## K-epiderivatives for set-valued functions and optimization

Giancarlo Bigi and Marco Castellani

### Summary

In the last years set-valued optimization problems have been considered by many researchers. Besides intrinsic interest, this type of programs arise quite naturally in the context of duality for vector optimization. Moreover, when the data of a single-valued optimization problem are not exactly known, it is reasonable to replace the values of the involved functions with sets representing their fuzzy outcomes. In order to study set-valued problems, some notion of derivative for set-valued functions is required. The first concept of derivative had been introduced by Aubin relying on the Bouligand contingent cone: the contingent derivative of a set-valued function H at a given point is the map whose graph equals the Bouligand contingent cone of the graph of H at the considered point. Even if it was originally employed within the context of differential inclusions, since then many applications also to the study of optimality conditions for vector and set-valued optimization problems have been provided. Recently, Jahn and Rauh introduced the contingent epiderivative of a set-valued function, extending the concept of ``upper contingent derivative'' of real-valued ones. The main difference between the definitions of contingent derivative and epiderivative is that the graph is replaced by the epigraph and the epiderivative is single-valued. Though single-valuedness seems useful to develop calculus rules, we believe that replacing the graph with the epigraph is even more important: approximating just the graph with the contingent cone may not preserve enough information about the function. In fact, as pointed out by Jahn and Rauh, necessary and sufficient optimality conditions based on the contingent derivative do not coincide under convexity assumptions. Moreover, sometimes the domain of the contingent derivative is reduced to only one point. We stress that both these derivatives rely on the well-known concept of Bouligand contingent cone to a set. Actually, several kinds of derivatives have been developed exploiting different types of concrete tangent cones. Moreover, relying on standard properties, general definitions of tangent cone have been proposed and employed to define generalized derivatives of real-valued functions. Following these ideas, we propose a definition of generalized epiderivative for set-valued functions and we employ it to achieve a general scheme for necessary optimality conditions of set-valued optimization problems. Finally, we show how already known conditions can be recovered within this scheme quite easily.

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