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Conjugate convex operators. (English) Zbl 0588.46027

The properties of convex operators with values in ordered t.v.s. with \(\infty\) adjoint are considered-continuity, semicontinuity etc. The convex l.s.c. operators are proved to be suprema of affine, continuous operators under the assumption that the positive cone has nonempty interior and in some special cases without this assumption. The results are applied to polarity theory.
Reviewer: T.Rzežuchowski

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46A55 Convex sets in topological linear spaces; Choquet theory
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[1] Anger, B.; Lembcke, J., Hahn-Banach type theorems, Math. Ann., 209, 127-175 (1974) · Zbl 0268.46006
[2] Borwein, J., Continuity and differentiability. Properties of convex operators, (Proc. London Math. Soc., 3 (1982)), 420-444, (44) · Zbl 0487.46026
[3] Borwein, J., A Lagrange multiplier theorem and a sandwich theorem for convex relations, Math. Scand., 48, 189-204 (1981) · Zbl 0468.49007
[4] Borwein, J., Convex relations in analysis and optimization, (Schaible, S.; Zeinba, W., NATO Advances Study Institute on Generalized Concavity in Optimization and Economics (1981), Academic Press: Academic Press New York), 335-377
[5] Brumelle, S. L., Convex operators and supports, Math. Oper. Res., 3, 2, 171-175 (1978) · Zbl 0408.46015
[6] Dieudonné, J., Sur la séparation des ensembles convexes, Math. Ann., 163, 1-3 (1966) · Zbl 0131.11401
[7] Duong, P. C.; Tuy, H., Stability, surjectivity, invertibility of non differentiable mappings, Acta Math. Vietnam., 3, 1, 80-105 (1978) · Zbl 0426.47041
[8] Ekeland, I.; Temam, R., Analyse convexe et problèmes variationnels (1973), Dunod: Dunod Paris
[9] Elster, K. H.; Nehse, R., Konjugierte operatoren und subdifferentiale, Math. Operationsforsch. Statist., 4, 641-657 (1975) · Zbl 0331.90050
[10] Fenchel, W., On conjugate convex functions, Canad. J. Math., 1, 73-77 (1949) · Zbl 0038.20902
[11] Gwinner, J., Closed images of convex multivalued mappings in linear topological spaces and applications, J. Math. Anal. Appl., 60, 75-86 (1977) · Zbl 0373.46012
[12] Hörmander, L., Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Ark. Mat., 3, 181-186 (1965) · Zbl 0064.10504
[13] Ioffe, A. D.; Levin, V. L., Subdifferential of convex functions, Trans. Moscow Math. Soc., 26, 1-72 (1972) · Zbl 0281.46039
[14] G. Isac; G. Isac · Zbl 0476.47042
[15] Jameson, G., Ordered Linear Spaces, (Lecture Notes in Mathematics No. 141 (1970), Springer-Verlag: Springer-Verlag New York) · Zbl 0196.13401
[16] Kutateladze, S. S., Formulas for computing subdifferentials, Soviet Math. Dokl., 18, 1, 146-148 (1977) · Zbl 0371.47055
[17] Kutateladze, S. S., Change of variables in the Young transformation, Soviet Math. Dokl., 18, 2, 545-548 (1977) · Zbl 0383.49011
[18] Larman, D. G.; Phelps, R., Gâteaux differentiability of convex functions on Banach spaces, J. London Math. Soc., 20, 115-127 (1979) · Zbl 0431.46033
[19] Laurent, P. J., Approximation et optimisation (1972), Hermann: Hermann Paris · Zbl 0238.90058
[20] Linke, Yu. E., On the support sets of sublinear operators, Soviet Math. Dokl., 13, 1561-1568 (1972) · Zbl 0272.47021
[21] Linke, Yu. E., Sublinear operators with values in spaces of continuous functions, Soviet Math. Dokl., 17, 3, 774-777 (1976) · Zbl 0349.47026
[22] McLinden, L., Affine minorants minimizing the sum of convex functions, J. Optim. Theory Appl., 24, 4, 569-583 (1978) · Zbl 0351.90055
[23] Malivert, C., A descent method for Pareto optimization, J. Math. Anal. Appl., 88, 610-631 (1982) · Zbl 0499.49015
[24] Malivert, C.; Penot, J. P.; Thera, M., Un prolongement du théorème de Hahn-Banach, C. R. Acad. Sci. Paris Ser. A, 165-168 (1978) · Zbl 0376.46003
[25] Moreau, J. J., Fonctionnelles convexes., (Equations aux dérivées partielles, Séminaire J. Leray (1966), Collège de France: Collège de France Paris) · Zbl 0137.31401
[26] Penot, J. P.; Thera, M., Semicontinuous mappings in general topology, Ark Mat., 38, 2, 158-166 (1982) · Zbl 0465.54019
[27] Penot, J. P.; Thera, M., Semi-continuité des applications et des multiapplications, C. R. Acad. Sci. Paris Ser. A, 288, 241-244 (1979) · Zbl 0421.54014
[28] Penot, J. P.; Thera, M., Polarité des applications convexes vectorielles, C. R. Acad. Sci. Paris Ser. A, 288, 419-422 (1979) · Zbl 0398.46010
[29] J. P. Penot and M. TheraAnn. Mat.; J. P. Penot and M. TheraAnn. Mat. · Zbl 0609.46021
[30] Peressini, A. L., Ordered Topological Vector Spaces (1967), Harper & Row: Harper & Row New York · Zbl 0169.14801
[31] Raffin, C., Contribution à l’étude des programmes convexes définis dans des espaces vectoriels topologiques, Ann. Inst. Fourier, 20, 1, 457-492 (1970) · Zbl 0195.49601
[32] Robinson, S., Regularity and stability for convex multivalued functions, Math. Oper. Res., 1, 2 (1976) · Zbl 0418.52005
[33] Rockafellar, R. T., Extension of Fenchel’s duality theorem for convex functions, Duke Math. J., 33, 81-90 (1960) · Zbl 0138.09301
[34] Rockafellar, R. T., Convex Functions and Duality Optimization Problems and Dynamics, (Lecture Notes in Oper. Res. and Math. Ec., Vol. 11 (1969), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0186.23901
[35] Rockafellar, R. T., Convex Analysis (1970), Princeton Univ. Press: Princeton Univ. Press Princeton, N. J · Zbl 0202.14303
[36] Rubinov, A. M., Sublinear operators and their applications, Russian Math. Surveys, 32, 4, 115-175 (1977) · Zbl 0384.47002
[37] Schaeffer, H. H., Topological Vector Spaces (1971), Springer-Verlag: Springer-Verlag New York
[38] Thera, M., Etude des fonctions convexes vectorielles semi-continues, Thèse de spécialité Pau (1978)
[39] Thera, M., Convex lower-semicontinuous vector-valued mappings and applications to convex analysis, Oper. Res. Verfahren, 31, 631-636 (1979) · Zbl 0403.46009
[40] Thera, M., Subdifferential calculus for convex operators, J. Math. Anal. Appl., 80, 1, 78-91 (1981) · Zbl 0467.90078
[41] Ursescu, Multifunctions with closed convex graph, Czechoslovak Math. J., 25, 100 (1975) · Zbl 0318.46006
[42] Valadier, M., Sous-différentiabilité des fonctions convexes à valeurs dans un espace vectoriel ordonné, Math. Scand., 30, 65-74 (1972) · Zbl 0239.46038
[43] Zalinescu, C., The Fenchel-Moreau duality theory of mathematical programming in order complete vector lattices and applications, (Preprint 45 (1980), Institutul de Matematica: Institutul de Matematica Bucuresti)
[44] Zowe, J., Linear maps majorized by a sublinear map, Arch. Math., 637-645 (1975) · Zbl 0326.46036
[45] Zowe, J., Subdifferentiability of convex functions with values in an ordered vector space, Math. Scand., 34, 69-83 (1974) · Zbl 0288.46007
[46] Zowe, J., A duality theorem for a convex programming problem in order complete vector lattices, J. Math. Anal. Appl., 50, 273-287 (1975) · Zbl 0314.90079
[47] Zowe, J., Sandwich theorems for convex operators with values in an ordered vector space, J. Math. Anal. Appl., 66, 292-296 (1978) · Zbl 0389.46003
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