Optimality of Globally Proper Effcient Solutions in Set-valued Optimization Involving Directional Derivatives

This note deals with the optimality conditions of set-valued unconstraint optimization problems in topological vector spaces. Based upon the concepts of directional derivatives and generalized cone-preinvexity for the set-valued mappings, the necessary and sufficient optimality conditions are established for globally proper minimizer respectively in set-valued optimization problems.

[1]  Aparna Mehra,et al.  Lagrangian Duality for Preinvex Set-Valued Functions☆ , 1997 .

[2]  B. Mond,et al.  Pre-invex functions in multiple objective optimization , 1988 .

[3]  Johannes Jahn,et al.  Optimality conditions for set-valued optimization problems , 1998, Math. Methods Oper. Res..

[4]  Cone-directed contingent derivatives and generalized preinvex set-valued optimization * * Supported , 2007 .

[5]  Guo Lin Yu,et al.  Globally proper saddle point in ic-cone-convexlike set-valued optimization problems , 2009 .

[6]  H. W. Corley,et al.  Optimality conditions for maximizations of set-valued functions , 1988 .

[7]  Sanyang Liu,et al.  Some vector optimization problems in Banach spaces with generalized convexity , 2007, Comput. Math. Appl..

[8]  H. P. Benson,et al.  An improved definition of proper efficiency for vector maximization with respect to cones , 1979 .

[9]  Shouyang Wang,et al.  Optimality conditions for proper efficient solutions of vector set-valued optimization ✩ , 2003 .

[10]  X. Q. Yang.  Directional derivatives for set-valued mappings and applications , 1998, Math. Methods Oper. Res..

[11]  Johannes Jahn,et al.  On Subdifferentials of Set-Valued Maps , 1999 .

[12]  X. Gong.  Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior , 2005 .

[13]  M. I. Henig.  Proper efficiency with respect to cones , 1982 .

[14]  Johannes Jahn,et al.  Contingent epiderivatives and set-valued optimization , 1997, Math. Methods Oper. Res..