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Duality and separation theorems in idempotent semimodules. (English) Zbl 1042.46004
Summary: We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.

46A20 Duality theory for topological vector spaces
06F07 Quantales
46A55 Convex sets in topological linear spaces; Choquet theory
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[1] Akian, M; Gaubert, S; Kolokoltsov, V, Invertibility of functional Galois connections, C.R. acad. sci. Paris ser. I, 335, 1-6, (2002) · Zbl 1022.06001
[2] Aliprantis, C.D; Border, K.C, Infinite dimensional analysis. A hitchiker’s guide, (1999), Springer · Zbl 0938.46001
[3] Baccelli, F; Cohen, G; Olsder, G.J; Quadrat, J.P, Synchronization and linearity–an algebra for discrete event systems, (1992), Wiley · Zbl 0824.93003
[4] Birkhoff, G, ()
[5] Blyth, T.S; Janowitz, M.F, Residuation theory, (1972), Pergamon Press · Zbl 0301.06001
[6] Bourbaki, N, Espaces vectoriels topologiques. éléments de mathématique, (1964), Livre V. Hermann · Zbl 0482.46001
[7] Cao, Z.Q; Kim, K.H; Roush, F.W, Incline algebra and applications, (1984), Ellis Horwood · Zbl 0541.06009
[8] Carré, B.A, An algebra for network routing problems, J. inst. math. appl., 7, 273-294, (1971) · Zbl 0219.90020
[9] Carré, B.A, Graphs and networks, (1979), The Clarendon Press, Oxford University Press New York
[10] Cohen, G; Dubois, D; Quadrat, J.P; Viot, M, A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing, IEEE trans. automat. control, 30, 210-220, (1985) · Zbl 0557.93005
[11] Cohen, G; Gaubert, S; Quadrat, J.P, Kernels, images and projections in dioids, () · Zbl 0957.93056
[12] Cohen, G; Gaubert, S; Quadrat, J.P, Linear projectors in the MAX-plus algebra, () · Zbl 0957.93056
[13] Cohen, G; Gaubert, S; Quadrat, J.P, MAX-plus algebra and system theory: where we are and where to go now, Annu. rev. control, 23, 207-219, (1999)
[14] Cohen, G; Gaubert, S; Quadrat, J.P, Separation theorem for MAX-plus semimodules, () · Zbl 1054.46500
[15] G. Cohen, S. Gaubert, J.-P. Quadrat, I. Singer, Max-plus convex sets and functions, ESI, Vienna, 2003, Preprint 1341. Also arXiv:math.FA/0308166 · Zbl 1093.26005
[16] R.A. Cuninghame-Green, Process synchronization in a steelworks–a problem of feasibility, in: J. Banbury, J. Maitland (Eds.), Proceedings of the 2nd International Conference on Operations research, Aix-en-Provence, France, 1961
[17] Cuninghame-Green, R.A, Describing industrial processes with interference and approximating their steady state behavior, Oper. res. quat., 13, 1, 95-100, (1962)
[18] Cuninghame-Green, R.A, Minimax algebra, Lecture notes in economics and mathematical systems, vol. 166, (1979), Springer · Zbl 0399.90052
[19] Cuninghame-Green, R.A, Minimax algebra and applications, Adv. imaging electron phys., (1995) · Zbl 0845.04007
[20] M.L. Dubreil-Jacotin, L. Lesieur, R. Croisot, Leçons sur la Théorie des Treillis, des Structures Algébriques Ordonnées, et des Treillis géométriques, vol. XXI of Cahiers Scientifiques, Gauthier Villars, Paris, 1953 · Zbl 0051.26005
[21] S. Gaubert, Théorie des systèmes linéaires dans les dioı̈des, Thèse, École des Mines de Paris, 1992
[22] Gaubert, S; Plus, M, Methods and applications of (MAX,+) linear algebra, ()
[23] Gierz, G; Hofmann, K.H; Keimel, K; Lawson, J.D; Mislove, M; Scott, D.S, A compendium of continuous lattices, (1980), Springer · Zbl 0452.06001
[24] Golan, J.S, The theory of semirings with applications in mathematics and theoretical computer science, vol. 54, (1992), Longman Sci & Tech · Zbl 0780.16036
[25] Gondran, M; Minoux, M, Valeurs propres et vecteurs propres dans LES dioı̈des et leur interprétation en théorie des graphes, EDF, bulletin de la direction des etudes et recherches, serie C, mathématiques informatique, 2, 25-41, (1977)
[26] Gondran, M; Minoux, M, Linear algebra in dioids: a survey of recent results, Ann. discrete math., 19, 147-164, (1984) · Zbl 0568.08001
[27] Gondran, M; Minoux, M, Graphes, dioı̈des et semi-anneaux, (2002), TEC & DOC Paris · Zbl 1025.90034
[28] ()
[29] Hasse, M, Über die behandlung graphentheoretischer probleme unter verwendung der matrizenrechnung, Wiss. Z. techn. univ. Dresden, 10, 1313-1316, (1961) · Zbl 0101.16605
[30] Kim, K.H, Boolean matrix theory and applications, (1982), Marcel Dekker New York
[31] V. Kolokoltsov, personal communication, 1999
[32] Kolokoltsov, V; Maslov, V, Idempotent analysis and applications, (1997), Kluwer Academic Publishers · Zbl 0941.93001
[33] Korbut, A.A, Extremal spaces, Dokl. akad. nauk SSSR, 164, 1229-1231, (1965) · Zbl 0135.34201
[34] Litvinov, G.L; Shpiz, G.B, Nuclear semimodules and kernel theorems in idempotent analysis: an algebraic approach, Dokl. math. sci., (2002), Also math.FA/0206026 · Zbl 1246.46063
[35] Litvinov, G.L; Maslov, V.P; Shpiz, G.B, Linear functionals on idempotent spaces: an algebraic approach, Dokl. math., 58, 3, 389-391, (1998) · Zbl 0970.46003
[36] Litvinov, G.L; Maslov, V.P; Shpiz, G.B, Idempotent functional analysis: an algebraic approach, Math. notes, 69, 5, 696-729, (2001), Also reprint arXiv:math.FA/0009128 · Zbl 1017.46034
[37] V. Maslov, S. Samborskiı̆ (Eds.), Idempotent Analysis, Adv. in Sov. Math., vol. 13, AMS, RI, 1992
[38] V.P. Maslov, Méthodes Opératorielles, Mir. Moscou. French Transl. 1987, 1973
[39] Max Plus, Linear systems in (max,+)-algebra, in: Proceedings of the 29th Conference on Decision and Control, Honolulu, 1990 · Zbl 0699.90094
[40] P. Moller, Théorie algébrique des Systèmes à Événements Discrets, Thèse, École des Mines de Paris, 1988
[41] Romanovskiı̆, I.V, Optimization of stationary control of discrete deterministic process in dynamic programming, Kibernetika, 3, 2, 66-78, (1967)
[42] Samborskiı̆, S.N; Shpiz, G.B, Convex sets in the semimodule of bounded functions, (), 135-137 · Zbl 0896.47030
[43] Vorob’ev, N.N, An extremal matrix algebra, Dokl. akad. nauk SSSR, 152, 24-27, (1963) · Zbl 0168.02602
[44] Vorob’ev, N.N, Extremal algebra of positive matrices, Elektron. informationsverarbeit. kybernetik, 3, 39-71, (1967) · Zbl 0168.02603
[45] Vorob’ev, N.N, Extremal algebra of non-negative matrices, Elektron. informationsverarbeit. kybernetik, 6, 303-311, (1970)
[46] Wagneur, E, Moduloids and pseudomodules. 1. dimension theory, Discrete math., 98, 57-73, (1991) · Zbl 0757.06008
[47] E. Wagneur, personal communication, 1991
[48] Yoeli, M, A note on a generalization of Boolean matrix theory, Amer. math. monthly, 68, 552-557, (1961) · Zbl 0115.02103
[49] K. Zimmermann, Extremálnı́ Algebra, Ekonomický ùstav C̆SAV, Praha (in Czech), 1976
[50] Zimmermann, K, A general separation theorem in extremal algebras, Ekonom.-mat. obzor, 13, 2, 179-201, (1977)
[51] Zimmermann, U, Linear and combinatorial optimization in ordered algebraic structures, (1981), North Holland · Zbl 0466.90045
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