The main aim of this paper is to obtain optimality conditions for a constrained set-valued optimization problem. The concept of Clarke epiderivative is introduced and is used to derive necessary optimality conditions. In order to establish sufficient optimality criteria we introduce a new class of set-valued maps which extends the class of convex set-valued maps and is different from the class of invex set-valued maps.2000 Mathematics subject classification: primary 49J53, 90C30.

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Lagrange multiplier rules characterizing for nonsmooth programming problems on a real Hilbert space are established in terms of the limiting subgradients.