A "joint+marginal" approach to parametric polynomial optimization

TL;DR

This paper introduces a hierarchy of semidefinite relaxations for polynomial optimization problems with parameters, enabling the approximation of optimal solutions, their distributions, and related functionals through moment-based methods.

Contribution

It develops a novel 'joint+marginal' approach that leverages moments of probability measures to approximate all solutions and functionals of parametric polynomial optimization problems.

Findings

Convergence of the relaxation hierarchy to the true moment vector.

Ability to approximate distributions and functionals of optimal solutions.

Provision of polynomial lower bounds to the optimal value function.

Abstract

Given a compact parameter set $Y\subset R^p$, we consider polynomial optimization problems $(P_y$) on $R^n$ whose description depends on the parameter $y\inY$. We assume that one can compute all moments of some probability measure $\phi$ on $Y$, absolutely continuous with respect to the Lebesgue measure (e.g. $Y$ is a box or a simplex and $\phi$ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions $x^*(y)$ of $P_y$. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their $\phi$-mean. In addition, using this knowledge on moments, the measurable function $y\mapsto x^*_k(y)$ of the $k$-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable $x_k$, one may approximate as closely as desired its persistency $\phi(\{y:x^*_k(y)=1\})$, i.e. the probability that in an optimal solution $x^*(y)$, the coordinate $x^*_k(y)$ takes the value 1. At last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp. piecewise polynomial) lower approximations with $L_1(\phi)$ (resp. almost uniform) convergence to the optimal value function.