Abstract
A a set-valued optimization problem min C F(x), x ∈X 0, is considered, where X 0 ⊂ X, X and Y are normed spaces, F: X 0 ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x 0,y 0), y 0 ∈F(x 0), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x 0, y 0) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubin JP, Frankowska H (1990) Set-valued analysis. Birkhäuser, Boston
Auslender A (1984) Stability in mathematical programming with nondifferentiable data. SIAM J Control Optim 22:239–254
Bednarczuk EM (2002) A note of lower semicontinuity of minimal points. Nonlinear Anal 50:285–297
Bednarczuk EM, Song W (1998) Contingent epiderivative and its applications to set-valued optimization. Control Cybernet 27(3):375–386
Bigi G, Castellani M (2002) K-epiderivatives for set-valued functions and optimization. Math Methods Oper Res 55(3):401–412
Flores-Bazán F (2001) Optimality conditions in non-convex set-valued optimization. Math Methods Oper Res 53(3):403–417
Ginchev I (2002) Higher order optimality conditions in nonsmooth vector optimization. In: Cambini A, Dass BK, Martein L (eds) Generalized convexity, generalized monotonicity, optimality conditions and duality in scalar and vector optimization. J Stat Manag Syst 5(1–3):321–339
Ginchev I, Guerraggio A, Rocca M (2003) First-order conditions for C 0,1 constrained vector optimization. In: Giannessi F, Maugeri A (eds) Variational analysis and applications. Proc. Erice, June 20– July 1, 2003, Springer, New York (2005)
Ginchev I, Guerraggio A, Rocca M (2005) From scalar to vector optimization. Appl Math (to appear)
Ginchev I, Guerraggio A, Rocca M (2004) Isolated minimizers, proper efficiency and stability for C 0,1 constrained vector optimization problems. Preprint 2004/9, Universitá dell’Insubria, Facoltá di Economia, Varese 2004 (http:// eco.uninsubria.it/ dipeco/ Quaderni/ files/ QF2004_9.pdf)
Ginchev I, Hoffmann A (2002) Approximation of set-valued functions by single-valued one. Discuss Math Diff Incl Control Optim 22:33–66
Henig MI (1982) Proper efficiency with respect to cones. J Optim Theory Appl 36:387–407
Hiriart-Urruty J-B (1979a) New concepts in nondifferentiable programming. Analyse non convexe. Bull Soc Math France 60:57–85
Hiriart-Urruty J-B (1979b) Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math Oper Res 4:79–97
Huang YW (2002) Generalized constraint qualifications and optimality conditions for set-valued optimization problems. J Math Anal Appl 265(2):309–321
Jahn J, Khan AA (2002) Generalized contingent epiderivatives in set-valued optimization: optimality conditions. Numer Funct Anal Optim 23(7–8):807–831
Jahn J, Khan AA (2003) Some calculus rules for contingent epiderivatives. Optimization 52(2):113–125
Jahn J, Rauh R (1997) Contingent epiderivatives and set-valued optimization. Math Methods Oper Res 46:193–211
Jiménez B (2002) Strict efficiency in vector optimization. J Math Anal Appl 265:264–284
Jiménez B (2003) Strict minimality conditions in nondifferentiable multiobjective programming. J Optim Theory Appl 116:99–116
Jiménez B, Novo V (2003) First and second order conditions for strict minimality in nonsmooth vector optimization. J Math Anal Appl 284:496–510
Khan AA, Raciti F (2003) A multiplier rule in set-valued optimisation. Bull Austral Math Soc 68(1):93–100
Lalitha CS, Dutta J, Govil MG (2003) Optimality criteria in set-valued optimization. J Aust Math Soc 75(2):221–231
Luc DT (1989) Theory of vector optimization. Springer, Berlin Heidelberg New York
Rubinov A (2000) Abstract convexity and global optimization. Kluwer, Dordrecht
Song W (1998) A generalization of Fenchel duality in set-valued vector optimization. Set-valued optimization. Math Methods Oper Res 48(2):259–272
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Crespi, G.P., Ginchev, I. & Rocca, M. First-order optimality conditions in set-valued optimization. Math Meth Oper Res 63, 87–106 (2006). https://doi.org/10.1007/s00186-005-0023-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-005-0023-7