997) for formal definitionsand further details.

The result for average optimization is given in thenext theorem; see the Appendix for a proof.

Theorem 2 Let α, ε ∈ (0, 1), let d be an upper bound onthe P-dim of F and let

N ≥ Nav.= 128

ε2

⌈ln

8

α+ d

(ln

32eε

+ ln ln32eε

)⌉.

Let x∗av be a minimizer of φav(x), and let x(N)

av be aminimizer of the empirical mean φ

(N)av (x). Then, it holds

with probability at least (1 − α) that

φav(x(N)av ) − φav(x∗

av)

V≤ ε.

That is, x(N)av is an ε-suboptimal solution (in the relative

scale), with high probability (1 − α). A solution x(N)av

such that the above holds is called an (1 − α)-probableε-near minimizer of φav(x), in the relative scale V.

4 Applications to robust truss topology design

In this section, we apply some of the techniques dis-cussed in the previous paragraphs to problems of trusstopology optimization under uncertainty on the loadpattern and/or on the material characteristics. SeeBen-Tal and Nemirovski (1997) for other referencesdealing with the worst-case approach to truss topologydesign.

We consider a truss structure with m nodes con-nected by n bars in a given and fixed geometry. Nodescan be static (constrained) or free. The allowable move-ments of the nodes define the number M of degrees offreedom of the structure, with M ≤ mν, being ν = 2 forplanar structures and ν = 3 for three-dimensional struc-tures. Let ζ ∈ R

M denote the vector of components ofexternal forces (loadings) applied on the free nodesof the structure, and let d ∈ R

M denote the vector ofcomponent node elastic displacements caused by theload ζ at equilibrium.

Denote with xi the cross-sectional area of the ith bar,by �i the bar length, and by ϑi the Young’s modulus

of the bar. For small displacements, the following lin-earized relation holds

ζ = K(x, ϑ)d, K(x, ϑ).= A

⎡

⎢⎣

ϑ1�1

x1

. . .ϑn�n

xn

⎤

⎥⎦ AT ,

where K(x, ϑ) is the stiffness matrix of the structure andA is an M × n matrix that describes the geometry ofthe truss, that is, the kth column ak of A is a vector ofdirection cosines such that aT

k d measures the elongationof the kth bar. The compliance of the truss is the workperformed by the truss, which is given by

f (x, ζ, ϑ) = 1

2ζ Td = 1

2ζ T K−1(x, ϑ)ζ. (10)

The compliance, thus, represents the elastic energystored in the structure. The optimization objective thatwe consider here is to minimize the compliance (10),hence maximizing the stiffness, subject to a constrainton the total volume of the structure

V(x) =n∑

i=1

�1xi = �T x ≤ Vmax (11)

and to upper and lower bounds on the cross-sections:

xlb ≤ x ≤ xup (12)

(vector inequalities are intended component-wise). Tothis end, notice first that minimizing f (x, ζ, ϑ) is equiv-alent to minimizing a new slack variable γ under the ad-ditional constraint f (x, ζ, ϑ) ≤ γ . Then, if the loadingpattern ζ and the elastic coefficients ϑi, i = 1, . . . , n areknown and fixed, the nonlinear optimization problemthat we intend to solve is

minγ,x

γ, subject to (11), (12), and ζ T K−1(x, ϑ)ζ ≤ γ.

Notice further that the Schur complements rule (see,e.g., Section 2.1 of Boyd et al. 1994) permits to convertthe nonlinear (and non-convex) constraint f (x, ζ, ϑ) ≤γ into the convex LMI constraint

[K(x, ϑ) ζ

ζ T γ

]� 0.

Therefore, for fixed loading pattern ζ and coefficientsϑi, i = 1, . . . , n, the problem simply amounts to the

G.C. Calafiore, F. Dabbene

solution of the following (convex) SDP in the designvariable x and slack variable γ :

minγ,x

γ subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup[

K(x, ϑ) ζ

ζ T γ

]� 0. (13)

This problem may also be recast as a second-order coneprogram, which allows for additional efficiency in thenumerical solution (see Ben-Tal and Bendsøe 1993;Ben-Tal and Nemirovski 1994; Lobo et al. 1998).

Remark 1 Notice that other types of truss design prob-lems can be formulated in SDP form. For instance,within the previous setting, one can minimize the to-tal volume V(x) of the structure under a constraintf (x, ζ, ϑ) ≤ γ on the compliance. This results in thefollowing convex SDP problem:

minx

�T x subject to:

xlb ≤ x ≤ xup[

K(x, ϑ) ζ

ζ T 2γ

]� 0.

Alternatively, the fundamental modal frequency ofthe structure can be optimized, or lower bounds on thisfrequency can be imposed. If M(x) (a symmetric, posi-tive, semi-definite matrix affine in x) is the mass matrixof the structure, the fundamental modal frequency ω isthe square root of the smallest generalized eigenvalueof the pair (M, K), that is

ω2 = λmin(M, K).

It is easy to show (see Vandenberghe and Boyd 1999)that, for a given � ≥ 0,

ω ≥ � ⇔ M(x)�2 − K(x) � 0,

and this latter condition is an LMI in x. For instance, theproblem of minimizing the total volume of the structureunder the condition that ω ≥ � is cast as the followingSDP in x:

minx

V(x) subject to:

xlb ≤ x ≤ xup

M(x)�2 − K(x) � 0.

On the other hand, there also exist important problemsin structural optimization that may not be cast in the

form of convex programs. A notable example is given,for instance, by stress-constrained problems in whichbounds on the compression and tension on the indi-vidual bars are imposed together with local bucklingconstraints (Stolpe and Svanberg 2003). When bucklingconstraints are not present, the stress-constrained prob-lem can be recast and solved by linear programming(see, for instance, Stolpe and Svanberg 2004). In thiscontext, uncertainty in the loading pattern and materialcharacteristic is usually dealt with using stochastic pro-gramming (see Marti 1999). For an overview of stress-constrained optimization, see Rozvany (2001) andreferences therein.

In the sequel, we consider a situation where theloading pattern and the material characteristics areuncertain in problem (13) and discuss robust designapproaches.

4.1 Worst-case design

We first follow a robust approach and look for adesign vector x that minimizes the worst-case (withrespect to allowable loading patterns and Young’smoduli) compliance of the structure. We consider threedifferent cases, namely, polytopic uncertainty on bothϑ and ζ , norm-bounded uncertainty on ζ only, and ageneral case.

4.1.1 Polytopic uncertainty

Suppose that the loading vector ζ and the Young’smoduli vector ϑ are only known to belong to a givenpolytope P

(ζ, ϑ) ∈ P .= co{(ζ, ϑ)1, . . . , (ζ, ϑ)v}

where co{} denotes the convex hull and (ζ, ϑ)i denotesthe ith vertex of P . Notice that this setup encompasses,for instance, the case when the elements of ζ and/or ϑ

belong to independent intervals. The following propo-sition holds; see the Appendix for a proof.

Proposition 1 A solution to the robust optimizationproblem

minx

max(ζ,ϑ)∈P ζ T K−1(x, ϑ)ζ subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup

Optimization under uncertainty

is obtained by solving the following convex SDP

minx,γ

γ subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup

[K(x, ϑi) ζi

ζ Ti γ

]� 0, i = 1, . . . , v. (14)

4.1.2 Norm-bounded uncertainty on loadings

Suppose ϑ is known while the loading pattern ζ isassumed to belong to an ellipsoid E of center ζ c andshape matrix W, i.e.,

E = {ζ = ζ c + Wz, ‖z‖ ≤ 1}. (15)

For instance, in Ben-Tal and Nemirovski (1997), this el-lipsoid is used to represent the envelope of the primaryloadings and small “occasional loads.” The followingproposition holds; see the Appendix for a proof.

Fig. 1 a Design resultingfrom nominal load. b Robustdesign under intervaluncertainty on the loadingof node 6

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b

G.C. Calafiore, F. Dabbene

Proposition 2 A solution to the robust optimizationproblem

minx

maxζ∈E ζ T K−1(x, ϑ)ζ subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup (16)

is obtained by solving the following convex SDP

minx,γ,τ

γ subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup⎡

⎣K(x, ϑ) − τWWT ζ c 0M,1

ζ cT γ 1

01,M 1 τ

⎤

⎦ � 0.

4.1.3 Generic uncertainty on loadingsand Young’s moduli

In the case when both ζ and ϑ vary over generic sets, itis not possible in general to obtain exact reformulationof the worst-case design problem in terms of SDPs.However, the problem can still be posed in a proba-bilistically approximate sense by following the sampling

approach described in Section 3.1. Specifically, let ζ ∈Z and ϑ ∈ �, and assume that these uncertainties areof random nature and that samples ζ (i) ∈ Z , ϑ(i) ∈ �,and i = 1, . . . , N can be generated according to theunderlying probability distribution P over Z × �. Oncethe probabilistic levels ε, β are fixed, Theorem 1 pro-vides a bound on the required number of samples, andan approximate solution to the robust design problemcan be obtained by solving the following SDP:

minx,γ

γ subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup[

K(x, ϑ(i)) ζ (i)

ζ (i)T γ

]� 0, i = 1, . . . , N. (17)

4.2 Average design

In an average design approach, we look for a solu-tion vector x that minimizes the expected value of thecompliance. We assume that ϑ (the Young’s moduli)is exactly known, whereas ζ (the loading) is randomwith known expected value ζ

.= E{ζ } and covariancematrix

�.= E{(ζ − ζ )(ζ − ζ )T} = E{ζ ζ T} − ζ ζ T .

Fig. 2 Worst-case com-pliance as a function ofuncertainty radius r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 104

radius of uncertainty

wors

t-ca

se c

om

pli

ance

Optimization under uncertainty

In this case, we have explicitly

Eζ {ζ T K−1(x, ϑ)ζ } = tr Eζ {K−1(x, ϑ)ζ ζ T}= tr

(K−1(x, ϑ)(� + ζ ζ T)

)

= ζ T K−1(x, ϑ)ζ + tr(K−1(x, ϑ)�

).

Introducing two slack variables γ and X = XT , theproblem

minx

Eζ {ζ T K−1(x, ϑ)ζ } subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup (18)

can be equivalently rewritten as the following convexSDP problem (see Vandenberghe and Boyd 1999, fordetails on the derivation):

minx,γ,X

γ + tr �X subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup[

K(x, ϑ) ζ

ζ γ

]� 0

[K(x, ϑ) I

I X

]� 0.

Fig. 3 a Optimal design fornominal load. b Probabilis-tically robust designresulting from the scenariosolution under sphericaluncertainty on loadingsof nodes 3, 4, 5, and 6

a

b

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50

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G.C. Calafiore, F. Dabbene

4.2.1 Approximate solution for generic uncertainty

In a general situation of joint random uncertainty on ζ

and ϑ , we may resort to sampling approximation of theexpectation, as discussed in Section 3.2. In practice, theobjective in (18) is replaced by the sample mean, and itcan be shown that the resulting approximate problemcan be cast in SDP form as follows:

minx,t

1N

∑Ni=1 ti subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup[

K(x, ϑ(i)) ζ (i)

ζ (i)T ti

]� 0, i = 1, . . . , N,

where ϑ(i) and ζ (i) are the randomly extracted samplesof the uncertainties.

5 Numerical examples

In this section, we present some simple numerical ex-amples of robust truss optimization. We first considera planar ten-bar example from Marti (1999), assum-ing xlb = 0, xub = 100 mm2, Vmax = 105 mm3, Young’smoduli θi = 104 N/mm2, and loading intensity P =1, 000 N.

The solution of problem (13) under nominal loadpattern is depicted in Fig. 1a.

The optimal compliance resulted to be γ nom =6, 480 N mm, and the optimal bar sections

xnom = [27.78 39.28 0 83.33 0 27.78 0 39.28 27.78 0

]T.

Next, we “robustified” this design by considering aset of loading patterns where the loading on node 6is (P, δP), with δ ∈ [−r, r]. This situation falls in thepolytopic setting discussed in Section 4.1.1. Solving theSDP (14) with uncertainty level r = 0.5, we obtainedthe solution depicted in Fig. 1b. The worst-case optimalcompliance resulted to be γ wc = 12, 701 N mm, and theoptimal bar sections

xwc =[39.68 28.06 14.03 69.44 0 19.84 14.03 28.06 29.76 0

]T.

Figure 2 shows the increase in the worst-case compli-ance as the uncertainty radius r varies from 0 (nominalload pattern) to 1.

Finally, we assume a nominal loading pattern as theone considered in Ben-Tal and Nemirovski (1997). Theoptimal solution of problem (13) under this nominalload is depicted in Fig. 3a. The optimal compliance

resulted to be γ nom = 4, 147 N mm, and the optimalbar sections

xnom =[

69.44 0 0 69.44 0 34.72 0 49.10 34.72 0]T

.

On this configuration, we next considered the fol-lowing uncertainty on the loadings: In addition to thenominal loading, each node is subject to a force havingrandom direction and intensity 50 N. This worst-casedesign problem was solved using the sampling tech-nique discussed in Section 4.1.3. Specifically, settingrobustness level ε = 0.05 and confidence β = 10−10, thebest bound in Calafiore and Campi (2006) requiresN = 1, 651 samples. Problem (17) then yielded theprobabilistically robust solution depicted in Fig. 3b,having probable worst-case compliance γ (N)

wc = 9, 527and bar sections

xwc(N) =

[70.16 0.05 4.36 66.68 0 32.42 2.26 45.83 34.23 0

]T.

6 Conclusions

In this paper, we discussed worst-case and average ap-proaches for convex programming under uncertainty.Optimal solutions can be determined efficiently in somespecial situations, whereas the general case can betackled by means of sampling-based probabilistic re-laxations. These methods have the advantage of beingconceptually simple and easy to apply, thus represent-ing an appealing design option also for the practitioner.As shown in Section 4, robust and sampling-basedoptimization methods can be applied with success todesign problems arising in structural mechanics. Themethods presented here are all based on batch solu-tions to SDP problems with many constraints. Thismay represent a problem in practice for large-scaleapplications. Alternative iterative (i.e., sequential, non-batch) techniques have been developed for feasibilityproblems (see Calafiore and Dabbene 2007), whileextension to optimization problems is the subject ofcurrent research.

Acknowledgements The authors are grateful to Mauro Baldifor his help in producing the Matlab code used in the numericalexamples. This work has been supported by FIRB funds from theItalian Ministry of University and Research.

Appendix

Proof of Lemma 1 The implication from left to rightin (4) is obvious; we, hence, consider only the reverseimplication. Let x be such that the right-hand side of(4) holds, and let δ be any point in the polytope �.

Optimization under uncertainty

Then, there exist α1, . . . , αv ≥ 0,∑v

j=1 α j = 1 such thatδ = ∑v

j=1 α jδ j. Since F(x, δ) is affine in δ, we may writeit as

F(x, δ) = F(x, 0) + F(x, δ),

where F(x, δ) is linear in δ. Therefore,

F(x, δ) = F

⎛

⎝x,

v∑

j=1

α jδ j

⎞

⎠

= F(x, 0) + F

⎛

⎝x,

v∑

j=1

α jδ j

⎞

⎠

= F(x, 0) +v∑

j=1

α j F(x, δ j)

=v∑

j=1

α jF(x, 0) +v∑

j=1

α j F(x, δ j)

=v∑

j=1

α jF(x, δ j) � 0,

which proves the statement. ��

Proof of Theorem 2 Consider the function family Ggenerated by the functions

g(x, δ).= f (x, δ) − f ∗(δ)

V,

as x varies over X . The family G is a simple rescalingof F and maps � into the interval [0, 1]; therefore, theP-dimension of G is the same as that of F . Define

φg(x).= Eδ[g(x, δ)] = φav(x) − K

V, (19)

and

φ(N)g (x)

.= 1

N

N∑

i=1

g(x, δ(i)) = φ(N)av (x) − K

V,

where

K .= Eδ[ f ∗(δ)], K .= [ f ∗(δ)] = 1

N

N∑

i=1

f ∗(δ(i)).

Notice that a minimizer xav of φ(N)av (x) is also a mini-

mizer of φ(N)g (x). Then, Theorem 2 in Vidyasagar (2001)

guarantees that, for α, ν ∈ (0, 1),

Pr

{supg∈G

∣∣∣∣∣Eδ[g(δ)] − 1

N

N∑

i=1

g(δ(i))

∣∣∣∣∣ > ν

}≤ α,

provided that

N ≥ 32

ν2

[ln

8

α+ P-dim (G)

(ln

16eν

+ ln ln16eν

)].

Applying this theorem with ν = ε/2 and using thebound P-dim (G) = P-dim (F) ≤ d, we have that, for allx ∈ X , it holds with probability at least (1 − α) that

|φg(x) − φ(N)g (x)| ≤ ε

2. (20)

From (20), evaluated in x = x∗av, it follows that

φg(x∗av) ≥ φ(N)

g (x∗av) − ε

2≥ φ(N)

g (x(N)av ) − ε

2, (21)

where the last inequality follows because x(N)av is a mini-

mizer of φg. From (20), evaluated in x = x(N)av , it follows

that

φ(N)g (x(N)

av ) ≥ φg(x(N)av ) − ε

2,

which substituted in (21) gives

φg(x(N)av ) ≥ φg(x(N)

av ) − ε.

From the last inequality and (19), it follows that

φ(x(N)av ) − φ(x∗

av) ≤ εV,

which concludes the proof. ��

Proof of Proposition 1 First, notice that minimizingmax(ζ,ϑ)∈P ζ T K−1(x, ϑ)ζ is equivalent to minimizing aslack variable γ , subject to the additional constraintmax(ζ,ϑ)∈P ζ T K−1(x, ϑ)ζ ≤ γ . In turn, this latter con-straint is equivalent to imposing

ζ T K−1(x, ϑ)ζ ≤ γ, ∀(ζ, ϑ) ∈ P,

which is rewritten in robust SDP form using Schurcomplements:[

K(x, ϑ) ζ

ζ T γ

]� 0, ∀(ζ, ϑ) ∈ P.

A direct application of Lemma 1 then easily concludesthe proof. ��

Proof of Proposition 2 Following the same steps as inthe proof of Proposition 1, we have that problem (16) isequivalent to

minx,γ,τ

γ subject to:

�T x ≤ Vmax

xlb ≤ x ≤ xup

[K(x, ϑ) ζ

ζ T γ

]� 0, ∀ζ ∈ E .

G.C. Calafiore, F. Dabbene

Substituting (15), the latter constraint becomes[

K(x, ϑ) ζc

ζ Tc γ

]+

[W

0M,1

]z

[01,M 1

] + (last term)T � 0,

∀ζ : ‖z‖ ≤ 1.

The statement now follows from direct application ofLemma 2 to the latter expression. ��

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