Abstract
This note deals with approximate solutions in vector optimization involving a generalized cone-invex set-valued mapping. First, a new class of generalized cone-invex set-valued maps, called cone-subinvex set-valued maps, is introduced. Then the sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity. Finally, it also reveals the closed relationships between a weakly approximate minimizer of a cone-subinvex set-valued optimization problem and a weakly approximate solution of a kind of vector variational inequality.
Highlights
There has been an increasing interest in the extension of vector optimization to set-valued optimization
6 Conclusions and remarks In this paper, we focus on the approximate solutions in set-valued optimization
A sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity
Summary
There has been an increasing interest in the extension of vector optimization to set-valued optimization. The derivative of set-valued maps is most important for the formulation of optimality conditions. Aubin and Frankowsa [ ] introduced the notion of a contingent derivative of a set-valued map as an extension of the concept of Fréchet differentiability. Various approaches have been followed in defining the concept of derivative for set-valued maps. Among these notions, a meaningful and useful concept is the contingent epiderivative, which was given by Jahn and Rauh [ ]. Much attention has been paid to characterizing optimality conditions for set-valued optimization and related problems by utilizing contingent epiderivatives; for example, see [ – ].
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