Border Basis relaxation for polynomial optimization

TL;DR

This paper introduces a border basis relaxation method that enhances polynomial optimization efficiency, offers a new stopping criterion, and provides algorithms for reconstructing measures and computing minimizer ideals, with demonstrated improvements on benchmarks.

Contribution

It presents a novel border basis relaxation approach, a sparse flat extension stopping criterion, and algorithms for measure reconstruction and minimizer ideal computation.

Findings

Improved efficiency over Lasserre's approach.

Effective stopping criterion for relaxation sequences.

Successful application to benchmark problems.

Abstract

A relaxation method based on border basis reduction which improves the efficiency of Lasserre's approach is proposed to compute the optimum of a polynomial function on a basic closed semi algebraic set. A new stopping criterion is given to detect when the relaxation sequence reaches the minimum, using a sparse flat extension criterion. We also provide a new algorithm to reconstruct a finite sum of weighted Dirac measures from a truncated sequence of moments, which can be applied to other sparse reconstruction problems. As an application, we obtain a new algorithm to compute zero-dimensional minimizer ideals and the minimizer points or zero-dimensional G-radical ideals. Experimentations show the impact of this new method on significant benchmarks.