Weak Clarke epiderivative in set-valued optimization

Abstract

A new notion of weak Clarke epiderivative for a set-valued map is introduced using the concept of Clarke tangent cone. The existence, characterization and properties of weak Clarke epiderivative are then studied. Finally optimality criteria are established for a constrained set-valued optimization problem in terms of weak Clarke epiderivative.

Higher order weak epiderivatives and applications to duality and optimality conditions

“…Recently, Jahn et al [10] introduced second-order contingent epiderivative and generalized contingent epiderivative for set-valued maps and obtained some second-order optimality conditions based on these concepts. Very recently, Lalitha and Arora [11] introduced a notion of weak Clarke epiderivative for a set-valued map by using the concept of Clarke tangent cone and established optimality conditions for a constrained set-valued optimization problem in terms of weak Clarke epiderivative. On the other hand, various kinds of differentiable type dual problems for set-valued optimization problems, such as Mond-Weir type and Wolfe type dual problems have been investigated, for example, see [12][13][14] and so on.…”

“…Motivated by the work reported in [15][16][17]11], we introduce the concepts of higher order weak contingent epiderivative and higher order weak adjacent epiderivative for set-valued maps. Based on higher order weak adjacent (contingent) epiderivatives and Henig efficiency, we investigate higher order Mond-Weir type duality, higher order Wolfe type duality and higher order Kuhn-Tucker type optimality conditions to a constrained set-valued optimization problem (SOP).…”

A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems

“…Definition 2. [10,21,22] (i) The cone C is called Daniell if any decreasing sequence in Y that has a lower bound converges to its infimum. (ii) A subset M of Y is said to be minorized if there is a y ∈ Y such that…”

First and second-order optimality conditions without differentiability in multivalued vector optimization

“…To apply [29] with the weak Clarke epiderivative, we need the condition that G(0) = {y ∈ R|(0, y) ∈ T (epiF, (x 0 , y 0 ))} is pointed, which is seen violated in this case. Hence, Theorem 5.2 of [29] is not applicable either.…”

“…For example, in [27,30] C-star-shapedness and pseudoconvexity are used along with variational sets. In [29], using the weak Clarke epiderivative, cone semilocal convexlikeness assumptions are imposed. Here we make use of C-arcwise-connectedness.…”