An Approximation Bound Analysis for Lasserre's Relaxation in Multivariate Polynomial Optimization

Abstract

Suppose f,g1,...,gm are multivariate polynomials in x∈ℝn and their degrees are at most 2d. Consider the problem: Minimize f(x) subject to g1(x) ≥ 0, ... ,gm(x) ≤ 0. Let fmin (resp., fmax) be the minimum (resp., maximum) of f on the feasible set S, and fsos be the lower bound of fmin given by Lasserre's relaxation of order d. This paper studies its approximation bound. Under a suitable condition on g1,...,gm, we prove that (fmax - fsos) ≤ Q(fmax - fmin) with Q a constant depending only on g1,...,gm but not on f. In particular, if S is the unit ball, Q = O(n2d); if S is the boolean set, Q = O(nd). © 2013 Operations Research Society of China, Periodicals Agency of Shanghai University, and Springer-Verlag Berlin Heidelberg.