Spectraèdres tropicaux : application à la programmation semi-définie et aux jeux à paiement moyen Mateusz Skomra To cite this version: Mateusz Skomra. Spectraèdres tropicaux : application à la programmation semi-définie et aux jeux à paiement moyen. Optimization and Control [math.OC]. Université Paris Saclay (COmUE), 2018. English. NNT : 2018SACLX058. tel-01958741 HAL Id: tel-01958741 https://pastel.archives-ouvertes.fr/tel-01958741 Submitted on 18 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Tropical spectrahedra: Application to semidefinite programming and mean payoff games These` de doctorat de l’Universite´ Paris-Saclay prepar´ ee´ a` L’Ecole´ polytechnique Ecole doctorale n◦574 Mathematiques´ Hadamard (EDMH) NNT : 2018SACLX058 Specialit´ e´ de doctorat : Mathematiques´ appliquees´ These` present´ ee´ et soutenue a` Palaiseau, le 5 decembre´ 2018, par M. MATEUSZ SKOMRA Composition du Jury : M. Ilia Itenberg Professeur, Sorbonne Universite´ President´ M. Mohab Safey El Din Professeur, Sorbonne Universite´ et INRIA Rapporteur M. Thorsten Theobald Professeur, Goethe-Universitat¨ Rapporteur M. Marcin Jurdzinski´ Associate Professor, University of Warwick Examinateur M. Jean Bernard Lasserre Directeur de Recherche, LAAS-CNRS et Universite´ de Toulouse Examinateur Mme Cynthia Vinzant Assistant Professor, North Carolina State University Examinatrice M. Stephane´ Gaubert Directeur de Recherche, INRIA et Ecole´ polytechnique Directeur de these` M. Xavier Allamigeon Charge´ de Recherche, INRIA et Ecole´ polytechnique Co-directeur de these` ` ese de doctorat Th Acknowledgments I would like to express my supreme gratitude to my advisors, Stéphane Gaubert and Xavier Allamigeon, for the years of guidance. It was a tremendous pleasure to both work and spend time with them during this period, and I profited greatly form their experience. This thesis would not have happened without the help of Stéphane, who supported me even before I came to France. His erudition, vast mathematical knowledge, and intuition had a great impact on this dissertation. The same has to be said about the countless hours that I spent discussing research with Xavier. I cannot thank him enough for his encouragement, attention to detail, and sense of humor. I am grateful to Mohab Safey El Din and Thorsten Theobald for devoting their time to writing the reviews of this dissertation. I also wish to thank Ilia Itenberg, Marcin Jurdziński, Jean Bernard Lasserre, and Cynthia Vinzant for expressing their interest in my work as well as taking part in the jury. I thank the administrative teams of CMAP, INRIA, and DIM RDM-IdF, notably Nasséra Naar, Alexandra Noiret, Jessica Gameiro, Corinne Petitot, and Hanadi Dib, for their kindness and helpfulness. Thanks should also go to numerous current and former members of the INRIA research team Tropical for their encouragement and support. I especially thank Marianne Akian and Jean Bernard Eytard for the enjoyable time we spent together during various conferences. It was a great pleasure to work at École polytechnique. I thank all the fellow PhD students from CMAP for the friendly atmosphere in the lab. Last but not least, I am deeply grateful to my friends and family for the continuous aid and assistance they gave me during the years I spent in Paris. This work was funded by a grant from Région Ile-de-France and also partially supported by INRIA. Contents 1 Introduction 7 1.1 Context of this work .................................. 7 1.2 Our contribution .................................... 12 1.3 Organization of the manuscript ............................ 17 1.4 Notation ......................................... 18 2 Preliminaries 19 2.1 Polyhedra and polyhedral complexes ......................... 19 2.2 Real closed fields .................................... 20 2.3 Puiseux series and tropical semifield ......................... 23 2.4 Tropical polynomials .................................. 26 2.5 Valued fields ...................................... 30 2.6 Model theory ...................................... 33 2.6.1 Theories with quantifier elimination ..................... 37 2.6.2 Model completeness .............................. 42 2.7 Markov chains ..................................... 43 I Tropical spectrahedra 47 3 Tropicalization of semialgebraic sets 49 3.1 Real analogue of the Bieri–Groves theorem ..................... 51 3.2 Puiseux series with rational exponents ........................ 57 4 Tropical spectrahedra 61 4.1 Tropical Metzler spectrahedra ............................. 64 4.2 Non-Metzler spectrahedra ............................... 68 4.3 Genericity conditions .................................. 74 4.4 Valuation of interior and regions of strict feasibility ................. 78 5 Tropical analogue of the Helton–Nie conjecture 83 5.1 Tropical convexity ................................... 84 5.2 Real tropical cones as sublevel sets of dynamic programming operators ...... 88 5.3 Description of real tropical cones by directed graphs ................ 91 5.4 Tropical Helton–Nie conjecture for real tropical cones ............... 95 5.5 Tropical Helton–Nie conjecture for Puiseux series .................. 99 5.6 Extension to general fields ............................... 103 II Relation to stochastic mean payoff games 107 6 Introduction to stochastic mean payoff games 109 6.1 Optimal policies and Shapley operators ....................... 110 6.2 Sublevel sets, dominions, and the Collatz–Wielandt property ........... 114 6.3 Bipartite games, their operators, and graphs ..................... 117 7 Equivalence between games and tropical spectrahedra 125 7.1 From stochastic mean payoff games to tropical spectrahedra ............ 126 7.2 From tropical spectrahedra to stochastic mean payoff games ............ 130 7.2.1 Games obtained from well-formed linear matrix inequalities ........ 132 7.2.2 Preprocessing .................................. 133 7.3 Extension to non-Metzler matrices .......................... 136 7.4 Nonarchimedean semidefinite feasibility problems .................. 137 8 Condition number of stochastic mean payoff games 139 8.1 Archimedean feasibility problems ........................... 140 8.1.1 Geometric interpretation of the condition number .............. 142 8.2 Value iteration for constant value games ....................... 147 8.2.1 Oracle-based approximation algorithms ................... 150 8.3 Estimates of condition number ............................ 154 8.4 Parametrized complexity bounds for stochastic mean payoff games ........ 162 8.4.1 Solving constant value games ......................... 163 8.4.2 Finding the states with maximal value .................... 166 8.5 Application to nonarchimedean semidefinite programming ............. 170 8.5.1 A class of ergodic problems .......................... 172 8.5.2 Experimental results .............................. 173 9 Perspectives 175 Bibliography 179 A Fields of convergent power series 195 A.1 Basic definitions .................................... 195 A.2 Structure of ordered field ............................... 198 A.3 Theorem of division, henselianity, and real closedness ................ 200 5 B Additional proofs 205 B.1 Model theory ...................................... 205 B.2 Ergodic theorem for Markov chains with rewards .................. 208 B.3 Kohlberg’s theorem ................................... 211 C Résumé en français 215 C.1 Contexte et motivations ................................ 215 C.2 Contributions ...................................... 220 6 CHAPTER 1 Introduction 1.1 Context of this work Semidefinite programming (SDP) is one of the fundamental tools of convex optimization. It consists in minimizing a linear function over a spectrahedron, which is a set defined by a single linear matrix inequality of the form n (0) (1) (n) S := {x ∈ R : Q + x1Q + ··· + xnQ < 0} , where Q(0),...,Q(n) ∈ Rm×m is a sequence of symmetric matrices, and < is the Loewner order on symmetric matrices. By definition, A < B if A − B is positive semidefinite. Because of its expressive power, SDP has found numerous applications. For instance, SDP relaxations can be used to obtain polynomial-time approximations for some NP-hard problems in combinato- rial optimization, such as the Max-Cut Problem [GW95]. Another classical application of SDP in the area of combinatorial optimization is the Lovász theta function [Lov79]—this func- tion is computable in polynomial-time by SDP and is sandwiched between the clique number and the chromatic number, which are both NP-hard to compute. We refer to [GM12, LR05] for more information about applications of SDP in combinatorial optimization. SDP is also a major tool in the area of polynomial optimization. Even though polynomial optimization problems are not convex in general, Lasserre [Las01, Las02] and Parrilo [Par03] have shown that a large class of these problems can be solved to arbitrary precision using a hierarchy of SDP relaxations. More information about the use of SDP in polynomial optimization can be found in the books