A "joint+marginal" algorithm for polynomial optimization

TL;DR

This paper introduces a novel 'joint+marginal' semidefinite relaxation algorithm for polynomial optimization, iteratively solving univariate problems to approximate the global optimum with promising initial results.

Contribution

It develops a new hierarchical approach combining joint and marginal relaxations for polynomial optimization, enabling iterative univariate approximations of the optimal value function.

Findings

Converges to the optimal value function as the hierarchy level increases.

Allows reduction of multivariate problems to univariate polynomial approximations.

Preliminary numerical results show potential effectiveness.

Abstract

We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the associated (n-1)-variable (x2,...,xn) problem P(x1) where the parameter x1 is fixed and takes values in some interval Y1 with some probability uniformly distributed on Y1. Then one considers the hierarchy of what we call "joint+marginal" semidefinite relaxations, whose duals provide a sequence of univariate polynomial approximations that converges to the optimal value function J(x1) of problem P(x1), as k increases. Then with k fixed a priori, one computes a minimizer of the univariate polynomial pk(x1) on the interval Y1, which reduces to solving a single semidefinite program. One iterates the procedure with now an (n-2)-variable problem P(x2) with parameter x2 in some new interval Y2, etc. The quality of the approximation depends on how large k can be chosen (in general for significant size problems, k=1 is the only choice). Preliminary numerical results are provided