An algorithm for semi-infinite polynomial optimization

TL;DR

This paper introduces a two-step polynomial approximation method for semi-infinite polynomial optimization problems, combining a joint+marginal approach with semidefinite relaxations to approximate the optimal value.

Contribution

The paper develops a novel polynomial approximation technique for semi-infinite constraints and demonstrates convergence of the approximations to the true optimal value.

Findings

Polynomial approximations converge to the true function as degree increases.

Semidefinite relaxations effectively solve the approximated polynomial optimization problems.

The method allows practical adjustment of approximation accuracy via a relaxation parameter.

Abstract

We consider the semi-infinite optimization problem: $f^*:=\min_{x\in X}\:\{f(x): g(x,y)\,\leq \,0,\:\forally\in Y_x\}$, where $f,g$ are polynomials and $X\subset R^n$ as well as $Y_\x\subset R^p$, $x\in X$, are compact basic semi-algebraic sets. To approximate $f^*$ we proceed in two steps. First, we use the "joint+marginal" approach of the author to approximate from above the function $x\mapsto\Phi(x)=\sup \{g(x,y): y\in Y_x\}$ by a polynomial $\Phi_d\geq\Phi$, of degree at most $2d$, with the strong property that $\Phi_d$ converges to $\Phi$ for the $L_1$-norm, as $d\to\infty$ (and in particular, almost uniformly for some subsequence $(d_\ell)$, $\ell\in\N$). Then we solve the polynomial optimization problem $f^*_d=\min_{x\in X} \{f(x): \Phi_d(x)\leq0\}$ via a (by now standard) hierarchy of semidefinite relaxations. It turns out that the optimal value $f^*_d\geq f^*$ converges to $f^*$ as $d\to\infty$. In practice we let $d$ be fixed, small, and relax the constraint $\Phi_d\leq0$ to $\Phi_d(x)\leq\epsilon$ with $\epsilon>0$, allowing to change $\epsilon$ dynamically.