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An extension of d. c. duality theory, with an appendix on \(*\)-subdifferentials. (English) Zbl 0893.90143

Summary: We present some results extending d.c. duality theory to the generalized convex setting. Given a binary operation \(*\) on the extended real line, we consider the problem of minimizing \(f*-h\), with \(f\) and \(h\) being functions defined on an arbitrary set. For this problem, we obtain results which encompass, as particular cases, some well-known duality theorems for the infimum of the difference of two functions and the infimum of the maximum of two functions, as well as some well-known formulae for the conjugate of the difference of two functions and the conjugate of type Lau of the maximum of two functions. Instead of considering the Fenchel conjugation operator, our results are expressed with the aid of dualities which are associated to the operation \(*\). We also study, for such dualities, the corresponding generalized subdifferential.

MSC:

90C26 Nonconvex programming, global optimization
49J52 Nonsmooth analysis
06A15 Galois correspondences, closure operators (in relation to ordered sets)
49N15 Duality theory (optimization)
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[1] DOI: 10.1137/0315022 · Zbl 0366.90103
[2] DOI: 10.1007/BF00941076 · Zbl 0568.90076
[3] DOI: 10.1007/BF01582120 · Zbl 0405.90067
[4] Martinez-Legaz J.-E., Econ. Math. Systems 345 pp 168– (1990)
[5] DOI: 10.1017/S0004972700036984 · Zbl 0742.90071
[6] Martinez-Legaz J.-E., Optimization 21 pp 483– (1990)
[7] DOI: 10.1080/02331939108843691 · Zbl 0728.06007
[8] DOI: 10.1080/02331939408843993 · Zbl 0815.49026
[9] DOI: 10.1007/BF01415676 · Zbl 0838.90111
[10] Martinez-Legaz J.-E., R.J.Conv.Anal 2 pp 185– (1995)
[11] Moreau J.J. Fonctionnelles convexes Collége de France Paris 1966–1967 Sémin. Eq. Dériv. Part 2
[12] Moreau .J.-J., J. Math. Pures Appl 49 pp 109– (1970)
[13] DOI: 10.1007/BF01899228 · Zbl 0047.26402
[14] Pshenichnyi B. N., Contrôle optimal et jeux differentiels (1971)
[15] DOI: 10.1017/S0004972700010844 · Zbl 0404.90101
[16] Singer I., Math 3 pp 235– (1980)
[17] Singer I., Lecture Notes Econ. Math. Systems 226 pp 80– (1984)
[18] DOI: 10.1016/0022-247X(86)90021-1 · Zbl 0601.46043
[19] DOI: 10.1007/BF01774294 · Zbl 0638.06006
[20] Singer I., Lecture Notes Pure Appl.Math 107 pp 253– (1987)
[21] DOI: 10.1016/0022-247X(78)90243-3 · Zbl 0403.90066
[22] Volle M., Thése. Lrniv. de Pau (1986)
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