Optimization of Polynomials on Compact Semialgebraic Sets

Abstract

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. In the case where f possesses a unique minimizer x * , we prove that every sequence of "nearly" optimal solutions of the successive relaxations gives rise to a sequence of points in R n converging to x * .

Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

“…for any stable set S in G of size r. (24) In order to prove Theorem 2, we have to show that, for 1 ≤ r ≤ min(α(G)−1, 6),…”

“…As was already proved by Hilbert in 1888 not every nonnegative multivariate polynomial can be written as a sum of squares (see Reznick [19] for a nice survey on this topic). However, some representation theorems have been proved ensuring the existence of certain sums of squares decompositions under some assumption, like positivity of the polynomial on a compact basic closed semi-algebraic set (see, e.g., [24] for an exposition of such results). An early such result is due to Pólya [18] who showed that, if p(x) is a homogeneous polynomial which is positive on R n + \ {0}, then ( n i=1 x i ) r p(x) has only nonnegative coefficients [and thus ( n i=1 x 2 i ) r p(x 2 1 , .…”

A note on the representation of positive polynomials with structured sparsity

“…In 1991, Schmüdgen [S] proved a very surprising partial converse to these facts (for example, if M M ⊆ M , then M is archimedean if and only if S is compact). See [Sch,Section 1] and [PD] for this and a discussion when M is archimedean. Shortly after this groundbreaking work of Schmüdgen, Putinar [Put] proved the following theorem (note that f > 0 on S should mean that f is strictly positive on S):…”

“…The idea is to maximize λ such that f − λ lies in a certain finitedimensional subset of M that can be expressed in a semidefinite program (SDP for short) whose size depends on the size of the chosen subset of M . Since M can be exhausted by such subsets, the results of Schmüdgen and Putinar say that the accuracy of this method is arbitrarily good for large SDPs [L1,Sch].…”

On parameter-dependent Lyapunov functions for robust stability of linear systems

“…It can be shown [23] that if problem (4) has a unique global minimizer (which is a generic property) then lim k→∞ p k = p . In particular if p < 0, then necessarily p k < 0 for all k ≥ k and some finite relaxation order k.…”