Sums of squares, moment matrices and optimization over polynomials

Journal of Global Optimization, 2009

We briefly review the duality between moment problems and sums of squares (s.o.s.) representations of positive polynomials, and compare s.o.s. versus nonnegative polynomials. We then describe how to use such results to define convergent semidefinite programming relaxations in polynomial optimization as well as for the two related problems of computing the convex envelope of a rational function and finding all zeros of a system of polynomial equations.

2003

A polynomial SDP (semidefinite programs) minimizes a polynomial objective function over a feasible region described by a positive semidefinite constraint of a symmet- ric matrix whose components are multivariate polynomials. Sums of squares relaxations developed for polynomial optimization problems are extended to propose sums of squares relaxations for polynomial SDPs with an additional constraint for the variables to be in

Optimization Methods and Software, 2015

We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by (f * − ǫ) and from above by f * + ǫ(n + 1), where f * is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

SIAM Journal on Optimization, 2005

A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g 1 ≥ 0, . . . , gm ≥ 0. We give a short introduction to Lasserre's method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f * of f on S which is constructive and elementary.

2009

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide two algebraic certificate of convexity which can be checked numerically. The second is simpler and holds if a sufficient (and almost necessary) condition is satisfied, it also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie [6]. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures.

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.

2012

There are a wide variety of mathematical problems in different areas which are classified under the title of Moment Problem. We are interested in the moment problem with polynomial data and its relation to real algebra and real algebraic geometry. In this direction, we consider two different variants of moment problem.

Mathematical Programming, 2007

This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let E and E+ be a finite dimensional real vector space and a symmetric cone embedded in E; examples of E and E+ include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a

2008

Journal of Pure and Applied Algebra, 2009

Let f , g i , i = 1, . . . , l, h j , j = 1, . . . , m, be polynomials on R n and S := {x ∈ R n | g i (x) = 0, i = 1, . . . , l, h j (x) ≥ 0, j = 1, . . . , m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.