Solving generic nonarchimedean semidefinite programs using stochastic game algorithms. (March 2018)

• Record Type: Journal Article
• Authors: Allamigeon, Xavier; Gaubert, Stéphane; Skomra, Mateusz
• Is Part Of: Journal of symbolic computation. Volume 85(2018)
• Page Start: 25
• Page End: 54
• Publication Date: 2018-03
• Subjects: Semidefinite programming -- Nonarchimedean fields -- Tropical geometry -- Stochastic mean payoff games; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.07.002
• Languages: English
• ISSNs: 0747-7171

Abstract

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.