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Conjugaison par tranches. (French) Zbl 0581.49007
We are interested here in extended real-valued functions whose level sets are closed with respect to a given closure operator. This class of functions is closed under pointwise suprema so that a regularization can be defined. By using the notion of polarity we decompose the closure operator and introduce a (bi) conjugation for the real extended valued functions f such that the biconjugate of f is just the regularized of f. We apply this theory to many forms of quasiconvex dualities and to mathematical programming in the general form.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49N15 Duality theory (optimization)
90C25 Convex programming
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B23 Complete lattices, completions
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[1] Araoz, J.; Edmonds, J.; Griffin, V. J., Polarities given by systems of bilinear inequalities, Math. Op. Res., vol. 8, n. 1, 34-41 (1983) · Zbl 0517.90085
[2] M.Atteia - A. ElQortobi,Quasiconvex duality, Lecture Notes in Control,30, Springer Verlag (1981), pp. 3-8.
[3] Balder, S. J., An extension of duality stability relations to non convex optimization problems, Siam, J. Contr. Opt., 15, 329-343 (1977) · Zbl 0366.90103
[4] G.Birkhoef,Lattice theory, Providence, A.M.S., 1966.
[5] Y.Chabrillac,Thèse de 3-èmecycle, Clermont II, 1982.
[6] Choquet, G., Sur les notions de filtre et de grille, C.R.A.S. Paris, 884, 171-173 (1947) · Zbl 0029.07602
[7] J. P.Crouzeix,Contribution à l’étude des fonctions quasiconvexes, Thèse, Clermont II, 1977.
[8] Crouzeix, J. P., Duality between direct and indirect utility functions, J. Math. Eco., vol. 12, n. 2, 139-165 (1983) · Zbl 0527.90004
[9] Dolecki, S., Remarks on semicontinuity, Bull. Ac. Pol. Ser. Math., 85, 863-867 (1977) · Zbl 0372.54008
[10] Dolecki, S., Abstract study of optimality conditions, J. Math. Anal. Appl., 73, 24-58 (1980) · Zbl 0438.49018
[11] S.Dolecki - G.Greco,Cyrtologies and convergence, I, mimeo Université de Trente. 1983.
[12] Durier, Communication personnelle.
[13] A. ElQortobi,These de 3-ème cycle, Toulouse, 1980.
[14] Elster, K. H.; Nehse, R., Zur theorie der polarfunktionale, Math. Oper. forsch. Stat., 5, 3-21 (1974) · Zbl 0283.90049
[15] Everett, C. J., Closure operators and Galois theory in lattices, Trans. Amer. Math. Soc., vol. 55, 514-525 (1944) · Zbl 0060.06205
[16] J. J. M.Evers - H. V.Maaren,Duality principles in mathematics and their relations to conjugate functions, Dpt of applied mathematics, Twente Univ. of Tech., memorandum336 (1981). · Zbl 0598.49009
[17] Fenchel, W., On conjugate convex functions, Can. J. Math., 1, 73-77 (1949) · Zbl 0038.20902
[18] W.Fenchel,Convex cones, sets and functions, mimeo Lectures Notes, Princeton University, 1951.
[19] Flachs, J.; Pollatschek, Duality theorems for certain programs involving minimum or maximum operations, Math. Prog., 16, 348-370 (1979) · Zbl 0405.90067
[20] Flachs, J., Global saddle point duality for quasiconcave programs, Math. Prog., 80, 327-347 (1981) · Zbl 0461.90058
[21] Flachs, J., Global saddle point duality for quasiconcave programs, II, Math. Prog., 84, 326-345 (1982) · Zbl 0493.90070
[22] Greenberg, H. J.; Pierskalla, W. P., Quasiconjugate functions and surrogate duality, Cahier Centre Etude, Rech. Oper., 15, 437-448 (1973) · Zbl 0276.90051
[23] Ivanof, E. H.; Nehse, R., Relations between generalized concepts of convexity and conjugacy, Math. Oper. Stat. Ser. Opt., 13, 9-18 (1982) · Zbl 0503.90077
[24] Kukateladze, S. S.; Rubinov, A. M., Minkowski’s duality and its applications (1976), Nauka: Novosibirsk, Nauka
[25] P. O.Lindberg,A generalization of Fenchel conjugation, Proc. IX Int. Symp. on Math. Prog., vol. 2, Akad Kiado Budapest, 1979.
[26] J. E.Martinez Legaz,Exact quasiconvex conjugation, Symp. Math. Prog., Bonn, 1982. · Zbl 0522.90069
[27] J. E.Martinez Legaz,A new approach to symmetric quasiconvex conjugacy, Sym. Oper. Res. Karlsruhe, 1983. · Zbl 0543.90085
[28] J. E.Martinez Legaz,Conjugacion associada a un grafo, Jornadas Matematicas Hispano-Lusas, 1982.
[29] J. J.Moreau,Fonctionnelles convexes, Collège de France, 1966.
[30] Moreau, J. J., itInf convolution, sous additivité, convexité des fonctions numériques, J. Math. Pures. Appl., 49, 109-154 (1970) · Zbl 0195.49502
[31] E. H.Moore,Introduction to a form of general analysis, Amer. Math. Soc. Coll. Pub., vol. 2, (1910). · JFM 41.0376.01
[32] Ore, O., Galois connexions, Trans. Amer. Math. Soc., vol. 55, 493-513 (1944) · Zbl 0060.06204
[33] O.Ore,Theory of graphs, A.M.S., coll. Pub., vol. XXXVIII, (1962). · Zbl 0105.35401
[34] U.Passy - E. Z.Prisman,On quasiconvex functions and their conjugate, mimeo 297, Fac. of Ind. and Mang. Eng. Technion Haifa Israel, 1981.
[35] Passy, U.; Prisman, E. Z., Duality in quasiconvex programming, mimeo, 308, 123-456 (1981)
[36] Passy, U.; Prisman, E. Z., Saddle functions and min-max problems, mimeo, 303, 123-456 (1981)
[37] J. P.Penot - M.Volle,On quasiconvex duality, en préparation. · Zbl 0662.49009
[38] Pickert, G., Bemerkungen über Galois-Verbindungen, Arch. Math., 3, 285-289 (1952) · Zbl 0047.26402
[39] R. T.Rockafellar,Convex analysis, Princeton, 1970. · Zbl 0193.18401
[40] I.Singer,Pseudoconjugate functionals and pseudoduality, Math. Meth. Op. Res. Sofia, (1981), pp. 115-134.
[41] Volle, M., Multiapplications duales et problèmes d’optimisation en dualité, C.R.A.S. Paris, 896, 11-15 (1983) · Zbl 0528.49010
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