On subdifferentials of set-valued maps. (English) Zbl 0922.90118

Baier, J.; Jahn, J.

J. Optimization Theory Appl. 100, No. 1, 233-240 (1999).

Summary: Using the concept of contingent epiderivative, we generalize the notion of subdifferential to a cone-convex set-valued map. Properties of the subdifferential are presented and an optimality condition is discussed.

MSC:

90C29	Multi-objective and goal programming
49J52	Nonsmooth analysis

Keywords:

convex analysis; set-valued analysis; subdifferentials; vector optimization

References:

[1]	Jahn, J., Introduction to the Theory of Nonlinear Optimization, Springer, Berlin, Germany, 1996. · Zbl 0855.49001
[2]	Jahn, J., and Rauh, R., Contingent Epiderivatives and Set-Valued Optimization, Mathematical Methods of Operations Research, Vol. 46, pp. 193–211, 1997. · Zbl 0889.90123
[3]	Yang, Q. X., A Hahn-Banach Theorem in Ordered Linear Spaces and Its Applications, Optimization, Vol. 25, pp. 1–9, 1992. · Zbl 0834.46006
[4]	Chen, G. Y., and Jahn, J., Optimality Conditions for Set-Valued Optimization Problems, Mathematical Methods of Operations Research, Vol. 48, 1998. · Zbl 0927.90095
[5]	Jahn, J., Mathematical Vector Optimization in Partially-Ordered Linear Spaces, Peter Lang, Frankfurt, Germany, 1986. · Zbl 0578.90048
[6]	Aubin, J. P., Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions, Mathematical Analysis and Applications, Part A, Edited by L. Nachbin, Academic Press, New York, New York, pp. 160–229, 1981. · Zbl 0484.47034
[7]	Baier, J., Subdifferentiale für mengenwertige Abbildungen, Diplom Thesis, University of Erlangen-Nürnberg, 1997.
[8]	Zowe, J., Konvexe Funktionen und konvexe Dualitätstheorie in geordneten Vektorräumen, Habilitation Thesis, University of Würzburg, 1976.