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.

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revisions + benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign patterns + some results have been replaced by references to the companion work arXiv:1610.06746

References in corpus (3)

• Hyperfields for Tropical Geometry I. Hyperfields and dequantization
• On the Turing model complexity of interior point methods for semidefinite programming
• Tropical spectrahedra

Cited by in corpus (7)

• Tropical spectrahedra
• Tropical totally positive matrices
• The tropical analogue of the Helton-Nie conjecture is true
• Symmetric Polynomials in Tropical Algebra Semirings
• Spectral inequalities for nonnegative tensors and their tropical analogues
• Convex geometry over ordered hyperfields
• Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming