On The Study Of Many Problems About Supper Efficiency Solution And Strong Efficiency Solution For Set-Valued Optimization Problems

The theory of set-valued optimization is one of focal point problems in the vector optimization field and finds wide applications in fixed point theory, variation problems, differential inclusions, control theory and mathematical economics, and the optimality conditions for set-valued optimization problems in the sense of various solutions are its important components and are the important base of developing modern algorithms. On the other hand, the concept of convexity plays important roles in the optimization theory, hence each of generalizations of convexity receives researcher's attentions. In this paper, the set-valued optimization problem with constraints is considered in the sense of super efficiency in real normed linear space. Under the assumption of the ic-cone-convexlikeness, by applying separation theorem for convex sets, Kuhn-Tucker and Lagrange necessary conditions are derived respectively, and the new saddle-points optimality condition is obtained . The concept of the generalized gradient in sense of strong efficiency is introduced by epiderivative for a set-valued map in ordered Banach spaces . Under the condition of lower C semicontinuous, its existence is proved by the separation theorem for convex sets; Thus the optimality condition of strong-efficient solution of set-valued optimization problems is established in the sense of generalized gradient.