GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality (LMI) relaxations of non-convex optimization problems with multivariate polynomial objective function and constraints, based on the theory of moments. In contrast with the dual sum-of-squares 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

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.

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...

Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. We show that the minimum-degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colorability, we found only graphs with Nullstellensatz certificates of degree four.

real radical ideals

Abstract. 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. 1.

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.

Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, 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 large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.

Abstract. Let β ≡ β (2n) be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rank-preserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, which is r-atomic, with suppµ equal to V(M(n + 1)), the algebraic variety of M(n + 1). Further, β has an r-atomic (minimal) representing measure supported in a semi-algebraic set KQ subordinate to a family Q ≡ {qi} m i=1 ⊆ R[t1,..., tN] if and only if M(n) is positive semidefinite and admits a rank-preserving extension M(n + 1) for which the associated localizing matrices Mqi (n + [ 1+deg qi]) are positive semidefinite (1 ≤ i ≤ m); in this case, µ (as 2 above) satisfies supp µ ⊆ KQ, and µ has precisely rank M(n) − rank Mqi (n + [ 1+deg qi]) atoms in 2 Z(qi) ≡ { t ∈ R N: qi(t) = 0} , 1 ≤ i ≤ m.

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 (Karush-Kuhn-Tucker) 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), Karush-Kuhn-Tucker (KKT) system, KKT ideal. 1