Analysis of Non Convex Polynomial

Programs by the Method of Moments

OMEVA RESEARCH GROUP

Departamento de Matematicas, Universidad de Castilla la Mancha,

13071, Ciudad Real, Spain

http://matematicas.uclm.es/omeva/

Rene J. MeziatDepartamento de Matematicas, Universidad de los Andes

Carrera 1 este No 18A-10, Bogota, Colombia

Phone: +57 1 - 339 49 49 ext. 3586 Fax: +57 1 - 332 43 40

Email: rmeziat@uniandes.edu.co

www.uniandes.edu.co

March 21, 2003

Abstract

In this work we propose a general procedure for estimating the globalminima of mathematical programs given in the general form as follows:min P0 (t) s.t. Pi (t) ≤ 0 for i = 1, . . . , k (Π) , where the func-tions Pi : Rn → R are n-dimensional polynomials which are supposed tobe non convex. The theory behind the Method of Moments guaranteesthat all global minima of the non convex program (Π) can be estimatedby solving an equivalent convex program. One difficult question about theMethod of Moments is how to characterize one particular convex set V ofmoment vectors. In this work we solve the Multidimensional TruncatedMoment Problem in algebraic compact sets. Later we use this result tofind the global minima of non convex polynomial programs (Π) by us-ing only one equivalent semidefinite program. In particular, we solve thegeneral non convex quadratic program. Theory and several examples areexplained in full detail.

1 Introduction

The Method of Moments is a general method for treating non convex optimiza-tion problems, which is particularly well suited to cope with global optimization

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problems which come from several research areas in Optimization Theory likeCalculus of Variations, Control Theory and Mathematical Programming. It hasbeen successfully applied to non convex variational problems and one dimen-sional polynomial programs. See [1], [4], [5] and [6] for different applications ofthe Method of Moments. The Method of Moments takes a proper formulationin probability measures of a non convex optimization problem. Thus, when theproblem can be stated in terms of polynomial expressions, we can transform themeasures into algebraic moments to obtain a new convex program defined inalgebraic moments. This procedure has been successfully employed for treatingnon convex variational problems when we use their generalized formulation inYoung measures. See [1], [5] and [6].

The Method of Moments provides an alternative way for studying globaloptimization problems under no convexity assumptions. Indeed, it has beensuccessfully applied to the analysis of general non convex one dimensional poly-nomial programs. See a complete review of this application in [4]. The purposeof this work is to extend the applications of the Method of Moments for treatingnon convex polynomial programs in several dimensions given in the general form

min P0 (t) s.t. Pi (t) ≥ 0, for i = 1, . . . , k (1)

where every function Pi (t) is a n-dimensional polynomial. It is very importantto observe here, that the general non convex quadratic program is included inthis family of polynomial problems.

In order to properly use the Method of Moments, we must solve some par-ticular Problem of Moments for every non convex mathematical program weare interested in. These problems are really difficult to solve, and they makepart of the classical repertory of famous problems in contemporary mathemat-ics. A a short review on the classical treatment of the Problem of Momentscan be found in [2]. In this paper, we describe how to solve general momentproblems in arbitrary semi algebraic compact sets Ω in Rn, by using the gen-eral characterization of positive polynomials in Ω given by Putinar in [9]. Inthis way, we will prove that the solution of a particular Problem of Momentsin a multidimensional domain Ω, is obtained by constructing a correspondingset of positive semidefinite quadratic forms from the inequalities which definethe feasible set Ω of the polynomial program (1) . From this point of view, theMultidimensional Moment Problem extends the classical one dimensional mo-ment problems. Indeed, all of them are solved by characterizing a particularconvex cone of positive functions with the help of a corresponding set of positivesemidefinite quadratic forms.

In order to solve a particular non convex polynomial program (1) , we canapply the Method of Moments by transforming it into an equivalent convexprogram. This new problem has a linear objective function, which is defined bythe coefficients of the objective function P0; and a convex feasible set, definedby the set of positive semidefinite quadratic forms that characterize the multi-dimensional algebraic moments of probability measures supported in Ω. In thisway, we obtain a semidefinite relaxation of the original non convex program.

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The reader can find a good introduction for semidefinite programming in [7] and[10]. Many authors call these relaxations LMI relaxations as they are derivedby using a single linear matrix inequality.

We would like to emphasize that this paper is not meant to be a pioneerpaper on semidefinite relaxations of polynomial programs as several accountson the subject have already been published i.e. [11],[8]. However our objectiveis to present a different point of view which improves previous proposals. Theapproach followed in this paper relies on the analysis of the duality between aparticular moment cone and the corresponding cone of positive functions, whichallows us to apply the theory of the Method of Moments for characterizing theglobal minima of a particular non convex polynomial program. Other works onthis subject use the duality of the theory of semidefinite programming in order todefine a hierarchy of semidefinite relaxations whose sequence of optimal valuesconverges to the global optimum of one particular polynomial program, see [11].

In this paper, we will see that we need only one semidefinite program toestimate the global minima of any non convex polynomial program like (1) .

The present paper is organized as follows. In Section 2 we will review thegeneral theory behind the Method of Moments, and will deduce the most gen-eral results for global optimization of non convex mathematical programs. InSection 3, we will explain how to solve the classical one dimensional momentproblems. Indeed, we will see how to solve the Hamburger’s Moment Problem,the Trigonometric Moment Problem, the Stieltjes Moment Problem and theHausdorff’s Moment Problem by reducing all of them to the classical charac-terizations of one dimensional positive polynomials. Then we will apply theseresults for analyzing one dimensional non convex polynomial programs. In Sec-tion 4, we will treat the Multi-dimensional Moment Problem and we will useit for solving general non constrained polynomial programs in several variables.In Section 5, we will show how to solve arbitrary multi-dimensional momentproblems in arbitrary semi-algebraic domains in Rn. Moreover, we will applythese techniques for solving arbitrary polynomial programs. In particular, wewill analyze the family of non convex quadratic programs. Finally, we will givesome comments and remarks in Section 6.

2 General Theory of the Method of Moments

In this paper we are concerned with the search of all global minima of a con-tinuous function f : Ω → R, defined in some arbitrary closed set Ω ⊂ Rn. Thisproblem is particularly difficult when no assumptions about the convexity off or Ω are available. Nonetheless, progress in this kind of problems can beachieved by using convex analysis and measure theory. We will see here howwe can analyze global optimization problems when its objective function f isexpressed as a linear combination of simpler functions. This is the general ideaof many applications of the Method of Moments.

Theorem 1 Let P (Ω) be the set of all regular Borel probability measures sup-ported in a closed set Ω ⊂ Rn, and let f be a bounded from below continuous

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function f : Ω → R, then

infµ∈P (Ω)

〈f, µ〉 = inft∈Ω

f (t) .

Proof. By elementary integration we have that m ≤ 〈f, µ〉 for every prob-ability measure µ, where m = inft∈Ω f (t) . On the other hand, it is clear that〈f, δtn〉 → m, where tn ⊂ Ω is a minimizing sequence for the function f, andδt is the Dirac’s measure supported in the point t.

Theorem 2 Let G be the set of all global minima of the function f in Ω, then

〈f, µ∗〉 = infµ∈P (Ω)

〈f, µ〉

if and only if the support of µ∗ is contained in G.

Proof. It is easy to see that 〈f, µ∗〉 = m when µ∗ ∈ P (G) . We will verifythat µ∗ ∈ P (G) when 〈f, µ∗〉 = m. Let us assume that t ∈ supp (µ) ∩Gc, thenf (t) > m. Since f is continuous, there exists a neighborhood U of t such thatf ≥ γ > m in U. On the other hand, µ (U) > 0 because t ∈ supp (µ) . So,〈f, µ〉 =

∫U

fdµ +∫

Uc fdµ ≥ γµ (U) + mµ (U c) > m.From these results, it follows that we should use the generalized optimization

problem in measuresmin

µ∈P (Ω)〈f, µ〉 (2)

as an alternative formulation of the global optimization problem

mint∈Ω

f (t) . (3)

Thus, we obtain a new optimization problem with two significant features inoptimization, namely, a linear objective function: µ → 〈f, µ〉 and a convexfeasible set: P (Ω) . In addition, this formulation includes all the informationabout the solution of the standard global optimization problem (3) . Indeed, theset P (G) , which is composed of all probability measures supported in G, is thesolution set for the generalized problem (2) .

When the objective function f can be expressed by a linear combination ofsimpler functions

f =k∑

i=1

ciψi (4)

where ψ1, . . . , ψk is a basis of continuous functions in Ω, then every integralin (2) can be expressed by an elementary dot product in Rk

〈f, µ〉 =k∑

i=1

cixi = c · x

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whose factors are the coefficients vector c from the linear combination (4) andthe moment vector x whose entries

xi =∫

ψidµ, for i = 1, . . . , k

are the moments of the measure µ with respect to the basis functions ψ1, . . . , ψk .Every moment vector x may also be expressed by integration in the followingmanner:

x = 〈T, µ〉 =∫

Tdµ,

where T is the non linear transformation T : Ω → Rk defined by the expres-sion T (t) = (ψ1 (t) , . . . , ψk (t)) . For convenience, henceforth we assume thatapplication T is one to one.

With the help of the transformation T, we can easily define the set

V = 〈T, µ〉 , µ ∈ P (Ω)

which consists of all moment vectors of probability measures supported in Ω.Since the application

µ → 〈T, µ〉 : P (Ω) → Rk

is linear, we immediately observe that V is a convex set in Rk. In this way, wecan represent every measure µ ∈ P (Ω) by its respective moment vector x ∈ V.By using this representation, we can transform the generalized optimizationproblem (2) into the equivalent convex program

minx∈V

c · x (5)

whose solution set is the convex set

W = 〈T, µ〉 , µ ∈ P (G)

where G is the set of all global minima of the function f in Ω.It is remarkable that problem (5) is a convex mathematical program which

encloses the information about the global minima of the objective function fin Ω. In fact, it gives a non trivial characterization of the global minima of thefunction f in the set Ω. Before we show the solution to the global optimiza-tion problem (3) given by the convex program (5) , we have to must introduceadditional results that link convex analysis and measure theory.

Since the image T (Ω) is contained in the euclidean space Rk, its convexenvelope can be expressed as

co (T (Ω)) = 〈T, µ〉 , µ ∈ Q (Ω)

where Q (Ω) is the family of all finitely supported probability measures in Ω.This means that co (T (Ω)) is just the set of moment vectors of finitely supportedmeasures in Ω. Then, co (T (Ω)) ⊂ V.

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The theory of the Method of Moments uses a convenient interplay betweenmeasure theory and convex analysis for describing the moment vector set V. Infact, from Carathedory’s Theorem we know that every point of the convex hullco (T (Ω)) may be represented by a convex combination with less than k + 2terms. On the other hand, every regular Borel probability measure can be ap-proximated by discrete probability distributions whose moment vectors belongto the convex hull co (T (Ω)) . In this way, we obtain a method for approximatingevery point in V by convex combinations with a fixed number of terms. This isthe reason that explains why the Method of Moments works particularly wellin many practical applications. The following theorems confirm this remark.

Theorem 3 The convex hull co (T (Ω)) is dense in the moment vectors set V,so V = co (T (Ω)).

Proof. For a µ-measurable positive function f, its integral 〈f, µ〉 is usuallydefined as the supremum of all integrals 〈s, µ〉 , where every s is a simple functionsatisfying s ≤ f. A simple function has the form s =

∑ji=1 ciχAi

where everyAi is a Borel set with characteristic function χAi . The integral of s with respectto the measure µ is defined by the elementary sum 〈s, µ〉 =

∑ji=1 ciµ (Ai) .

Since f is continuous, the definition of the integral 〈f, µ〉 does not change if weconsider simple functions with the form s =

∑ji=1 f (ti)χAi , where ti ∈ Ai for

every index i. Thus we have 〈s, µ〉 =∑j

i=1 f (ti) µ (Ai) =⟨f,

∑ji=1 µ (Ai) δti

⟩.

Finally, we extend this conclusion to every basis function ψi.

Theorem 4 When the domain Ω is compact, so are the convex hulls co (T (Ω))and co (T (G)) .

Proof. Convex hulls of compact sets are compact.

Theorem 5 The convex hull co (T (Ω)) and the set V of moment vectors arenot necessarily closed.

Proof. By taking Ω = R, ψ1 (t) = t and ψ2 (t) = e−t2 , we found thatco (T (Ω)) fails to be closed. In order to verify that V does not need to be closed,we will exhibit an example suggested by Pedregal in [1]. Take the probabilitymeasure µ = λδt1 + (1− λ) δt2 where

t1 = −(

1− λ

λ

) 14

, t2 =(

λ

1− λ

) 14

, 0 < λ < 1.

Notice that∫ (

1, t, t2, t3, t4)dµ → (1, 0, 0, 0, 1) when λ → 1, which means that

the five algebraic moments of µ converge to the vector (1, 0, 0, 0, 1) when λ → 1.However, it is easy to see that there is no positive measure on the real line withvalues (1, 0, 0, 0, 1) as its five first algebraic moments.

Theorem 6 Let Qk+1 (Ω) be the set of all probability measures in Ω supportedin k + 1 points at most, then co (T (Ω)) = 〈T, µ〉 , µ ∈ Qk+1 (Ω) when T is aone to one application.

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Proof. Apply Caratheodory´s theorem from Convex Analysis.

Theorem 7 Let µε ∈ Q (Ω) for every ε > 0 and let y = limε→0 〈T, µε〉 . Ify does not belong to co (T (Ω)) , then there exists an unbounded subsequencetεn ⊂ Ω such that every tεn belongs to the support of the measure µεn .

Proof. From Theorem 6, we can express every moment vector xε = 〈T, µε〉by using a convex combination with at most k + 1 terms:

xε = λε1T (tε1) + · · ·+ λε

k+1T(tεk+1

)

where points tε1, . . . , tεk+1 form the support of the measure µε. If every family

tεi : ε > 0 is bounded, then there exists a converging sub sequence xεn suchthat

y = limn→∞

xεn = λ1T(t1

)+ · · ·+ λk+1T

(tk+1

)= 〈T, µ〉 ,

where limn→∞ λεni = λi, limn→∞ tεn

i = ti, for every i = 1, . . . , k + 1. Thus, theentries of the vector y ∈ Rk are the moments of the finitely supported measureµ =

∑k+1i=1 λiδti

and this result contradicts the assumptions about y.Although Theorem 3 states that the moment vector set V is closed in the

particular cases where the feasible set Ω is bounded, Theorem 4 claims thatconvex set V does not need to be closed in general, so we should replace program(5) by the extended program

minx∈V

c · x . (6)

It is remarkable that any solution for this program is linked with a minimizingsequence of the function f. Let us assume that x∗ is a solution to the extendedprogram (6) , and x∗ ∈ co (T (Ω)) , then x∗ = 〈T, µ∗〉 being µ∗ a finitely sup-ported measure in G. In this way, we can obtain a finite set of global minimaof the function f by analyzing the support of µ∗. In the opposite case whenx∗ /∈ co (T (Ω)) , we can obtain a minimizing sequence for f by using a familyof finitely supported measures µε whose moments xε = 〈T, µε〉 approach x∗ inRk.

Theorem 8 If x∗ is a solution to the extended program (6) in co (T (Ω)) , thenx∗ ∈ co (T (G)) .

Proof. Since c·x∗ = m (≡ infΩ f) for x∗ = 〈T, µ∗〉 with µ∗ ∈ Q (Ω) , we have〈f, µ∗〉 = m. From Theorem 2, the support of measure µ∗ consists of finitelymany points in G.

Theorem 9 Let xε = 〈T, µε〉 : ε > 0 be a family of moments in co (T (Ω))such that xε → x∗, where x∗ is a solution to the extended program (6) . Then,there exists a minimizing sequence tεn ⊂ Ω for the function f, such that everyterm tεn belongs to the support of the measure µεn .

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Proof. From Theorem 6, we can write every moment vector xε by using theconvex combination

xε = 〈T, µε〉 = λε1T (tε1) + · · ·+ λε

k+1T(tεk+1

)

where every measure µε ∈ Qk+1 (Ω) . Since x∗ is a solution for program (6) ,then

c · xε = 〈f, µε〉 = λε1f (tε1) + · · ·+ λε

k+1f(tεk+1

) → m(≡ inf

Ωf)

. (7)

Let us assume that the family tεi : ε > 0, i = 1, . . . , k + 1 does not contain anyminimizing sequences of the function f. Then f (tεi ) > m + γ with γ > 0, whichprevents 〈f, µε〉 from converging to m.

The success of the theory of the Method of Moments relies on the importantfact that it provides an alternative characterization of the global minima of thefunction f.

Theorem 10 Let us assume that there exist no unbounded minimizing sequencesof f. If x∗ ∈ Rk is a solution for the extended program (6) , then there existfinitely many points t1, . . . , tρ ∈ G and positive values λ1, . . . , λρ such that

x∗ = λ1T (t1) + · · ·+ λρT (tρ) (8)1 = λ1 + · · ·+ λρ

where ρ may be chosen to be less than k + 2.

Proof. Since f has no unbounded minimizing sequence, Theorem 9 impliesthat x∗ ∈ co (T (Ω)) . Hence, x∗ = 〈T, µ∗〉 where µ∗ is finitely supported. FromTheorem 2, we can verify that every point in the support of µ∗ is a globalminimum of the objective function f. Thus we also have that µ∗ ∈ P (G) because〈f, µ∗〉 = c · x∗ = m (≡ infΩ f) . Finally, from Theorem 6 we have x∗ = 〈T, µ〉where µ is supported in k + 1 points at the most. And applying Theorem 2again, we conclude that µ is supported in G.

Corollary 11 If finitely many points t1, . . . , tρ ∈ Ω satisfy (8) , then every ti isa global minimum of f in Ω.

Proof. By taking the measure µ∗ =∑ρ

i=1 λiδti we found that 〈f, µ∗〉 =c · x∗ = m, then the support of µ∗ is contained in G because of Theorem 2.

Theorem 12 Let us assume that f does not have any unbounded minimizingsequence. If x∗ ∈ Rk is an extreme point of the solution set of the convexprogram (6) , then there exists a global minimum t ∈ Ω, of the objective functionf, satisfying the set of k non linear equations

x∗ = T (t) . (9)

Proof. If ρ > 1 in (8) , then x∗ could not be an extreme point of the solutionset of program (6) .

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Corollary 13 Every point t ∈ Ω satisfying the set of k non linear equations (8)is a global minimum of the objective function f .

Proof. Take µ∗ = δt and observe that f (t) = 〈f, µ∗〉 = m, then t ∈ Gbecause t is the support of µ∗.

Theorem 14 For arbitrary points t1, . . . , tρ ∈ G and positive values λ1, . . . , λρ

satisfying λ1 + · · ·+ λρ = 1, the point x∗ ∈ Rk defined by

x∗ = λ1T (t1) + · · ·+ λρT (tρ) (10)

is a solution of the extended program (6) .

Proof. Take c · x∗ = λ1f (t1) + · · ·+ λρf (tρ) = m (≡ infΩ f) .

Theorem 15 Every solution of program (6) can be expressed in the form (10) .

Proof. Apply Theorem 10.From these results we conclude that a necessary and sufficient condition for

a finite number of points t1, . . . , tρ ∈ Ω be a set of global minima of f , is thatthey satisfy the k non linear equations

x∗ = λ1T (t1) + · · ·+ λρT (tρ) (11)

for some positive λ1, . . . , λρ with λ1 + · · ·+ λρ = 1, where x∗ ∈ Rk is a momentvector that solves the extended program (6) . In order to estimate a particularset of global minima of the function f, we must solve equations (11) for aparticular solution of the program (6) . However, this question is equivalent tolooking for a finitely supported measure µ∗ whose moments (with respect to thebasis ψ1, . . . , ψk) are the optimal values x∗1, . . . , x

∗k. Notice that we are mostly

interested in finding the support of µ∗ rather than determining the measure µ∗.The answer to this question comes again from the Problem of Moments where itis clarified how to recover a measure from its moments. Then, in order to applythe Method of Moments on specific problems, we need a proper characterizationof the set of all moment vectors V and a practical method for recovering everyfinitely supported measure from its moments.

3 One Dimensional Moment Problems

The Problem of Moments consists in determining the conditions which guar-antee that values x1, . . . , xk are the moments of an arbitrary positive measureµ with respect to a particular basis of functions ψ1, . . . , ψk, which is definedin some domain Ω. The solution of the Problem of Moments also should pro-vide techniques for recovering the measure µ from the sequence of momentsx1, . . . , xk. This is a classical problem in modern mathematics in which greatmathematicians have been involved since the nineteenth century. For a classicalintroductory review on the Problem of Moments see [2]. Here we solve several

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moment problems using a powerful tool from convex analysis. To attain thistask, we will use the classical duality between the moment cone and the cone ofthe corresponding positive functions. This tool is introduced in [3] and can betraced back to the seminal works of Markov and Tchebishev.

Let us define

M =

x ∈ Rk : xi =∫

ψidµ, i = 1, . . . k, µ positive mesure supported in Ω

as the set of moment vectors of all positive measures supported in Ω. We caneasily see that M is a convex cone in Rk. We also define P as follows:

P =

c ∈ Rk :

k∑

i=1

ciψi (t) ≥ 0, ∀t ∈ Ω

,

where the vectors c in Rk determine non-negative functions in Ω. It is also easyto check out that P is a closed convex cone in Rk. The usual way for solvingmoment problems is to analyze the cone P, since its dual is exactly the closureof the moment cone M.

Theorem 16 The dual of the cone P is the closure of the cone M.

Proof. For arbitrary vectors c ∈ P and x ∈ M, we have

c · x =∫ (

k∑

i=1

ciψi

)dµ ≥ 0.

Thus, P ⊂ M∗, and M ⊂ P ∗. If there exists a point t0 ∈ Ω such that

k∑

i=1

ciψi (t0) < 0,

then∑k

i=1 cix0i < 0, where x0 is the moment vector of the Dirac measure δt0 .

Thus, M∗ ⊂ P, and P ∗ ⊂ M.By using the duality statement of Theorem 16, we can find the answer to

many classical moment problems, provided we can properly characterize thecorresponding family of positive functions P.

3.1 Hamburger’s Moment Problem

When the function basis is the algebraic system 1, t, . . . , t2r, and the domain Ωis the real line, the moment problem is referred to as the Hamburger’s MomentProblem. It is well known, from elementary algebra, that every positive poly-nomial

∑2ri=0 cit

i on the real line can be expressed as the sum of two squares,that is

2r∑

i=0

citi =

(r∑

i=0

aiti

)2

+

(r∑

i=0

biti

)2

. (12)

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However, it will be useful to express (12) by using quadratic forms, so we claimthat every positive polynomial

∑2ri=0 cit

i on the real line can be expressed in thefollowing form:

2r∑

i=0

citi =

r∑

i=0

r∑

j=0

aiajti+j +

r∑

i=0

r∑

j=0

bibjti+j . (13)

For solving the classical Hamburger’s Moment Problem, we only need toapply previous duality statement and the decomposition (13) for positive poly-nomials. If values x0, . . . , x2r are the algebraic moments of a positive measuresupported on the real line, then x ∈ P ∗, and thus

2r∑

i=0

cixi ≥ 0

for the coefficients c of every positive polynomial∑2r

i=0 citi. In particular, for

arbitrary values a0, . . . , ar we haver∑

i=0

r∑

j=0

aiajxi+j ≥ 0

due to(∑r

i=0 aiti)2 ≥ 0. Thus, we conclude that a necessary condition for a

vector x ∈ R2k+1 to be a moment vector, is that its components form a positivesemidefinite Hankel matrix H = (xi+j)

ri,j=0 .

On the other hand, assuming that the entries of a vector x ∈ R2r+1 com-pose a positive semidefinite Hankel matrix H = (xi+j)

ri,j=0 , we can see from

expression (13) that2r∑

i=0

cixi =r∑

i=0

r∑

j=0

aiajxi+j +r∑

i=0

r∑

j=0

bibjxi+j ≥ 0

for the coefficients c of every positive polynomial∑2r

i=0 citi on the real line.

Therefore, x ∈ P ∗ = M and we conclude that x is a moment vector or at leastit is a limit point of a sequence of moment vectors. This procedure may beapplied to obtain the characterization of the moment vectors for other basesand domains.

3.2 Trigonometric Moment Problem

From the Riesz-Fejer Theorem in complex analysis, we know that every positivetrigonometric polynomial

∑ri=−r cie

ijt can be expressed as

r∑

i=−r

cieijt =

∣∣∣∣∣r∑

k=0

akekjt

∣∣∣∣∣

2

=r∑

l=0

r∑

k=0

akalej(k−l)t. (14)

By using the quadratic form (14) and the arguments explained above, we easilysolve the Trigonometric Moment Problem.

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The closure of the cone M of all moment vectors of positive mea-sures supported in the unitary circumference S1, with respect tothe trigonometric system e−rjt, . . . , erjt, is the set of all vectorsx ∈ C2r+1 whose entries form a positive semidefinite Toeplitz matrixT = (xk−l)

rk,l=0 .

3.3 Stieltjes and Hausdorff’s Moment Problem

The Stieltjes Moment Problem arises when we consider the algebraic system1, t, . . . , tr on the semi-axis Ω = [0,∞) of the real line. If we restrict the domainto a bounded interval Ω = [a, b] , we obtain the Hausdorff’s Moment Problem.

3.3.1 Solution to the Stieltjes Problem - Even Case

The closure of the cone M of all moment vectors of positive measuressupported in the semiaxis [0,∞), with respect to the algebraic system1, t . . . , t2r, is the set of all vectors x ∈ R2r+1 whose entries formtwo positive semidefinite Hankel matrices given by

H1 = (xi+j)ri,j=0 H2 = (xi+j+1)

r−1i,j=0 .

Proof. It is known that positive polynomials q (t) on the semiaxis t ≥ 0,can be expressed by q (t) = q2

1 (t) + t q22 (t) , where qi are polynomials. Then,

we can express an arbitrary, even degree, non negative polynomial∑2r

i=0 citi on

the semiaxis [0,∞) in the form

2r∑

i=0

citi =

(r∑

i=0

aiti

)2

+ t

(r−1∑

i=0

biti

)2

which may be written by the following couple of quadratic forms

2r∑

i=0

citi =

r∑

i=0

r∑

j=0

aiajti+j +

r−1∑

i=0

r−1∑

j=0

bibjti+j+1.

Then, we can repeat the arguments used in the proof of Hamburger´s momentProblem.

3.3.2 Solution to the Stieltjes Problem - Odd Case

The closure of the cone M of all moment vectors of positive measuressupported in the semiaxis [0,∞), with respect to the algebraic system1, t . . . , t2r+1, is the set of all vectors x ∈ R2r+2 whose entries formtwo positive semidefinite Hankel matrices with the following form

H1 = (xi+j)ri,j=0 H2 = (xi+j+1)

ri,j=0 .

12

Proof. For odd degree, positive polynomials in [0,∞), we have the analo-gous expression

2r+1∑

i=0

citi =

(r∑

i=0

aiti

)2

+ t

(r∑

i=0

biti

)2

which can be written as the sum of two quadratic forms:

2r+1∑

i=0

citi =

r∑

i=0

r∑

j=0

aiajti+j +

r∑

i=0

r∑

j=0

bibjti+j+1.

3.3.3 Solution to the Hausdorff’s Problem - Even Case

The closure of the cone M of all moment vectors of positive mea-sures supported in the bounded interval [κ1, κ2] , with respect to thealgebraic system 1, t . . . , t2r, is the set of all vectors x ∈ R2r+1 whoseentries make positive semidefinite the following symmetric matrices:

H1 = (xi+j)ri,j=0 H2 = ((κ1 + κ2)xi+j+1 − κ1κ2xi+j − xi+j+2)

r−1i,j=0 .

Proof. From Markov-Luckas Theorem [3], we can express every even degreepositive polynomial on the bounded interval Ω = [κ1, κ2] as

2r∑

i=0

citi =

(r∑

i=0

aiti

)2

+ (t− κ1) (κ2 − t)

(r−1∑

i=0

biti

)2

.

Then we can write this expression by using quadratic forms:

r∑

i=0

r∑

j=0

aiajti+j +

r−1∑

i=0

r−1∑

j=0

bibj

((κ1 + κ2) ti+j+1 − κ1κ2t

i+j − ti+j+2)

and repeat the arguments for the Hamburger´s Moment Problems.

3.3.4 Solution to the Hausdorff’s Problem - Odd Case

The closure of the cone M of all moment vectors of positive measuressupported in the bounded interval [κ1, κ2] , with respect to the alge-braic system 1, t . . . , t2r+1, is the set of all vectors x ∈ R2r+2 whoseentries form two positive semidefinite symmetric matrices given by

H1 = (xi+j+1 − κ1xi+j)ri,j=0 H2 = (κ2xi+j − xi+j+1)

ri,j=0 .

Proof. Again, from Markov-Luckas Theorem, we can express every odddegree positive polynomial on the bounded interval Ω = [κ1, κ2] as

2r+1∑

i=0

citi = (t− κ1)

(r∑

i=0

aiti

)2

+ (κ2 − t)

(r∑

i=0

biti

)2

.

13

Hence we have the quadratic form expression

r∑

i=0

r∑

j=0

aiaj

(ti+j+1 − κ1t

i+j)

+r∑

i=0

r∑

j=0

bibj

(κ2t

i+j − ti+j+1).

At this stage we should note the important fact that every one dimensionalmoment problem was solved by using a particular set of quadratic forms comingfrom the classical characterizations of positive polynomials on intervals. InSection 4, we will extend this procedure to solve multidimensional momentproblems defined on algebraic domains in Rγ .

3.4 Measure Recovery

The second question behind a particular moment problem is about the construc-tion of a measure µ from a set of values x1, . . . , xk which are supposed to be themoments of µ. Once again, this a very difficult problem in modern mathematics.However, for one dimensional algebraic and trigonometric moment problems wehave the right answer. In Section 5 of [4] the reader can find the proper meth-ods for obtaining a finitely supported measure µ from a finite sequence of onedimensional moments.

These results are briefly recalled in the following. If we take the valuesx0, . . . , x2r as the algebraic moments of a positive measure µ supported on thereal line, its supporting points can be estimated by finding the roots of thepolynomial

P (t) =

∣∣∣∣∣∣∣∣

x0 x1 · · · xj

· · ·xj−1 xj · · · x2j−1

1 t · · · t2j

∣∣∣∣∣∣∣∣where j is linked with the rank of the Hankel matrix H = (xi,j)

ri,j=0 . See [4]

p. 23. Observe here that we only need to know the supporting points of µ todetermine the global minima of f by the Method of Moments.

3.5 One Dimensional Polynomial Programs

The solutions presented here to one dimensional moment problems allow us toapply the theory of the Method of Moments for solving mathematical programsinvolving one-dimensional polynomials. Since we have characterized the onedimensional algebraic moments of positive measures on the line, and becausewe have a practical method to estimate its supporting points from its momentsequence, then we can fruitfully apply the general theory of the Method ofMoments for analyzing arbitrary, non convex, one dimensional polynomial pro-grams. A detailed exposition of the application of the Method of Moments toone dimensional polynomial programs may be found in [4].

14

For instance, to estimate the global minima of a particular one-dimensionalpolynomial given by

f (t) =2r∑

i=0

citi (15)

we should solve the corresponding semidefinite program:

minx

2r∑

i=0

cixi s.t. (xi+j)ri,j=0 ≥ 0 and x0 = 1. (16)

As the polynomial f has no unbounded minimizing sequence, we conclude thatevery solution of the convex program (16) provides a set of global minima ofthe polynomial f in R.

Theorem 17 For every solution x∗ ∈ R2r+1 of the semidefinite program (16) ,there exist finitely many points t1, . . . , tρ ∈ G and positive values λ1, . . . , λρ

satisfying the equations

x∗j = λ1tj1 + · · ·+ λρt

jρ, j = 0, . . . , 2r

where G is the set of all global minima of the polynomial f given in (15) . Hereρ may be chosen to be less than 2r + 3.

Proof. Apply Theorem 10 and the solution of the Hamburger´s MomentProblem.

Theorem 18 A necessary and sufficient condition for finitely many points

t1, . . . , tρ ∈ R

to be global minima of the polynomial f given in (15) , is that the following 2r+1equations

x∗j = λ1tj1 + · · ·+ λρt

jρ, j = 0, . . . , 2r

hold true for some solution x∗ ∈ R2r+1 of the semidefinite program (16) andpositive values λ1, . . . , λρ.

Proof. Apply Corollary 11, Theorem 14 and the solution of the Ham-burger´s Moment Problem.

Corollary 19 If x∗ is an extreme point of the solution set of program (16) ,then x∗1 is a global minimum in R of the polynomial f given by the expression(15) .

Proof. Apply Theorem 12 and the solution of the Hamburger’s MomentProblem.

Since we have obtained an explicit method for determining a finitely sup-ported measure from a sequence of its algebraic moments, we can find the global

15

minima of any one dimensional algebraic polynomial with the form (15) . Letus assume that x∗ ∈ R2r+1 is a solution of the semidefinite program (16) , thenthe roots of the polynomial

P ∗ (t) =

∣∣∣∣∣∣∣∣

x∗0 x∗1 · · · x∗j· · ·

x∗j−1 x∗j · · · x∗2j−1

1 t · · · t2j

∣∣∣∣∣∣∣∣(17)

are global minima of the polynomial f in (15) . See the proof of this fundamentalresult in [4].

Each one of the classical one dimensional moment problems allows us tosolve a general family of non convex one dimensional polynomial programs.The following results illustrate this statement.

Theorem 20 A necessary and sufficient condition for finitely many points

z1, . . . , zρ ∈ S1 ≡ z ∈ C : |z| = 1to be global minima of the trigonometric polynomial

f (z) =r∑

i=−r

cizi

is that the equations

x∗j = λ1zj1 + · · ·+ λρz

jρ, j = −r, . . . , r

hold true for some solution x∗ ∈ C2r+1 of the semidefinite program

minx

r∑

i=−r

cixi s.t. (xk−l)rk,l=0 ≥ 0 and x0 = 1

and positive values λ1, . . . , λρ.

Proof. Apply Corollary 11, Theorem 14 and the solution of the Trigono-metric Moment Problem.

The procedure for recovering a finitely supported measure from its trigono-metric moments is explained in [4].

We can settle similar results for global optimization of one dimensional poly-nomial programs defined in arbitrary intervals of the real line. By using the evencases of Stieltjes and Hausdorff’s Moment Problems we obtain the following the-orems. The reader can infer the analogous results for odd cases.

Theorem 21 A necessary and sufficient condition for points t1, . . . , tρ ≥ 0 tobe global minima of the even degree, algebraic polynomial f given by (15) , isthat the equations

x∗j = λ1tj1 + · · ·+ λρt

jρ, j = 0, . . . , 2r

16

hold true for some solution x∗ ∈ R2r+1 of the semidefinite program:

minx

2r∑

i=0

cixi s.t. (xk+l)rk,l=0 ≥ 0, (xk+l+1)

r−1k,l=0 ≥ 0 and x0 = 1

and positive values λ1, . . . , λρ.

Proof. Apply Corollary 11, Theorem 14 and the solution for the even caseof the Stieltjes Moment Problem.

Theorem 22 A necessary and sufficient condition for points t1, . . . , tρ to beglobal minima in the interval [κ1, κ2] of the even degree, algebraic polynomial fgiven by (15) , is that the equations

x∗j = λ1tj1 + · · ·+ λρt

jρ, j = 0, . . . , 2r

hold true for some solution x∗ ∈ R2r+1 of the semidefinite program:

minx

2r∑

i=0

cixi s.t. (xk+l)rk,l=0 ≥ 0

((κ1 + κ2) xk+l+1 − κ1κ2 xk+l − xk+l+2)r−1k,l=0 ≥ 0

and

x0 = 1

and positive values λ1, . . . , λρ.

Proof. Apply Corollary 11, Theorem 14 and the solution of the even caseof the Hausdorff’s Moment problem.

In this section, we have applied the Method of Moments for transforming aone dimensional polynomial program into an equivalent semidefinite program.The key in this procedure is to find a convenient quadratic form which charac-terizes the algebraic moments of positive measures on intervals. We use a similarprocedure for characterizing the multidimensional algebraic moments of positivemeasures supported on multidimensional domains Ω ⊂ Rn. This characteriza-tion will allow us to apply the Method of Moments for solving arbitrary nonconvex polynomial programs. We will explain this procedure in the followingsections. Finally, because we are transforming non convex polynomial programsinto semidefinite programs, we can use many existing results in software andliterature regarding this kind of optimization problems. See [7] for a review onsemidefinite programming.

4 Multi-Dimensional Moment Problem

At first glance, the Multidimensional Moment Problem seems to be increasinglymore complex to handle than its one-dimensional counterpart. This is owed

17

to the fact that we must characterize the family of all positive polynomials inRγ ; and it is well known from Hilbert’s work, that we can not express everypositive, two variables polynomial as a finite sum of squares, see [8]. Thus, wecan not reproduce the arguments based on duality and quadratic forms givenabove. Nevertheless, we propose to elaborate on recently published results in al-gebra, which actually provide for the right characterization of many dimensionalpositive polynomials in compact semi algebraic sets.

Our starting point is Putinar’s work [9], where the characterization of posi-tive polynomials in a compact semi-algebraic set

K = t ∈ Rn : Pi (t) ≥ 0, i = 1, . . . , kis given. There, the author defines the convex cone V of all finite sums of squaresof polynomials in Rγ , and characterizes the family of positive polynomials inK as all finite sums whose terms are formed by taking one factor from V andthe others (without duplicity) from the family of polynomials Pi. Although thischaracterization may be improved for particular cases, it is sufficient for ourpurposes here.

From this characterization, we can see that every positive polynomial in Kcan be expressed as a proper quadratic form which allows us to apply the dualityresult from Theorem 12. For example, consider the unit circle on the plane

K =t ∈ R2 : 1− t21 − t22 ≥ 0

,

from Putinar’s work we know that every positive polynomial in K can be ex-pressed as ∑

i

q2i (t1, t2) +

(1− t21 − t22

) ∑

j

q2j (t1, t2) (18)

where qi and qj are finite sets of two variables polynomials. Making use of thequadratic form

q2i (t1, t2) =

∑

0≤r+s≤2n

∑

0≤r′+s′≤2n

cir,sc

ir′,s′t

r+r′1 ts+s′

2

which represents the square of a single polynomial qi, we can see that everypositive polynomial in the unitary circle K, can be given by the expression

(19)

∑

i

∑

0≤r+s≤2n

∑

0≤r′+s′≤2n

cir,sc

ir′,s′t

r+r′1 ts+s′

2 +

∑

j

∑

0≤r+s≤2n

∑

0≤r′+s′≤2n

cjr,sc

jr′,s′

(tr+r′1 ts+s′

2 − tr+r′+21 ts+s′

2 − tr+r′1 ts+s′+2

2

).

Then we can apply the duality result of Theorem 12, in order to characterizethe algebraic moments of a positive measure supported in the unitary circle K.

18

Theorem 23 Let M be the convex cone composed of the algebraic, bidimen-sional, moment vectors of all positive measures supported in the unitary circleK. Then, the bi-indexed vector (xr,s)0≤r+s≤2n ∈ R

(n+1)(n+2)2 belongs to the clo-

sure of M, if and only if its entries make positive semidefinite the followingcouple of symmetric matrices:

H1 = (xr+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n

H2 = (xr+r′,s+s′ − xr+r′+2,s+s′ − xr+r′,s+s′+2)0≤r+s≤2n−2,0≤r′+s′≤2n−2

Proof. When the matrices H1 and H2 are positive semidefinite and thevector c = (cr,s)0≤r+s≤2n consists of the coefficients of a non negative polynomialf in K, then f + ε > 0 in K for arbitrary ε > 0. Therefore f + ε can beexpressed in the form (19) . Hence cε · x ≥ 0, where cε is the coefficient vectorof the polynomial f + ε. Thus we conclude that c · x ≥ 0 and that x ∈ P ∗ = M.On the other hand, assuming that c is the coefficient vector of a non-negativepolynomial in K, and x the bidimensional, moment vector of a positive measuresupported in K, we obtain c · x ≥ 0. Finally, we conclude that H1 and H2 arepositive semidefinite matrices by choosing the particular cases

∑

0≤r+s≤2n

cr,str1t

s2

2

,(1− t21 − t22

) ∑

0≤r+s≤2n−2

cr,str1t

s2

2

with arbitrary coefficients cr,s.

Corollary 24 The characterization of the two-dimensional algebraic moments(xr,s)0≤r+s≤2n of a positive measure supported on the circle

Kd =t ∈ R2 : d2 − t21 − t22 ≥ 0

,

is given by the positivity of the matrices

H1 = (xr+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n (20)

Hd2 =

(d2 xr+r′,s+s′ − xr+r′+2,s+s′ − xr+r′,s+s′+2

)0≤r+s≤2n−2,0≤r′+s′≤2n−2

.

At this point it is very important to remark that we have not characterizedthe algebraic moments in R2. We will show that we can dismiss matrix H2 inthe characterization of the algebraic, bidimensional moments in R2.

First we express matrix Hd2 in (20) as

Hd2 = d2H1 −H0

where the matrix H0 is given by

H0 = (xr+r′+2,s+s′ + xr+r′,s+s′+2)0≤r+s≤2n−2,0≤r′+s′≤2n−2 .

19

We have denoted the matrix

(xr+r′,s+s′)0≤r+s≤2n−2,0≤r′+s′≤2n−2

as H1 because it is a sub-matrix of matrix H1 in (20) . Thus, as we have seen,the entries in the bi-indexed vector (xr,s)0≤r+s≤2n are the algebraic momentsof a positive measure supported in the circle Kd, if and only if the matrices H1

and d2H1 −H0 are positive semidefinite.On the other hand, we can exhibit a particular bi-indexed vector

(yr,s)0≤r+s≤2n

whose entries make positive definite the matrix:

A1 = (yr+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n .

Notice that matrices A1 and H1 have the same structure. From A1 we defineA0 and A1, in the same way that we define H0 and H1 from H1.

Now, let us assume that matrix H1 in (20) is positive semidefinite. Then,we claim that we can found d so that matrix

d2(H1 + cA1

)− (H0 + cA0) (21)

is positive definite for any c > 0 arbitrary. Observe that quadratic forms arehomogeneous functions, so their positivity is defined on the unitary sphere.Since matrix H1 is positive semidefinite, the matrix d2

(H1 + cA1

)is positive

definite. Therefore, its corresponding quadratic form takes a positive maximumvalue on the unitary sphere, which can be arbitrarily increased by taking highervalues of d. On the other hand, the quadratic form of the matrix H0 + cA0

takes a fixed maximum on the unitary sphere. Thus for a properly chosen d,the matrix (21) is positive definite. Since H1 + cA1 is also positive definite, weconclude that the bi-indexed vector

(xr,s + cyr,s)0≤r+s≤2n

consists of the algebraic moments of a positive measure supported in the circleKd. This argument shows that any bi-indexed vector (xr,s)0≤r+s≤2n satisfy-ing the matrix inequality H1 ≥ 0 stated in (20) may be approximated by themoment vectors of positive measures supported in the plane.

4.1 Two Dimensional Moment Problem

Now we can characterize easily the two dimensional algebraic moments of pos-itive measures supported in the plane and use them for treating unboundedpolynomial programs in two variables.

Theorem 25 A necessary and sufficient condition for a bi-indexed vector

(xr,s)0≤r+s≤2n ∈ R(n+1)(n+2)

2

20

to be a bidimensional, algebraic moment vector of a positive measure supportedin the plane (or at least, it is a limit point of them), is that its entries form apositive semidefinite quadratic form given by

H1 = (xr+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n .

Proof. We have seen above the case when H1 is positive semidefinite. Con-versely, assuming that (xr,s)0≤r+s≤2n is a bidimensional moment vector from ameasure µ, let us define the values xd

r,s by the integration on the circle Kd

xdr,s =

∫

Kd

tr1ts2 dµ

so it is clear that(xd

r+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n

→ H1 when d → ∞, and we

deduce that H1 is positive semidefinite. In fact, the values xdr,s are the moments

of a positive measure in the circle Kd.

4.1.1 Two Dimensional Unbounded Polynomial Programs

Now we can apply the solution of the Two Dimensional Moment Problem foranalyzing unbounded non convex polynomial programs in two variables. Let usassume that the two variables polynomial

f (t1, t2) =∑

0≤r+s≤2n

cr,str1t

s2 (22)

is bounded from below. We first apply the general theory of the Method ofMoments for the global optimization of the polynomial f in R2. Since f hasno unbounded minimizing sequence, Theorem 10 shows us that the extendedprogram (6) attains an optimal solution in a moment vector corresponding toa finitely supported measure µ∗ whose supporting points are global minima off. The solution of the Two Dimensional Moment Problem allow us to posethe extended program (6) as a particular semidefinite program whose solutionsprovide the global minima of the non convex polynomial f in R2.

Theorem 26 A necessary and sufficient condition for finitely many points

w1, . . . , wρ ∈ R2

to be global minima of f, is that they satisfy the equations

x∗ = λ1T (w1) + · · ·+ λρT (wρ)

for positive values λ1, . . . , λρ, where the bi-indexed vector x∗ ∈ R(n+1)(n+2)

2 is asolution of the semidefinite program

minx

∑

0≤r+s≤2n

cr,sxr,s s.t. (xr+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n ≥ 0 with x00 = 1

(23)

21

and the transformation T : R2 → R(n+1)(n+2)

2 is defined by

T (t) = (tr1ts2)0≤r+s≤2n .

Proof. As usual, apply Corollary 11, Theorem 14 and the solution of theTwo Dimensional Moment Problem.

Corollary 27 If x∗ is an extreme point of the solution set of the semidefiniteprogram (23) , the point

(x∗1,0, x∗0,1

) ∈ R2 is a global minimum of the bidimen-sional polynomial f.

Proof. Apply Theorem 12.Before extending these results to higher dimensions, we give some comments

about the construction of two dimensional measures from their moments.

4.1.2 Measure Recovery

Recovering a bidimensional, finitely supported measure from a sequence of bidi-mensional moments is a very challenging problem. See [12]. Fortunately, in an-alyzing two dimensional polynomial programs, we can use the results obtainedfor the one dimensional situation. See [4].

Let us assume that values(x∗r,s

)0≤r+s≤2n

solve the semidefinite program(23) , then from Theorem 10, we know that values x∗r,s are the bidimensionalalgebraic moments of a discrete probability distribution µ∗ supported on theset G of global minima of the two variables polynomial f in (22) . Obviously,(x∗r,0

)and

(x∗0,s

)are the one dimensional algebraic moments of the marginal

distributions of µ∗ with respect to the Cartesian axes of the plane R2. Then, wecan apply twice the equation (17) in order to estimate the coordinates of thefeasible global minima of the polynomial f.

4.1.3 Multi-Dimensional Unbounded Polynomial Programs

We can easily see that the multidimensional algebraic moments of a positivemeasure in Rγ are characterized by the positivity of the quadratic form

H =(xs1+s′1,...,sγ+s′γ

)0≤s1+···+sγ≤2n;0≤s′1+···+s′γ≤2n

(24)

which comes from taking the square of the general polynomial

f (t1, . . . , tγ) =∑

0≤s1+···+sγ≤2n

cs1,...,sγts11 · · · tsγ

γ . (25)

By the same arguments used for two dimensional cases, we claim that all globalminima of any γ-dimensional polynomial in (25) can be estimated by solvingthe semidefinite program

minx

∑

0≤s1+···+sk≤2n

cs1,...,skxs1 · · ·xsk

22

where the variables x are constrained by the positivity of the quadratic form Hin (24) and the constraint x0,...,0 = 1.

5 Non Convex Quadratic Programs

The most impressive achievement of the Method of Moments in global optimiza-tion is to allow us to analyze arbitrary non convex quadratic programs. Thegeneral form of a quadratic program is given by

mint

t ·A0t + b0 · t + c0 s.t. t ·Ait + bi · t + ci ≥ 0, for i = 1, . . . , j (26)

where t ∈ Rk, Ai ∈ Rk×k, bi, ci ∈ Rk. The first question to be answered here isabout the existence of feasible points for program (26) . The second question isabout the optimal solutions for problem (26) . Both questions are easily solvedby applying the Method of Moments. The essential features of the Method ofMoments remain when analyzing quadratic problems, however the notation maybecome much more cumbersome. For the sake of simplicity in the presentation,we choose to describe the method by solving a particular example.

Let us consider the quadratic program

mint

−4 + 4t1 − t21 − t22 s.t. 4− t21 − t22 ≥ 0, t21 + t22 − 1 ≥ 0 (27)

which has a very clear geometric meaning: find the farthest point from point(2, 0) , laying within the annulus Γ defined by the circles of radius 1 and 2 andcentered at the origin. In order to tackle this problem by the Method of Mo-ments, we need to characterize the algebraic moments of all probability measuressupported on Γ. Since this domain is defined by two inequalities involving twopolynomials, we can characterize the positive polynomials in Γ as:

(∑q2i

)+

(4− t21 − t22

) (∑q2i

)+

(t21 + t22 − 1

) (∑q2i

)

+(4− t21 − t22

) (t21 + t22 − 1

) (∑q2i

)

see [9] and expression (18) . Then, taking the product of the two constraints(4− t21 − t22

) (t21 + t22 − 1

)= −4 + 5t21 + 5t22 − 2t21t

22 − t41 − t42 ,

we deduce the characterization of the algebraic moments of positive measuressupported in Γ.

Theorem 28 A bi-indexed vector (xr,s)0≤r+s≤2n is a moment vector of somepositive measure supported in the annulus Γ, if and only if, the following matrices

23

are positive semidefinite:

H1 = (xr+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n (28)

H2 = (4 xr+r′,s+s′ − xr+r′+2,s+s′ − xr+r′,s+s′+2)0≤r+s≤2n−2,0≤r′+s′≤2n−2

H3 = (−xr+r′,s+s′ + xr+r′+2,s+s′ + xr+r′,s+s′+2)0≤r+s≤2n−2,0≤r′+s′≤2n−2

H4 = (−4 xr+r′,s+s′ + 5 xr+r′+2,s+s′ + 5 xr+r′,s+s′+2 − 2xr+r′+2,s+s′+2

−xr+r′+4,s+s′ − xr+r′,s+s′+4)0≤r+s≤2n−4,0≤r′+s′≤2n−4

At this stage, the reader should note the transcription of the constraints ofthe quadratic problem (27) into a proper arrangement of indexes imposed onthe entries of a set of quadratic forms (28) which must be positive semidefinite.

Now we can introduce our geometric, non convex program, as the semidefi-nite program:

minx

4x1,0 − x2,0 − x0,2 , (29)

subjected to the constraints guaranteeing that the matrices H1,H2,H3 and H4

in (28) are all positive semidefinite. By applying Theorem 10, we conclude thatprogram (29) has the unique optimal solution

(x∗r,s

)0≤r+s≤4

given as

x∗r,s = ((−2)r, 0) for 0 ≤ r + s ≤ 4.

Now it should be clear how to apply this technique to general polynomialprograms, where we must minimize an arbitrary polynomial expression subjectto several non linear constraints given by polynomials. It is important to stressupon the fact that the higher the number of restrictions, the higher the numberof quadratic forms which need be included into the equivalent convex program.On the other hand, it is important to observe that the Method of Momentscan transform the feasibility analysis of a particular non convex quadratic pro-gram, or even a general polynomial program, into the feasibility analysis of aLinear Matrix Inequality which is a known problem analyzed in recent workson semidefinite programming, see [10].

Theorem 29 The feasible set of a particular quadratic program is not empty,if and only if the feasible set of its corresponding semidefinite relaxation is notempty.

Proof. When Ω is the domain defined by all inequalities in (26) , take theDirac measure δt0 for t0 ∈ Ω. Conversely, take a point on the support of µ,where the moments of the measure µ satisfy the positivity restrictions on a setof quadratic forms coming from a feasible semidefinite relaxation.

We now illustrate the essentials of the Method of Moments by investigatinga particular example.

5.1 Example

24

Let us analyze the particular two variables polynomial program given by

mint

f (t) =(1− t21

)2+

(1− t22

)2s.t. 1− t31 + t2 ≥ 0 (30)

which clearly has three global minima located at the points (1, 1) and (−1,±1)of the plane R2. First we notice that the objective function f has no unboundedminimizing sequences. Then, we observe that the feasible set

Ω =t ∈ R2 : 1− t31 + t2 ≥ 0

(31)

is unbounded. However, as we did in the previous section, we can use a dummyconstraint to restrict the problem within the circle Kd. In this way, we can provethat only the defining constraint in (31) is necessary for the analysis.

In order to characterize the algebraic moments of positive measures sup-ported on the feasible set Ω, we use the general characterization of positivepolynomials in Ω given by

∑

i

q2i (t1, t2) +

(1− t31 + t2

)∑

j

q2j (t1, t2)

where qi and qj are finite sets of polynomials. See [9]. From the duality state-ment of Theorem 12, we conclude that the necessary and sufficient condition fora bi-indexed vector (xr,s)0≤r+s≤2n to be a moment vector of a positive measuresupported in Ω, is that the following quadratic forms be positive semidefinite

H1 = (xr+r′,s+s′)0≤r+s≤2n,0≤r′+s′≤2n

H5 = (xr+r′,s+s′ − xr+r′+3,s+s′ + xr+r′,s+s′+1)0≤r+s≤2n−2,0≤r′+s′≤2n−2

then the semidefinite relaxation for the program (30) takes the form

min 1− 2x2,0 + x4,0 + 1− 2x0,2 + x0,4

s.t

1 x1,0 x0,1 x2,0 x1,1 x0,2

x1,0 x2,0 x1,1 x3,0 x2,1 x1,2

x0,1 x1,1 x0,2 x2,1 x1,2 x0,3

x2,0 x3,0 x2,1 x4,0 x3,1 x2,2

x1,1 x2,1 x1,2 x3,1 x2,2 x1,3

x0,2 x1,2 x0,3 x2,2 x1,3 x0,4

≥ 0

[1− x3,0 + x0,1 x0,1 − x3,1 + x0,2

x0,1 − x3,1 + x0,2 x0,2 − x3,2 + x0,3

]≥ 0

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whose marginal solutions have the form

x∗1,0 = λ (−1) + (1− λ) (1)x∗2,0 = 1x∗3,0 = λ (−1) + (1− λ) (1)x∗4,0 = 1x∗0,1 = λ′ (−1) + (1− λ′) (1)x∗0,2 = 1x∗0,3 = λ′ (−1) + (1− λ′) (1)x∗0,4 = 1

where 0 < λ, λ′ < 1. By using equation (17) , we obtain the new feasible set(±1,±1) , which contains all the global minima of the objective polynomial f.

6 Concluding Remarks

In this paper we have seen how to treat non convex polynomial programs in sev-eral variables by reducing them to a single semidefinite program which enclosesthe information about the optimal solutions of the original problem. Althoughsemidefinite relaxations have been proposed in the past by other authors, we be-lieve that our approach also improves previous works as it clarifies how to extendone dimensional results to multidimensional cases. We have actually attainedthis task by focussing on the particular quadratic form that solves every multi-dimensional moment problem in arbitrary semi-algebraic sets. In addition, thispaper also shows that a properly posed single semidefinite program is sufficientfor analyzing every polynomial program. Moreover, the method proposed heremay be applied to the analysis of the general non convex quadratic problem. Infact, we can determine the feasible set and the optimal solutions of every nonconvex quadratic program.

References

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[3] Krein, M.G. and A.A. Nudel’man, The Markov Moment Problem and Ex-tremal Problems, Translations of Mathematical Monographs, vol. 50, AMS,1977.

[4] Meziat, R., The method of moments in global optimization, to appear inJournal of Mathematical Sciences, Kluwer.

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[5] Meziat, R., P. Pedregal and J.J. Egozcue, The method of moments fornon convex variational problems, Advances in Convex Analysis and GlobalOptimization, Non Convex Optimization and Its Applications Series, vol.54, p. 371-382, Kluwer, 2001.

[6] Meziat, R., Two dimensional non convex variational problems, to appearin the proceedings of the International Workshop in Control and Optimiza-tion, Erice, Italy, 2001.

[7] Ben-Tal, A. and A. Nemirovski, Lectures on Modern Convex Optimization,MPS-SIAM, 2001.

[8] Shor, N.Z., Nondifferentiable Optimization and Polynomial Problems,Kluwer, 1998.

[9] Putinar, M., Positive polynomials on compact semi-algebraic sets, IndianaUniversity Mathematics Journal, vol. 42, No. 3, pages 969-984, 1993.

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[11] Lasserre, J., Semidefinite programming vs lp relaxations for polynomialprogramming, Mathematics of Operations Research Journal, vol. 27, No 2,pp. 347-360, 2002.

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