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131
Global Optimization with Polynomials and the Problem of Moments
 SIAM Journal on Optimization
, 2001
"... We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear mat ..."
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Cited by 320 (33 self)
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We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems. A notion of KarushKuhnTucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided. Key words. global optimization, theory of moments and positive polynomials, semidefinite programming AMS subject classifications. 90C22, 90C25 PII. S1052623400366802 1.
Semidefinite Programming Relaxations for Semialgebraic Problems
, 2001
"... A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The mai ..."
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Cited by 222 (18 self)
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A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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Optimal inequalities in probability theory: A convex optimization approach
 SIAM Journal of Optimization
"... Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provi ..."
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Cited by 66 (10 self)
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Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provide optimal bounds on P (X ∈ S), when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and Ω are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NPhard to find tight bounds for k ≥ 4 and Ω = Rn and for k ≥ 2 and Ω = Rn +, when the data in the problem is rational. For k =1andΩ=Rn + we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and Ω are unions of convex sets, over which linear functions can be optimized efficiently. For the case k =2andΩ=Rn, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently. Key words. optimization probability bounds, Chebyshev inequalities, semidefinite optimization, convex
The truncated complex Kmoment problem
 Trans. Amer. Math. Soc
"... on the occasion of his eightyseventh birthday Abstract. Let γ ≡ γ (2n) denote a sequence of complex numbers γ00,γ01,γ10,...,γ0,2n,...,γ2n,0 (γ00> 0,γij = ¯γji), and let K denote a closed subset of the complex plane C. The Truncated Complex KMoment Problem for γ entails determining whether there ex ..."
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Cited by 48 (5 self)
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on the occasion of his eightyseventh birthday Abstract. Let γ ≡ γ (2n) denote a sequence of complex numbers γ00,γ01,γ10,...,γ0,2n,...,γ2n,0 (γ00> 0,γij = ¯γji), and let K denote a closed subset of the complex plane C. The Truncated Complex KMoment Problem for γ entails determining whether there exists a positive Borel measure µ on C such that γij = ∫ ¯z izj dµ (0 ≤ i + j ≤ 2n) and supp µ ⊆ K. For K ≡ KP a semialgebraic set determined by a collection of complex polynomials P = {pi (z, ¯z)} m i=1, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n)(γ) and the localizing matrices Mp i. We prove that there exists a rank M (n)atomic representing measure for γ (2n) supported in KP if and only if M (n) ≥ 0andthereissomerankpreserving extension M (n +1)forwhichMp i (n + ki) ≥ 0, where deg pi =2ki or 2ki − 1(1 ≤ i ≤ m). 1.
Optimization of polynomials on compact semialgebraic sets
 SIAM J. Optim
"... Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter conve ..."
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Cited by 43 (5 self)
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Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ∗ of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x ∗ , we prove that every sequence of “nearly ” optimal solutions of the successive relaxations gives rise to a sequence of points in R n converging to x ∗. 1. Introduction to Lasserre’s method Throughout the paper, we suppose 1 ≤ n ∈ N and abbreviate (X1,..., Xn) by ¯X. We let R [ ¯ X] denote the ring of real polynomials in n indeterminates. Suppose we are given a so called basic closed semialgebraic set, i.e., a set S: = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0}
Sums of squares of regular functions on real algebraic varieties
 Tran. Amer. Math. Soc
, 1999
"... Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a ..."
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Cited by 43 (8 self)
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Abstract. Let V be an affine algebraic variety over R (or any other real closed field R). We ask when it is true that every positive semidefinite (psd) polynomial function on V is a sum of squares (sos). We show that for dim V ≥ 3 the answer is always negative if V has a real point. Also, if V is a smooth nonrational curve all of whose points at infinity are real, the answer is again negative. The same holds if V is a smooth surface with only real divisors at infinity. The “compact ” case is harder. We completely settle the case of smooth curves of genus ≤ 1: If such a curve has a complex point at infinity, then every psd function is sos, provided the field R is archimedean. If R is not archimedean, there are counterexamples of genus 1.
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming prob ..."
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Cited by 37 (7 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.
Positivity sums of squares and the multidimensional moment problem
 II, Adv. Geom
, 2005
"... Abstract. Let K be the basic closed semialgebraic set in Rn defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f apolynomialinnvariables nonnegative on K and real ɛ>0, we have that f + ɛ ∈ T. In particular, the KMoment Problem has a positive solution. ..."
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Cited by 27 (7 self)
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Abstract. Let K be the basic closed semialgebraic set in Rn defined by some finite set of polynomials S and T, the preordering generated by S. For K compact, f apolynomialinnvariables nonnegative on K and real ɛ>0, we have that f + ɛ ∈ T. In particular, the KMoment Problem has a positive solution. In the present paper, we study the problem when K is not compact. For n = 1, we show that the KMoment Problem has a positive solution if and only if S is the natural description of K (see Section 1). For n ≥ 2, we show that the KMoment Problem fails if K contains a cone of dimension 2. On the other hand, we show that if K is a cylinder with compact base, then the following property holds: (‡) ∀f ∈ R[X],f ≥ 0onK ⇒∃q ∈ T such that ∀ real ɛ>0,f + ɛq ∈ T. This property is strictly weaker than the one given in Schmüdgen (1991), but in turn it implies a positive solution to the KMoment Problem. Using results of Marshall (2001), we provide many (noncompact) examples in hypersurfaces for which (‡) holds. Finally, we provide a list of 8 open problems.