Solving generic nonarchimedean semidefinite programs using stochastic game algorithms

TL;DR

This paper introduces a novel approach to solving nonarchimedean semidefinite feasibility problems by leveraging tropical geometry and stochastic game algorithms, enabling combinatorial solutions for large instances.

Contribution

It establishes a correspondence between tropical spectrahedra and stochastic games, providing a new combinatorial method for nonarchimedean semidefinite programming.

Findings

Successfully solves nonarchimedean semidefinite feasibility problems.

Establishes a link between tropical spectrahedra and stochastic games.

Algorithms are scalable for large problem instances.

Abstract

A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean. We provide a solution for a class of semidefinite feasibility problems given by generic matrices. Our approach is based on tropical geometry. It relies on tropical spectrahedra, which are defined as the images by the valuation of nonarchimedean spectrahedra. We establish a correspondence between generic tropical spectrahedra and zero-sum stochastic games with perfect information. The latter have been well studied in algorithmic game theory. This allows us to solve nonarchimedean semidefinite feasibility problems using algorithms for stochastic games. These algorithms are of a combinatorial nature and work for large instances.