We consider the problem of minimizing a polynomial over a semi-algebraic 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.

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 NP-complete, 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.

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 maximum-satisfiability problems. The approaches presented in some detail include the ground-breaking approximation algorithm of Goemans and Williamson for MAX-2-SAT, 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 3-SAT, and conclude with a review of the most recent results in the application of semidefinite programming to SAT and MAX-SAT.

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 SDO-based 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 rank-one 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

This work is concerned with the proof-complexity of certifying that optimization problems do not have good solutions. Specifically we consider bounded-degree “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 n-variable degree-d proof can be found in time n O(d). Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the “d/2-round Lasserre value ” of an optimization problem is equal to the best bound provable using a degree-d 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 Unique-Games 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 Balanced-Separator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot–Vishnoi Max-Cut 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 constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?