×

Quasiconvex duality theory by generalized conjugation methods. (English) Zbl 0671.49015

Summary: We survey duality theories for quasiconvex optimization problems, based on notions of generalized conjugation. Some of them are obtained from Moreau’s generalized conjugation, while the others are special cases of the so-called H-duality. The relationship existing between the two kinds of approaches is described.

MSC:

49N15 Duality theory (optimization)
26B25 Convexity of real functions of several variables, generalizations
90C30 Nonlinear programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1137/0315022 · Zbl 0366.90103
[2] Ben-Tal, A. and Ben-Israel, A. 1981.F-convex functions: properties and applications, Edited by: Schaible, S. and Ziemba, W.T. 301–334. New York: Acad. Press. Generalized concavity in optimization and economics · Zbl 0535.90074
[3] DOI: 10.1007/BF01917114 · Zbl 0494.90089
[4] Chae S.B., Holomorphy and calculus in normed spaces (1985) · Zbl 0571.46031
[5] DOI: 10.1007/BF00934352 · Zbl 0452.90055
[6] Crouzeix J.P., Coiuples Tlendus des Seances de l 279 pp 955– (1974)
[7] Crouzeix, J.P. 1977.Conjugacy in quasiconvex analysis, 66–99. Berlin, New York: Springer-Verlag. Convex analysis and its applications,Heidelberg · Zbl 0362.90096
[8] Crouzeix J.P., These, Univer-site de Clermont (1977)
[9] Crouzeix J.P., Generalized concavity in optimization and economics pp 207– (1981)
[10] DOI: 10.1007/BF02591908 · Zbl 0526.90083
[11] Deumlich R., Mathematische Ope-rationsforschung und Statistik, Series Optimization 8 pp 151– (1977)
[12] Deumlich R., Survey of Mathematical Programming. I pp 193– (1979)
[13] Deumlich R., Operations Research in Progress pp 27– (1982)
[14] DOI: 10.1080/02331938308842841 · Zbl 0524.90081
[15] Deumlich R., Mathematische Operationsforschung und Statistik, Series Optimization 15 pp 415– (1984)
[16] DOI: 10.1080/02331938508843077 · Zbl 0605.90104
[17] Dolecki S., Comptes Rendus des Seances de l 283 pp 91– (1976)
[18] DOI: 10.1137/0316018 · Zbl 0397.46013
[19] DOI: 10.1007/BF01581647 · Zbl 0464.90067
[20] Ekeland I., Analyse convexe et problemes variationnels (1974) · Zbl 0281.49001
[21] El Qortobi A., Applications. Thése (1980)
[22] Evers J.J.M., Nieuw Archiv voor Wiskunde 3 pp 23– (1985)
[23] Fenchel W., Comm. Sém. Math. Univ. Lund 3 pp 82– (1952)
[24] DOI: 10.1007/BF01589356 · Zbl 0461.90058
[25] DOI: 10.1007/BF01585114 · Zbl 0493.90070
[26] DOI: 10.1287/opre.13.6.879 · Zbl 0163.41301
[27] DOI: 10.1287/opre.16.4.741 · Zbl 0165.22602
[28] DOI: 10.1287/opre.18.5.924 · Zbl 0232.90059
[29] Greenberg H.J., Cahiers du Centre d’etudes de Recherche Operationelle 15 pp 437– (1973)
[30] GutierrezDiez J.M., Fisieas y Naturales de Madrid 78 pp 523– (1984)
[31] DOI: 10.1007/BF02889704 · Zbl 0735.90053
[32] DOI: 10.1215/S0012-7094-55-02209-2 · Zbl 0064.16601
[33] Hammer, P.C. 1963.Semispaces and the topology of convexity, Edited by: Kler, V.L. 305–316. Providence: American Mathematical Society. · Zbl 0151.29303
[34] Klee V.L., Mathematica Scandinavica 4 pp 54– (1956) · Zbl 0070.39203
[35] Linberg P.O., Survey of Mathematical Programming pp 249– (1979)
[36] Lindberg P.O., Royal Institute of Technology (1981)
[37] DOI: 10.1137/0116088 · Zbl 0212.23905
[38] Martínez-Legaz J.E., Aplicacion a las functiones quasiconvexas. Tesis (1981)
[39] Martínez-Legaz J.E., Actas IX Jornadas Mate-maticasHispano-Lusas pp 837– (1982)
[40] Martínez-Legaz J.E., Optimization, Theory and Algorithms pp 45– (1983)
[41] DOI: 10.1007/BF01916921 · Zbl 0522.90069
[42] Martínez-Legaz J.E., Selected Topics in Operations Research and Mathematical Economics pp 42– (1984)
[43] DOI: 10.1007/BFb0008893
[44] Martínez-Legaz J.E., Methods of Operations Research 49 pp 47– (1985)
[45] Martínez-Legaz J.E., Internationale Tagung Mathematische Optimierung-Theorie und Anwendungen pp 176– (1986)
[46] Martínez-Legaz J.E., Trends in Mathematical Optimization pp 197– (1988)
[47] DOI: 10.1016/0024-3795(87)90312-0 · Zbl 0614.52003
[48] DOI: 10.1080/01630568708816247 · Zbl 0609.49012
[49] Martínez-Legaz J.E., Dualities between complete lattices (1987)
[50] Martínez-Legaz J.E., Linear Algebra and its Applications (1987) · Zbl 0586.49004
[51] Martínez-Legaz J.E., lexicographical index and linear operators (1987) · Zbl 0711.52002
[52] Martos B., Studia Sci, Math. Hung 2 pp 265– (1967)
[53] Mizukami K., Control and Cybernetics 13 pp 375– (1984)
[54] Moreau J.J., Journal de Mathematiques pures et appliqués 49 pp 109– (1970)
[55] DOI: 10.1007/BF02591881 · Zbl 0547.49008
[56] DOI: 10.1007/BF01582050 · Zbl 0606.49009
[57] Penot J.P., manuscript 82 (1986)
[58] DOI: 10.1007/978-3-0348-9297-1_17
[59] DOI: 10.1007/BF00938758 · Zbl 0542.90083
[60] Plastria F., Journal of Optimization Theory and Applications 48 (1985)
[61] DOI: 10.1007/BF02591773 · Zbl 0505.90069
[62] Rockafellar R.T., Convex Analysis (1970) · Zbl 0193.18401
[63] Sikorski J., Analysis and algorithms of optimization problems (1986)
[64] Singer I., Publ. House Acad. Soc. Rep. Romania (1970)
[65] Singer I., Mathematical methods in operations research pp 115– (1981)
[66] DOI: 10.1080/01630568208816118 · Zbl 0497.49022
[67] Singer I., Optimization, Theory and Algorithms pp 13– (1983)
[68] DOI: 10.1016/0362-546X(83)90020-2 · Zbl 0528.49007
[69] Singer I., Applicable Analysis 98 pp 31– (1984) · Zbl 0584.49006
[70] DOI: 10.1016/0022-247X(84)90277-4 · Zbl 0584.49006
[71] DOI: 10.1080/02331938408842904 · Zbl 0538.49004
[72] DOI: 10.1016/0362-546X(84)90033-6 · Zbl 0538.49005
[73] DOI: 10.1016/0022-247X(84)90002-7 · Zbl 0607.90089
[74] Singer, I. 1984.Selected Topics ir Operations Research and Mathematical Economics, Edited by: Hammer, G. and Pallaschke, D. 49–79. Berlin, New York: Springer-Verlag. Heidel berg
[75] Singer I., Ibidem pp 80– (1984)
[76] DOI: 10.1016/0022-247X(86)90021-1 · Zbl 0601.46043
[77] DOI: 10.1016/0022-247X(86)90046-6 · Zbl 0598.49011
[78] Singer I., Parametric optimization problems and related topics pp 355– (1987)
[79] DOI: 10.1287/opre.21.1.305 · Zbl 0264.26012
[80] DOI: 10.1007/BF01584248 · Zbl 0467.90061
[81] Volle M., Comptes Rendus des Seances de l 293 pp 265– (1981)
[82] Volle M., Comptes Rendus des Seances de l 296 pp 11– (1983)
[83] DOI: 10.1007/BF01766858 · Zbl 0581.49007
[84] DOI: 10.1007/BF01774294 · Zbl 0638.06006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.