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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 36 (6 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.
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.
TreewidthBased Conditions for Exactness Of The SheraliAdams and . . .
, 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, there are va ..."
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Cited by 9 (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 su# cient 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.
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 6 (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.
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 2 (1 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?
Improved SDO Relaxations for SAT The Tseitin Instances on Toroidal Grid Graphs
"... The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. We consider instances of SAT specified by a set of boolean variables x1,..., xn and a propositional formula Φ = m∧ Cj, with each clause Cj having the form j=1 Cj = ∨ xk ∨ ∨ ¯xk ..."
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The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. We consider instances of SAT specified by a set of boolean variables x1,..., xn and a propositional formula Φ = m∧ Cj, with each clause Cj having the form j=1 Cj = ∨ xk ∨ ∨ ¯xk where Ij, Īj ⊆ {1,..., n}, Ij ∩ Īj = ∅, k∈Ij k ∈ Īj and ¯xi denotes the negation of xi. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. This research is concerned with the application of Semidefinite Optimization (SDO) to the SAT problem [4], particularly for proving unsatisfiability. The ultimate goal is a practical SDObased algorithm for solving SAT. Let P denote the set of all nonempty sets I ⊆ {1,..., n} such that the term ∏ xi appears in the instance’s satisfiability i∈I conditions. Introduce new variables xI: = ∏ xi for each I ∈ P, define the vector v: = (1, xI1,..., xI) P  T, and define the rankone matrix Y: = vvT whose rows and columns are indexed by ∅ ∪ P. By construction, Y∅,I = xI for all I ∈ P. Furthermore, YI1,I2 = YI3,I4 whenever I1∆I2 = I3∆I4, where Ii∆Ij denotes the symmetric difference of Ii and Ij. The tradeoff involved in adding such constraints to the SDO is that as the number of constraints increases, the SDO problems become increasingly more demanding computationally. We add the smaller set of constraints: Y ∅,I1 = YI2,I3, Y ∅,I2 = YI1,I3, and Y ∅,I3 = YI1,I2 (1) i∈I Consider a toroidal grid graph of the form