Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
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.
Cite
@article{arxiv.1603.06916,
title = {Solving generic nonarchimedean semidefinite programs using stochastic game algorithms},
author = {Xavier Allamigeon and Stéphane Gaubert and Mateusz Skomra},
journal= {arXiv preprint arXiv:1603.06916},
year = {2018}
}
Comments
v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions + benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign patterns + some results have been replaced by references to the companion work arXiv:1610.06746