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Bibliography

Major publications by the team in recent years
  • 1M. Akian, S. Gaubert, R. Bapat.

    Non-archimedean valuations of eigenvalues of matrix polynomials, in: Linear Algebra and its Applications, June 2016, vol. 498, pp. 592–627, Also arXiv:1601.00438. [ DOI : 10.1016/j.laa.2016.02.036 ]

    https://hal.inria.fr/hal-01251803
  • 2M. Akian, S. Gaubert, A. Guterman.

    Tropical polyhedra are equivalent to mean payoff games, in: Internat. J. Algebra Comput., 2012, vol. 22, no 1, 1250001, 43 p.

    http://dx.doi.org/10.1142/S0218196711006674
  • 3M. Akian, S. Gaubert, R. Nussbaum.

    Uniqueness of the fixed point of nonexpansive semidifferentiable maps, in: Transactions of the American Mathematical Society, February 2016, vol. 368, no 2, Also arXiv:1201.1536. [ DOI : 10.1090/S0002-9947-2015-06413-7 ]

    https://hal.inria.fr/hal-00783682
  • 4M. Akian, S. Gaubert, C. Walsh.

    The max-plus Martin boundary, in: Doc. Math., 2009, vol. 14, pp. 195–240.
  • 5X. Allamigeon, P. Benchimol, S. Gaubert, M. Joswig.

    Combinatorial simplex algorithms can solve mean payoff games, in: SIAM J. Opt., 2015, vol. 24, no 4, pp. 2096–2117.
  • 6X. Allamigeon, P. Benchimol, S. Gaubert, M. Joswig.

    Log-barrier interior point methods are not strongly polynomial, in: SIAM Journal on Applied Algebra and Geometry, 2018, vol. 2, no 1, pp. 140-178, https://arxiv.org/abs/1708.01544 - This paper supersedes arXiv:1405.4161. 31 pages, 5 figures, 1 table. [ DOI : 10.1137/17M1142132 ]

    https://hal.inria.fr/hal-01674959
  • 7X. Allamigeon, S. Gaubert, E. Goubault, S. Putot, N. Stott.

    A scalable algebraic method to infer quadratic invariants of switched systems, in: Proceedings of the International Conference on Embedded Software (EMSOFT), 2015, Best paper award. The extended version of this conference article appeared in ACM Trans. Embed. Comput. Syst., 15(4):69:1–69:20, September 2016.
  • 8J. Bolte, S. Gaubert, G. Vigeral.

    Definable zero-sum stochastic games, in: Mathematics of Operations Research, 2014, vol. 40, no 1, pp. 171–191, Also arXiv:1301.1967.

    http://dx.doi.org/10.1287/moor.2014.0666
  • 9S. Gaubert, T. Lepoutre.

    Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model, in: J. Math. Biol., 2015.

    http://dx.doi.org/10.1007/s00285-015-0874-3
  • 10S. Gaubert, G. Vigeral.

    A maximin characterization of the escape rate of nonexpansive mappings in metrically convex spaces, in: Math. Proc. of Cambridge Phil. Soc., 2012, vol. 152, pp. 341–363, Also arXiv:1012.4765.

    http://dx.doi.org/10.1017/S0305004111000673
  • 11C. Walsh.

    The horofunction boundary and isometry group of the Hilbert geometry, in: Handbook of Hilbert Geometry, IRMA Lectures in Mathematics and Theoretical Physics, European Mathematical Society, 2014, vol. 22.

    https://hal.inria.fr/hal-00782827
Publications of the year

Doctoral Dissertations and Habilitation Theses

Articles in International Peer-Reviewed Journals

International Conferences with Proceedings

  • 27X. Allamigeon, S. Gaubert, R. Katz, M. Skomra.

    Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming, in: 23rd International Symposium on Mathematical Theory of Networks and Systems, Hong-Kong, France, July 2018, https://arxiv.org/abs/1802.07712 - 14 pages, 2 figures.

    https://hal.inria.fr/hal-01967555
  • 28S. Gaubert, D. Jones.

    Tropical cellular automata : why urban fires propagate according to polyhedral balls, in: Cellular Automata and Discrete Complex Systems, 24th IFIP WG 1.5 International Workshop, AUTOMATA 2018 Ghent, Belgium, June 20-22, 2018 (Exploratory Papers), Ghent, Belgium, June 2018.

    https://hal.inria.fr/hal-01967561
  • 29S. Gaubert, N. Stott.

    A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices, in: 23rd International Symposium on Mathematical Theory of Networks and Systems, Hong-Kong, France, July 2018, https://arxiv.org/abs/1805.03284 - 18 pages.

    https://hal.inria.fr/hal-01967552
  • 30P. Jacquot, C. Wan.

    Routing Game on Parallel Networks: the Convergence of Atomic to Nonatomic, in: CDC 2018 - IEEE 57th Conference on Decision and Control, Miami, United States, Proceedings of the 57th IEEE Conference on Decision and Control, IEEE, December 2018, vol. 1.

    https://hal.archives-ouvertes.fr/hal-01762547

Conferences without Proceedings

Scientific Books (or Scientific Book chapters)

  • 32M. Akian, E. Fodjo.

    From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton-Jacobi-Bellman equations, in: Hamilton-Jacobi-Bellman Equations: Numerical Methods and Applications in Optimal Control, D. Kalise, K. Kunisch, Z. Rao (editors), De Gruyter, August 2018, https://arxiv.org/abs/1709.09049.

    https://hal.inria.fr/hal-01675067
  • 33M. Akian, E. Fodjo.

    Probabilistic max-plus schemes for solving Hamilton-Jacobi-Bellman equations, in: Numerical Methods for Optimal Control Problems, M. Falcone, R. Ferretti, L. Grune, W. McEneaney (editors), INDAM Series, Springer, February 2019, https://arxiv.org/abs/1801.01780.

    https://hal.inria.fr/hal-01675068

Other Publications

References in notes
  • 46M. Akian, E. Fodjo.

    A probabilistic max-plus numerical method for solving stochastic control problems, in: 55th Conference on Decision and Control (CDC 2016), Las Vegas, United States, December 2016, Also arXiv:1605.02816.

    https://hal.inria.fr/hal-01425344
  • 47M. Akian, S. Gaubert.

    Spectral theorem for convex monotone homogeneous maps, and ergodic control, in: Nonlinear Anal., 2003, vol. 52, no 2, pp. 637–679.

    http://dx.doi.org/10.1016/S0362-546X(02)00170-0
  • 48M. Akian, S. Gaubert.

    Policy iteration for perfect information stochastic mean payoff games with bounded first return times is strongly polynomial, 2013, Preprint arXiv:1310.4953, 17 pages.

    http://hal.inria.fr/hal-00881207
  • 49M. Akian, S. Gaubert, A. Guterman.

    Linear independence over tropical semirings and beyond, in: Proceedings of the International Conference on Tropical and Idempotent Mathematics, G. Litvinov, S. Sergeev (editors), Contemporary Mathematics, American Mathematical Society, 2009, vol. 495, pp. 1-38.

    http://www.arxiv.org/abs/0812.3496
  • 50M. Akian, S. Gaubert, A. Guterman.

    Tropical polyhedra are equivalent to mean payoff games, in: Internat. J. Algebra Comput., 2012, vol. 22, no 1, 1250001, 43 p. [ DOI : 10.1142/S0218196711006674 ]

    http://arxiv.org/abs/0912.2462
  • 51M. Akian, S. Gaubert, A. Guterman.

    Tropical Cramer Determinants Revisited, in: Tropical and Idempotent Mathematics and Applications, G. Litvinov, S. Sergeev (editors), Contemporary Mathematics, AMS, 2014, vol. 616, 45 p, See also arXiv:1309.6298.

    https://hal.inria.fr/hal-00881203
  • 52M. Akian, S. Gaubert, M. Sharify.

    Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots, in: Linear Algebra and its Applications, 2017, Also arXiv:1304.2967. [ DOI : 10.1016/j.laa.2016.11.004 ]

    https://hal.inria.fr/hal-00881196
  • 53X. Allamigeon, P. Benchimol, S. Gaubert.

    The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average, in: ICALP 2014, Copenhagen, France, J. Esparza, P. Fraigniaud, T. Husfeldt, E. Koutsoupias (editors), 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, Springer, July 2014, vol. 8572, 12 p. [ DOI : 10.1007/978-3-662-43948-7_8 ]

    https://hal.inria.fr/hal-01096447
  • 54X. Allamigeon, P. Benchimol, S. Gaubert, M. Joswig.

    Long and winding central paths, May 2014, Preprint arXiv:1405.4161, v2 May 2015.

    https://hal.inria.fr/hal-01096452
  • 55X. Allamigeon, V. Boeuf, S. Gaubert.

    Performance evaluation of an emergency call center: tropical polynomial systems applied to timed Petri nets, in: 13th International Conference, Formal Modeling and Analysis of Timed Systems (FORMATS 2015), Madrid, Spain, Formal Modeling and Analysis of Timed Systems, Springer, September 2015, vol. 9268. [ DOI : 10.1007/978-3-319-22975-1_2 ]

    https://hal.inria.fr/hal-01248814
  • 56X. Allamigeon, V. Boeuf, S. Gaubert.

    Stationary solutions of discrete and continuous Petri nets with priorities, in: Performance Evaluation, August 2017, vol. 113, pp. 1 - 12, https://arxiv.org/abs/1612.07661. [ DOI : 10.1016/j.peva.2017.04.007 ]

    https://hal.inria.fr/hal-01674492
  • 57X. Allamigeon, S. Gaubert, E. Goubault.

    Inferring Min and Max Invariants Using Max-plus Polyhedra, in: Proceedings of the 15th International Static Analysis Symposium (SAS'08), Valencia, Spain, LNCS, Springer, 2008, vol. 5079, pp. 189–204.

    http://dx.doi.org/10.1007/978-3-540-69166-2_13
  • 58X. Allamigeon, S. Gaubert, E. Goubault.

    Computing the Vertices of Tropical Polyhedra using Directed Hypergraphs, in: Discrete Comp. Geom., 2012, Published on line. [ DOI : 10.1007/s00454-012-9469-6 ]

    http://fr.arxiv.org/abs/0904.3436v3
  • 59X. Allamigeon, S. Gaubert, M. Skomra.

    Solving Generic Nonarchimedean Semidefinite Programs Using Stochastic Game Algorithms, in: ISSAC '16: International Symposium on Symbolic and Algebraic Computation, Waterloo, France, Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC'16), ACM, July 2016, Also arXiv:1603.06916. [ DOI : 10.1145/2930889.2930935 ]

    https://hal.inria.fr/hal-01422638
  • 60X. Allamigeon, S. Gaubert, M. Skomra.

    Tropical spectrahedra, October 2016, arXiv:1610.06746.

    https://hal.inria.fr/hal-01422639
  • 61P. Andy, W. Faisal, F. Bonnans.

    MIDAS: A Mixed Integer Dynamic Approximation Scheme, Inria, 2016.

    https://hal.inria.fr/hal-01401950
  • 62F. Baccelli, G. Cohen, G.-J. Olsder, J.-P. Quadrat.

    Synchronization and linearity: an algebra for discrete event systems, Wiley, 1992.
  • 63G. Barles, S. Mirrahimi, B. Perthame.

    Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result, in: Methods Appl. Anal., 2009, vol. 16, no 3, pp. 321–340.

    http://dx.doi.org/10.4310/MAA.2009.v16.n3.a4
  • 64R. Bhatia, S. Gaubert, T. Jain.

    Matrix versions of the Hellinger distance, 2019, To appear in Letters in Math. Physics.
  • 65V. Boeuf, P. Robert.

    A Stochastic Analysis of a Network with Two Levels of Service, August 2017, working paper or preprint.

    https://hal.archives-ouvertes.fr/hal-01583704
  • 66F. Bonnans, S. Gaubert.

    Recherche opérationnelle. Aspects mathématiques et applications, Ellipse, March 2016, 391 p.

    https://hal.inria.fr/hal-01422645
  • 67P. Butkovič.

    Max-algebra: the linear algebra of combinatorics?, in: Linear Algebra and its applications, 2003, vol. 367, pp. 313–335.
  • 68P. Butkovič.

    Max-linear systems: theory and algorithms, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2010, xviii+272 p.

    http://dx.doi.org/10.1007/978-1-84996-299-5
  • 69J. Cochet-Terrasson, G. Cohen, S. Gaubert, M. Mc Gettrick, J.-P. Quadrat.

    Numerical computation of spectral elements in max-plus algebra, in: Proc. of the IFAC Conference on System Structure and Control, Nantes, July 1998.
  • 70G. Cohen, S. Gaubert, J.-P. Quadrat.

    Max-plus algebra and system theory: where we are and where to go now, in: Annual Reviews in Control, 1999, vol. 23, pp. 207–219.
  • 71A. Connes.

    Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, in: Selecta Math. (N.S.), 1999, vol. 5, no 1, pp. 29–106.
  • 72A. Connes, C. Consani.

    The Arithmetic Site, in: Comptes Rendus Mathématiques, 2014, vol. Ser. I 352, pp. 971–975.
  • 73A. Connes, C. Consani.

    Geometry of the arithmetic site, 2015, arXiv:1502.05580.
  • 74A. Connes, C. Consani.

    Geometry of the arithmetic site, in: Adv. Math., 2016, vol. 291, pp. 274–329.
  • 75P. Cousot, R. Cousot.

    Abstract Interpretation: A unified lattice model for static analysis of programs by construction of approximations of fixed points, in: Principles of Programming Languages 4, 1977, pp. 238–252.
  • 76J. De Loera, B. Sturmfels, C. Vinzant.

    The central curve in linear programming, in: Foundations of Computational Mathematics, 2012, vol. 12, no 4, pp. 509–540.
  • 77J.-P. Dedieu, G. Malajovich, M. Shub.

    On the Curvature of the Central Path of Linear Programming Theory, in: Foundations of Computational Mathematics, 2005, vol. 5, no 2, pp. 145–171.
  • 78A. Deza, T. Terlaky, Y. Zinchenko.

    Polytopes and arrangements: diameter and curvature, in: Operations Research Letters, 2008, vol. 36, no 2, pp. 215–222.
  • 79A. Deza, T. Terlaky, Y. Zinchenko.

    Central path curvature and iteration-complexity for redundant Klee-Minty cubes, in: Advances in applied mathematics and global optimization, New York, Adv. Mech. Math., Springer, 2009, vol. 17, pp. 223–256.

    http://dx.doi.org/10.1007/978-0-387-75714-8_7
  • 80J. B. Eytard, M. Akian, M. Bouhtou, S. Gaubert.

    A bilevel optimization model for load balancing in mobile networks through price incentives, in: WiOpt 2017 - 15th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, Paris, France, IEEE, May 2017, pp. 1-8. [ DOI : 10.23919/WIOPT.2017.7959902 ]

    https://hal.inria.fr/hal-01649042
  • 81J.-B. Eytard.

    A tropical geometry and discrete convexity approach to bilevel programming : application to smart data pricing in mobile telecommunication networks, Université Paris-Saclay, November 2018.

    https://pastel.archives-ouvertes.fr/tel-01972391
  • 82A. Fahim, N. Touzi, X. Warin.

    A probabilistic numerical method for fully nonlinear parabolic PDEs, in: Ann. Appl. Probab., 2011, vol. 21, no 4, pp. 1322–1364.

    http://dx.doi.org/10.1214/10-AAP723
  • 83M. Farber.

    Invitation to Topological Robotics, Zurich lectures in advanced mathematics, European Mathematical Society, 2008.
  • 84M. Farber, E. Goubault, A. Sagnier.

    Directed topological complexity, 2018.

    https://hal.archives-ouvertes.fr/hal-01970569
  • 85A. Fathi, A. Siconolfi.

    Existence of C1 critical subsolutions of the Hamilton-Jacobi equation, in: Invent. Math., 2004, vol. 155, no 2, pp. 363–388.

    http://dx.doi.org/10.1007/s00222-003-0323-6
  • 86O. Fercoq, M. Akian, M. Bouhtou, S. Gaubert.

    Ergodic control and polyhedral approaches to PageRank optimization, in: IEEE Trans. Automat. Control, 2013, vol. 58, no 1, pp. 134–148.

    http://dx.doi.org/10.1109/TAC.2012.2226103
  • 87W. Fleming, W. McEneaney.

    A max-plus based algorithm for an HJB equation of non-linear filtering, in: SIAM J. Control and Opt., 2000, pp. 683–710.
  • 88S. Fomin, A. Zelevinsky.

    Cluster algebras. I. Foundations, in: J. Amer. Math. Soc., 2002, vol. 15, no 2, pp. 497–529.

    http://arxiv.org/abs/math.RT/0104151
  • 89S. Friedland, S. Gaubert, L. Han.

    Perron–Frobenius theorem for nonnegative multilinear forms and extensions, in: Linear Algebra Appl., 2013, vol. 438, no 2, pp. 738–749. [ DOI : 10.1016/j.laa.2011.02.042 ]

    http://hal.inria.fr/hal-00782755
  • 90S. Gaubert, E. Goubault, A. Taly, S. Zennou.

    Static Analysis by Policy Iteration in Relational Domains, in: Proceedings of the Proc. of the 16th European Symposium on Programming (ESOP'07), Braga (Portugal), LNCS, Springer, 2007, vol. 4421, pp. 237–252.

    http://dx.doi.org/10.1007/978-3-540-71316-6_17
  • 91S. Gaubert, J. Gunawardena.

    The Perron-Frobenius Theorem for Homogeneous, Monotone Functions, in: Trans. of AMS, 2004, vol. 356, no 12, pp. 4931-4950.

    http://www.ams.org/tran/2004-356-12/S0002-9947-04-03470-1/home.html
  • 92S. Gaubert, W. McEneaney, Z. Qu.

    Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms, in: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 11), Orlando, FL, USA, December 2011, pp. 1054-1061.

    http://arxiv.org/abs/1109.5241
  • 93S. Gaubert, M. Sharify.

    Tropical scaling of polynomial matrices, in: Positive systems, Berlin, Lecture Notes in Control and Inform. Sci., Springer, 2009, vol. 389, pp. 291–303.

    http://dx.doi.org/10.1007/978-3-642-02894-6_28
  • 94S. Gaubert, N. Stott.

    Tropical Kraus maps for optimal control of switched systems, in: Proceedings of the 57th IEEE Annual Conference on Decision and Control (CDC), Melbourne, Australia, 2017, https://arxiv.org/abs/1706.04471 - 15 pages.

    https://hal.inria.fr/hal-01674496
  • 95T. M. Gawlitza, H. Seidl, A. Adjé, S. Gaubert, E. Goubault.

    Abstract interpretation meets convex optimization, in: J. Symbolic Comput., 2012, vol. 47, no 12, pp. 1416–1446, Special issue on Invariant generation and reasoning about loops.

    http://dx.doi.org/10.1016/j.jsc.2011.12.048
  • 96I. M. Gelfand, M. M. Kapranov, A. V. Zelevinsky.

    Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008, x+523 p, Reprint of the 1994 edition.
  • 97M. Grandis.

    Directed Algebraic Topology, Models of non-reversible worlds, Cambridge University Press, 2009.
  • 98S. Hammarling, C. J. Munro, F. Tisseur.

    An algorithm for the complete solution of quadratic eigenvalue problems, in: ACM Trans. Math. Software, 2013, vol. 39, no 3, Art. 18, 19 p.

    http://dx.doi.org/10.1145/2450153.2450156
  • 99B. Heidergott, G. J. Olsder, J. van der Woude.

    Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications, Princeton, 2005.
  • 100H. Ishii, H. Mitake.

    Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians, in: Indiana Univ. Math. J., 2007, vol. 56, no 5, pp. 2159–2183.

    http://dx.doi.org/10.1512/iumj.2007.56.3048
  • 101I. Itenberg, G. Mikhalkin, E. Shustin.

    Tropical algebraic geometry, Oberwolfach Seminars, Birkhäuser Verlag, Basel, 2007, vol. 35, viii+103 p.
  • 102P. Jacquot, O. Beaude, S. Gaubert, N. Oudjane.

    Demand Side Management in the Smart Grid: an Efficiency and Fairness Tradeoff, in: 7th IEEE International Conference on Innovative Smart Grid Technologies, Torino, France, August 2017, https://arxiv.org/abs/1711.11129.

    https://hal.inria.fr/hal-01675658
  • 103H. Kaise, W. M. McEneaney.

    Idempotent expansions for continuous-time stochastic control: compact control space, in: Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, Dec. 2010.
  • 104V. Kolokoltsov, V. Maslov.

    Idempotent analysis and applications, Kluwer Acad. Publisher, 1997.
  • 105B. Lemmens, R. Nussbaum.

    Nonlinear Perron-Frobenius theory, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2012, vol. 189, xii+323 p.

    http://dx.doi.org/10.1017/CBO9781139026079
  • 106Q. Lu, M. Madsen, M. Milata, S. Ravn, U. Fahrenberg, K. G. Larsen.

    Reachability Analysis for Timed Automata using Max-Plus Algebra, in: J. Logic Alg. Prog., 2012, vol. 81, no 3, pp. 298-313.
  • 107V. Maslov.

    Méthodes Operatorielles, Edition Mir, Moscou, 1987.
  • 108W. McEneaney, A. Deshpande, S. Gaubert.

    Curse-of-Complexity Attenuation in the Curse-of-Dimensionality-Free Method for HJB PDEs, in: Proc. of the 2008 American Control Conference, Seattle, Washington, USA, June 2008.
  • 109W. M. McEneaney, H. Kaise, S. H. Han.

    Idempotent Method for Continuous-time Stochastic Control and Complexity Attenuation, in: Proceedings of the 18th IFAC World Congress, 2011, Milano, Italie, 2011, pp. 3216-3221.
  • 110W. M. McEneaney.

    Max-plus methods for nonlinear control and estimation, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 2006, xiv+241 p.
  • 111W. M. McEneaney.

    A curse-of-dimensionality-free numerical method for solution of certain HJB PDEs, in: SIAM J. Control Optim., 2007, vol. 46, no 4, pp. 1239–1276.

    http://dx.doi.org/10.1137/040610830
  • 112J.-F. Mertens, S. Sorin, S. Zamir.

    Repeated Games, Cambridge, 2015.
  • 113G. Mikhalkin.

    Enumerative tropical algebraic geometry in ℝ2, in: J. Amer. Math. Soc., 2005, vol. 18, no 2, pp. 313–377.

    http://dx.doi.org/10.1090/S0894-0347-05-00477-7
  • 114R. H. Möhring, M. Skutella, F. Stork.

    Scheduling with AND/OR precedence constraints, in: SIAM J. Comput., 2004, vol. 33, no 2, pp. 393–415.

    http://dx.doi.org/10.1137/S009753970037727X
  • 115A. Papadopoulos.

    Metric spaces, convexity and non-positive curvature, IRMA Lectures in Mathematics and Theoretical Physics, Second, European Mathematical Society (EMS), Zürich, 2014, vol. 6, xii+309 p.

    http://dx.doi.org/10.4171/132
  • 116M. V. F. Pereira, L. M. V. G. Pinto.

    Multi-stage stochastic optimization applied to energy planning, in: Math. Programming, 1991, vol. 52, no 2, Ser. B, pp. 359–375.

    http://dx.doi.org/10.1007/BF01582895
  • 117J.-E. Pin.

    Tropical Semirings, in: Idempotency, J. Gunawardena (editor), Publications of the Isaac Newton Institute, Cambridge University Press, 1998, vol. 11, pp. 50–69.
  • 118Z. Qu.

    Théorie de Perron-Frobenius non linéaire et méthodes numériques max-plus pour la résolution d'équations d'Hamilton-Jacobi, Ecole Polytechnique X, October 2013.

    http://hal.inria.fr/pastel-00927122
  • 119D. Reeb, M. J. Kastoryano, M. M. Wolf.

    Hilbert's projective metric in quantum information theory, in: J. Math. Phys., 2011, vol. 52, no 8, 082201, 33 p.

    http://dx.doi.org/10.1063/1.3615729
  • 120J. Richter-Gebert, B. Sturmfels, T. Theobald.

    First steps in tropical geometry, in: Idempotent mathematics and mathematical physics, Providence, RI, Contemp. Math., Amer. Math. Soc., 2005, vol. 377, pp. 289–317.
  • 121A. Sagnier.

    An arithmetic site of Connes-Consani type for imaginary quadratic fields with class number 1, 2018, submitted.
  • 122G. Sagnol, S. Gaubert, M. Bouhtou.

    Optimal monitoring on large networks by Successive c-Optimal Designs, in: Proceedings of the 22nd international teletraffic congress (ITC22), Amsterdam, The Netherlands, September, IEEE, 2010, http://dx.doi.org/10.1109/ITC.2010.5608717.
  • 123S. Sankaranarayanan, H. Sipma, Z. Manna.

    Scalable Analysis of Linear Systems using Mathematical Programming, in: VMCAI, LNCS, 2005, vol. 3385.
  • 124C. Scheiderer.

    Semidefinitely representable convex sets, 2016, arXiv:1612.07048v2.
  • 125R. Sepulchre, A. Sarlette, P. Rouchon.

    Consensus in noncommutative spaces, in: Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, USA, Dec 2010, pp. 6596-6601.

    http://dx.doi.org/10.1109/CDC.2010.5717072
  • 126A. Sidford, M. Wang, X. Wu, Y. Ye.

    Variance reduced value iteration and faster algorithms for solving Markov decision processes, in: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2018, pp. 770–787.

    https://doi.org/10.1137/1.9781611975031.50
  • 127I. Simon.

    Limited subsets of a free monoid, in: Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, NJ, 1978, pp. 143–150.
  • 128S. Smale.

    Mathematical problems for the next century, in: Math. Intelligencer, 1998, vol. 20, no 2, pp. 7–15.

    http://dx.doi.org/10.1007/BF03025291
  • 129H. L. Smith.

    Monotone dynamical systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995, vol. 41, x+174 p, An introduction to the theory of competitive and cooperative systems.
  • 130N. M. Tran, J. Yu.

    Product-Mix Auctions and Tropical Geometry, 2015.
  • 131O. Viro.

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