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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 52 (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.
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 19 (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?
TreewidthBased Conditions for Exactness Of The SheraliAdams and Lasserre Relaxations
, 2004
"... The SheraliAdams (SA) and Lasserre (LS) approaches are "liftandproject" methods that generate nested sequences of linear and/or semidefinite relaxations of an arbitrary 01 polytope . Although both procedures are known to terminate with an exact description of P after n steps, th ..."
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Cited by 14 (3 self)
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The SheraliAdams (SA) and Lasserre (LS) approaches are "liftandproject" methods that generate nested sequences of linear and/or semidefinite relaxations of an arbitrary 01 polytope . Although both procedures are known to terminate with an exact description of P after n steps, there are various open questions associated with characterizing, for particular problem classes, whether exactness is obtained at some step s n. This paper provides sufficient conditions for exactness of these relaxations based on the hypergraphtheoretic notion of treewidth. More specifically, we relate the combinatorial structure of a given polynomial system to an underlying hypergraph. We prove that the complexity of assessing the global validity of moment sequences, and hence the tightness of the SA and LS relaxations, is determined by the treewidth of this hypergraph. We provide some examples to illustrate this characterization.
An Improved Semidefinite Programming Relaxation for the Satisfiability Problem
, 2002
"... The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there ..."
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Cited by 13 (3 self)
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The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NPcomplete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable X is maximized (or minimized) subject to linear constraints on the elements of X and the additional constraint that X be positive semidefinite. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of “higher liftings” for constructing semidefinite programming relaxations of discrete optimization problems.
Interior point and semidefinite approaches in combinatorial optimization
, 2005
"... Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient ..."
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Cited by 9 (5 self)
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Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primaldual interiorpoint methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NPhard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to nonconvex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include nonconvex potential reduction methods, interior point cutting plane methods, primaldual IPMs and firstorder algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.
Semidefinite optimization approaches for satisfiability and maximumsatisfiability problems
 J. Satisf. Bool. Model. Comput
"... Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the res ..."
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Cited by 8 (3 self)
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Semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximumsatisfiability problems. The approaches presented in some detail include the groundbreaking approximation algorithm of Goemans and Williamson for MAX2SAT, the Gap relaxation of de Klerk, van Maaren and Warners, and strengthenings of the Gap relaxation based on the Lasserre hierarchy of semidefinite liftings for polynomial optimization problems. We include theoretical and computational comparisons of the aforementioned semidefinite relaxations for the special case of 3SAT, and conclude with a review of the most recent results in the application of semidefinite programming to SAT and MAXSAT.
Optimization over Polynomials: Selected Topics
"... Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of s ..."
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Cited by 1 (0 self)
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Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of squares of polynomials) and functional analysis (moments of measures) with semidefinite optimization. Sums of squares are used to certify positive polynomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this general methodology, illustrate how it applies to some combinatorial graph problems, and discuss links with other relaxation methods.
Improved SDO Relaxations for SAT The Tseitin Instances on Toroidal Grid Graphs
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