Subdifferentials with respect to dualities. (English) Zbl 0838.90111

Summary: Let \(X\) and \(W\) be two sets and \(\Delta: \overline R^X\to \overline R^W\) a duality (i.e., a mapping \(\Delta: f\in \overline R^X\to f^\Delta\in \overline R^W\) such that \((\inf_{i\in I} f_i)^\Delta= \sup_{i\in I} f^\Delta_i\) for all \(\{f_i\}_{i\in I}\subseteq \overline R^X\) and all index sets \(I\)). We introduce and study the subdifferential \(\partial^\Delta f(x_0)\) of a function \(f\in \overline R^X\) at a point \(x_0\in X\), with respect to \(\Delta\). We also consider the particular cases when \(\Delta\) is a (Fenchel-Moreau) conjugation, or a \(\vee\)-duality, or a \(\perp\)-duality.


90C30 Nonlinear programming
49J52 Nonsmooth analysis
Full Text: DOI


[1] Balder EJ (1977) An extension of duality-stability relations to nonconvex optimization problems. SIAM J Control Optim 15:329-343 · Zbl 0366.90103
[2] Dolecki S, Kurcyusz S (1978) On?-convexity in extremal problems. SIAM J Control Optim 16:277-300 · Zbl 0397.46013
[3] Elster K-H, Göpfert A (1990) Conjugation concepts in optimization. In: Methods of Oper Res vol 62 Rieder U, Gessner P, Peyerimhoff A, Radermacher FJ (Eds) 53-65. Hain A Meisenheim GmbH Frankfurt am Main
[4] Greenberg HJ, Pierskalla WP (1973) Quasi-conjugate functions and surrogate duality. Cahiers Centre d’Et Rech Opér 15:437-448
[5] Lindberg PO (1979) A generalization of Fenchel conjugation giving generalized Lagrangians and symmetric nonconvex duality. In: Survey of Mathematical Programming. Proc 9th Internat Math Progr Symposium, Budapest 1976 vol I, 249-267. North Holland Amsterdam
[6] Martínez-Legaz J-E (1988) Quasiconvex duality theory by generalized conjugation methods. Optimization 19:603-652 · Zbl 0671.49015
[7] Martínez-Legaz J-E, Singer I (1990) Dualities between complete lattices. Optimization 21:481-508 · Zbl 0728.90071
[8] Martínez-Legaz J-E, Singer I (1991) ? -dualities and ?-dualities. Optimization 22:483-511 · Zbl 0728.06007
[9] Moreau J-J (1966-1967) Fonctionnelles convexes. Sémin Eq Dériv Part Collège de France Paris No 2
[10] Moreau J-J (1970) Inf-convolution, sous-additivité, convexité des fonctions numériques. J Math Pures Appl 49:109-154
[11] Singer I (1983) Surrogate conjugate functionals and surrogate convexity. Appl Anal 16:291-327 · Zbl 0526.90097
[12] Singer I (1984) Conjugation operators. In: Selected topics in operations research and mathematical economics Hammer G, Pallaschke D (Eds) Lecture Notes in Econ and Math Systems 226:80-97. Springer-Verlag Berlin-Heidelberg-New York-Tokyo · Zbl 0545.49008
[13] Singer I (1986) Some relations between dualities, polarities, coupling functionals and conjugations. J Math Anal Appl 115:1-22 · Zbl 0601.46043
[14] Singer I (1987) Infimal generators and dualities between complete lattices. Ann Mat Pura Appl (4) 148:289-358 · Zbl 0638.06006
[15] Zabotin Ya I, Korablev AI, Habibullin RF (1973) Conditions for an extremum of a functional in the presence of constraints. Kibernetika 6:65-70
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.