FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

TL;DR

This paper presents an FPTAS for optimizing non-negative and arbitrary polynomials over mixed-integer points in convex polytopes, applicable when the number of variables is fixed, advancing computational efficiency in polynomial optimization.

Contribution

It establishes the existence of an FPTAS for polynomial optimization over mixed-integer sets in fixed dimensions, extending previous results to more general polynomial cases.

Findings

FPTAS exists for non-negative polynomial maximization in fixed dimensions.

FPTAS exists for arbitrary polynomial maximization/minimization in fixed dimensions.

Applicable to convex polytopes with mixed-integer points.

Abstract

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.