×

zbMATH — the first resource for mathematics

Generalized monotonicity and generalized convexity. (English) Zbl 0824.90124
Summary: Generalized monotonicity of bifunctions or multifunctions is a rather new concept in optimization and nonsmooth analysis. It is shown in the present paper how quasiconvexity, pseudoconvexity, and strict pseudoconvexity of lower semicontinuous functions can be characterized via the quasimonotonicity, pseudomonotonicity, and strict pseudomonotonicity of different types of generalized derivatives, including the Dini, Dini-Hadamard, Clarke, and Rockafellar derivatives as well.

MSC:
90C30 Nonlinear programming
26B25 Convexity of real functions of several variables, generalizations
49J52 Nonsmooth analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Avriel, M., Diewert, W. E., Schaible, S., andZiemba, W. T.,Generalized Concavity, Plenum Press, New York, New York, 1988.
[2] Arrow, K. J., andEnthoven, A. C.,Quasiconcave Programming, Econometrica, Vol. 29, pp. 779–800, 1961. · Zbl 0104.14302
[3] Diewert, W. E.,Alternative Characterizations of Six Kinds of Quasiconcavity in the Nondifferentiable Case with Applications to Nonsmooth Programming, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, pp. 51–95, 1981. · Zbl 0539.90088
[4] Karamardian, S.,Complementarity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976. · Zbl 0304.49026
[5] Hassouni, A,Sous-Differentiels des Fonctions Quasi-Convexes, Thése de 3{\(\deg\)} Cycle, Université Paul Sabatier, Toulouse, France, 1983.
[6] Hiriart-Urruty, J. B.,Miscellaneas on Nonsmooth Analysis and Optimization, Nondifferentiable Optimization: Motivations and Applications, Edited by V. F. Demyanov and D. Pallaschke, Springer Verlag, Berlin, pp. 8–25, 1985.
[7] Karamardian, S., andSchaible, S.,Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990. · Zbl 0679.90055
[8] Komlósi, S.,On Generalized Upper Quasidifferentiability, Nonsmooth Optimization: Methods and Applications, Edited by F. Giannessi, Gordon and Breach, London, England, pp. 189–201, 1992. · Zbl 1050.49509
[9] Komlósi, S.,Generalized Monotonicity of Generalized Derivatives, Proceedings of the Workshop on Generalized Concavity for Economic Applications, Pisa, Italy, 1992; Edited by P. Mazzoleni, Verona, Italy, pp. 1–7, 1992. · Zbl 1050.49509
[10] Komlósi, S.,Generalized Monotonicity in Nonsmooth Analysis, Generalized Convexity, Edited by S. Komlósi, T. Rapcsák, and S. Schaible, Springer Verlag, Heidelberg, Germany, pp. 263–275, 1994. · Zbl 0811.49013
[11] Luc, D. T.,Characterization, of Quasiconvex Functions, Bull. Austral. Math. Soc., Vol. 48, pp. 393–405, 1993. · Zbl 0790.49015
[12] Luc, D. T.,On Generalized Convex Nonsmooth Functions, Bull. Austral. Math. Soc., Vol. 49, pp. 139–149, 1994. · Zbl 0811.90096
[13] Schaible, S.,Generalized Monotone Maps, Nonsmooth Optimization: Methods and Applications, Edited by F. Giannessi, Gordon and Breach, London, England, pp. 392–408, 1992. · Zbl 1050.49504
[14] Schaible, S.,Generalized Monotone Maps: A Survey, Generalized Convexity, Edited by S. Komlósi, T. Rapcsák, and S. Schaible, Springer Verlag, Heidelberg, Germany, pp. 229–249, 1994. · Zbl 0802.90107
[15] Karamardian, S., Schaible, S., andCrouzeix, J. P.,Characterizations of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399–413, 1993. · Zbl 0792.90070
[16] Pini, R., andSchaible, S.,Some Invariance Properties of Generalized Monotone Maps, Proceedings of the Workshop on Generalized Concavity for Economic Applications, Pisa, Italy, 1992; Edited by P. Mazzoleni, Verona, Italy, pp. 87–89, 1992.
[17] Rockafellar, R. T.,The Theory of Subgradients and Its Applications to Problems of Optimization: Convex and Nonconvex Functions, Heldermann Verlag, Berlin, Germany, 1981. · Zbl 0462.90052
[18] Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley, New York, New York, 1983. · Zbl 0582.49001
[19] Penot, J. P.,On the Mean-Value Theorem, Optimization, Vol. 19, pp. 147–156, 1988. · Zbl 0661.49008
[20] Borwein, J. M., andWard, D. E.,Nonsmooth Calculus in FInite Dimension, SIAM Journal on Control and Optimization, Vol. 25, pp. 1312–1340, 1987. · Zbl 0633.46043
[21] Ioffe, A. D.,Approximate Subdifferentials and Applications, I: The Finite-Dimensional Theory, Transactions of the American Mathematical Society, Vol. 281, pp. 389–416, 1984. · Zbl 0531.49014
[22] Ellaia, R., andHassouni, A.,Characterization of Nonsmooth Functions through Their Generalized Gradients, Optimization, Vol. 22, pp. 401–416, 1991. · Zbl 0734.49005
[23] Mangasarian, O. L.,Pseudoconvex Functions, SIAM Journal on Control, Vol. 3, pp. 281–290, 1965. · Zbl 0138.15702
[24] Borde, J., andCrouzeix, J. P.,Continuity Properties of the Normal Cone to the Level Sets of a Quasiconvex Function, Journal of Optimization Theory and Applications, Vol. 66, pp. 415–429, 1990. · Zbl 0682.90079
[25] Komlósi, S.,Some Properties of Nondifferentiable Pseudoconvex Functions, Mathematical Programming, Vol. 26, pp. 232–237, 1983. · Zbl 0541.90081
[26] Hiriart-Urruty, J. B.,New Concepts in Nondifferentiable Programming, Bulletin de la Sociète Mathèmatique de France, Mémoires, Vol. 60, pp. 57–85, 1979.
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.