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First and second order optimality conditions for set-valued optimization problems. (English) Zbl 1096.49012

In this paper, the author establishes first and second order necessary and sufficient optimality conditions for set-valued constrained optimization problems in terms of Bouligand and Ursescu first and second order derivatives and tangents sets under nearly convexity assumptions imposed upon the objective and constraint mappings.
Reviewer: Do Van Luu (Hanoi)

MSC:

49J53 Set-valued and variational analysis
90C46 Optimality conditions and duality in mathematical programming
58E17 Multiobjective variational problems, Pareto optimality, applications to economics, etc.
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References:

[1] Aleman, A., On some generalization of convex sets and convex functions, Mathematica-Revue d’Analyse Numerique et de Théorie de l’Approximation, 14, 1-6 (1985) · Zbl 0578.46009
[2] Aubin, J.-P.; Frankowska, H., Set-valued analysis (1990), Basel, Berlin: Birkhäuser, Basel, Berlin · Zbl 0713.49021
[3] Bigi, G.; Castellani, M., K — epiderivatives for set-valued functions and optimization, Mathematical Methods of Operations Research, 55, 401-412 (2002) · Zbl 1042.49025 · doi:10.1007/s001860200187
[4] Breckner, W. W.; Kassay, G., A Systematization of Convexity Concepts for Sets and Functions, Journal of Convex Analysis, 4, 109-127 (1997) · Zbl 0885.52003
[5] Corley, H. W., Optimality Conditions for Maximization of Set-Valued Functions, Journal of Optimization Theory and Applications, 58, 1-10 (1988) · Zbl 0956.90509 · doi:10.1007/BF00939767
[6] Jahn, R.; Rauh, R., Contingent epiderivatives and set-valued optimization, Mathematical Methods of Operations Research, 46, 193-211 (1997) · Zbl 0889.90123 · doi:10.1007/BF01217690
[7] Jimenez, B.; Novo, V., A notion of local proper efficiency in the Borwein sense in vector optimization, ANZIAM Journal, 22, 75-89 (2003) · Zbl 1161.90478 · doi:10.1017/S144618110001316X
[8] Luc, D. T., Theory of Vector Optimization (1989), Berlin: Springer Verlag, Berlin
[9] Penot, J.-P., Differentiability of relations and differential stability perturbed optimization problems, SIAM Journal on Control and Optimization, 22, 529-551 (1984) · Zbl 0552.58006 · doi:10.1137/0322033
[10] Penot, J.-P., Metric estimates for the calculus of multimappings, Set-Valued Analysis, 5, 291-308 (1997) · Zbl 0905.54011 · doi:10.1023/A:1008625212506
[11] Taa, A., Set-valued derivatives of multifunctions and optimality conditions, Numerical Functional Analysis and Optimization, 19, 121-140 (1998) · Zbl 1009.90106 · doi:10.1080/01630569808816819
[12] Ursescu, C., Tangent sets’ calculus and necessary conditions for extremality, SIAM Journal on Control Optimization, 36, 563-574 (1982) · Zbl 0488.49009 · doi:10.1137/0320041
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