Positivity and optimization for semi-algebraic functions

TL;DR

This paper develops algebraic certificates of positivity for semi-algebraic functions, enabling the reduction of certain constrained optimization problems to polynomial optimization, supported by examples and numerical experiments.

Contribution

It introduces a method to certify positivity of semi-algebraic functions and reduces related optimization problems to polynomial optimization through a natural change of variables.

Findings

Certificates of positivity for semi-algebraic functions are established.

Optimization problems are effectively reduced to polynomial optimization.

Numerical experiments demonstrate the practical applicability of the approach.

Abstract

We describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of Borel measurable functions, with particular emphasis to algebras generated by semi-algebraic functions. In which case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization. A collection of simple examples and numerical experiments complement the theoretical parts of the article.