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Detecting global optimality and extracting solutions in GloptiPoly
 Chapter in D. Henrion, A. Garulli (Editors). Positive polynomials in control. Lecture Notes in Control and Information Sciences
, 2005
"... GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of po ..."
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Cited by 80 (12 self)
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GloptiPoly is a Matlab/SeDuMi addon to build and solve convex linear matrix inequality (LMI) relaxations of nonconvex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sumofsquares decompositions of positive polynomials, the theory of moments allows to detect global optimality of an LMI relaxation and extract globally optimal solutions. In this report, we describe and illustrate the numerical linear algebra algorithm implemented in GloptiPoly for detecting global optimality and extracting solutions. We also mention some related heuristics that could be useful to reduce the number of variables in the LMI relaxations. 1
Minimizing polynomials via sum of squares over the gradient ideal
 Math. Program
"... A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown ..."
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Cited by 51 (17 self)
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A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown to be SOS modulo its gradient ideal, provided the gradient ideal is radical or the polynomial is strictly positive on the gradient variety. This opens up the possibility of solving previously intractable polynomial optimization problems. The related problem of constrained minimization is also considered, and numerical examples are discussed. Experiments show that our method using the gradient variety outperforms prior SOS methods.
Theta Bodies for Polynomial Ideals
, 2009
"... Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s th ..."
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Cited by 43 (8 self)
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Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals.
Revisiting Two Theorems of Curto and Fialkow on Moment Matrices
, 2004
"... We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence... ..."
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Cited by 31 (4 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
Semidefinite characterization and computation of zerodimensional real radical ideals
, 2007
"... real radical ideals ..."
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Optimality conditions and finite convergence of Lasserres hierarchy
 Mathematical Programming
, 2013
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Representation of nonnegative polynomials, degree bounds and applications to optimization
, 2006
"... Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are giv ..."
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Cited by 22 (2 self)
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Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global optimization. In many cases, there are degree bounds for such presentations. These bounds are of theoretical interest, but they appear to be too large to be of much practical use at present. In the final section, other more concrete degree bounds are obtained which ensure at least that the feasible set of solutions is not empty.
Hilbert’s Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility
, 2008
"... Systems of polynomial equations over an algebraicallyclosed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution o ..."
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Cited by 19 (7 self)
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Systems of polynomial equations over an algebraicallyclosed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert’s Nullstellensatz certificates for polynomial systems arising in combinatorics and on largescale linearalgebra computations over K. We report on experiments based on the problem of proving the non3colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.
Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals
, 2006
"... This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R n: g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) ..."
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Cited by 19 (4 self)
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This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S = {x ∈ R n: g1(x) ≥ 0, · · · , gs(x) ≥ 0} modulo its KKT (KarushKuhnTucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)> 0 on S; furthermore, when the KKT ideal is radical, we have that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x) ≥ 0 on S. This is a generalization of results in [18], which discuss the SOS representations of nonnegative polynomials over gradient ideals. Key words: Polynomials, semialgebraic set, sum of squares (SOS), KarushKuhnTucker (KKT) system, KKT ideal. 1
Approximability and proof complexity
, 2012
"... This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Pa ..."
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Cited by 17 (6 self)
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This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any nvariable degreed proof can be found in time n O(d). Furthermore, the SDP is dual to the wellknown Lasserre SDP hierarchy, meaning that the “d/2round Lasserre value ” of an optimization problem is equal to the best bound provable using a degreed SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known “hard instances ” for the UniqueGames problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the BalancedSeparator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot–Vishnoi MaxCut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor.952 (>.878) using a constantdegree proof. These investigations also raise an interesting mathematical question: is there a constantdegree SOS proof of the Central Limit Theorem?