Bernstein Polynomial Relaxations for Polynomial Optimization Problems

TL;DR

This paper introduces Bernstein polynomial-based LP relaxations for polynomial optimization, providing tighter bounds and methods to improve relaxation accuracy for bounded domains.

Contribution

It develops a hierarchy of LP relaxations using Bernstein polynomials, enhancing the precision of polynomial optimization bounds over various domains.

Findings

Bernstein relaxations yield tighter lower bounds for POPs.

Higher degree relaxations improve solution accuracy.

The approach extends to polyhedral and semi-algebraic domains.

Abstract

In this paper, we examine linear programming (LP) relaxations based on Bernstein polynomials for polynomial optimization problems (POPs). We present a progression of increasingly more precise LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. The well-known bounds on Bernstein polynomials over the unit box combined with linear inter-relationships between Bernstein polynomials help us formulate "Bernstein inequalities" which yield tighter lower bounds for POPs in bounded rectangular domains. The results can be easily extended to optimization over polyhedral and semi-algebraic domains. We also examine techniques to increase the precision of these relaxations by considering higher degree relaxations, and a branch-and-cut scheme