Intrinsic complexity estimates in polynomial optimization

TL;DR

This paper introduces an intrinsic complexity measure for polynomial optimization problems and presents algorithms with quasi-polynomial complexity that efficiently solve point searching and global minimization in semialgebraic sets.

Contribution

It defines an intrinsic system degree to measure problem complexity and develops algorithms with quasi-polynomial complexity for solving these problems.

Findings

Intrinsic system degree is of order (n d)^{O(n)} in worst case.

Algorithms with intrinsic, quasi-polynomial complexity are proposed.

The methods improve understanding of the complexity landscape in polynomial optimization.

Abstract

It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using $(s\,d)^{O(n)}$ arithmetic operations, where $n$ and $s$ are the numbers of variables and constraints and $d$ is the maximal degree of the polynomials involved.\spar \noindent We associate to each of these problems an intrinsic system degree which becomes in worst case of order $(n\,d)^{O(n)}$ and which measures the intrinsic complexity of the task under consideration.\spar \noindent We design non-uniformly deterministic or uniformly probabilistic algorithms of intrinsic, quasi-polynomial complexity which solve these problems.