Minimizing Polynomial Functions

TL;DR

This paper compares various algorithms for global polynomial optimization, demonstrating that semidefinite programming relaxations significantly outperform traditional algebraic methods, thus broadening the scope of computational real algebraic geometry.

Contribution

It introduces the effectiveness of semidefinite programming relaxations for polynomial optimization, surpassing classical algebraic techniques like Gr"obner bases and homotopy methods.

Findings

Semidefinite programming relaxations outperform algebraic methods

Relaxation techniques enable broader applications in real algebraic geometry

Empirical results show significant computational advantages

Abstract

We compare algorithms for global optimization of polynomial functions in many variables. It is demonstrated that existing algebraic methods (Gr\"obner bases, resultants, homotopy methods) are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming. This opens up the possibility of using semidefinite programming relaxations arising from the Positivstellensatz for a wide range of computational problems in real algebraic geometry. This paper was presented at the Workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, held at DIMACS, Rutgers University, March 12-16, 2001.