Higher order weak epiderivatives and applications to duality and optimality conditions

Abstract

Higher order weak adjacent (contingent) epiderivative Higher order duality Higher order optimality conditions Henig efficiency Set-valued optimization a b s t r a c t In this paper, the notions of higher order weak contingent epiderivative and higher order weak adjacent epiderivative for a set-valued map are defined. By virtue of higher order weak adjacent (contingent) epiderivatives and Henig efficiency, we introduce a higher order Mond-Weir type dual problem and a higher order Wolfe type dual problem for a constrained set-valued optimization problem (SOP) and discuss the corresponding weak duality, strong duality and converse duality properties. We also establish higher order Kuhn-Tucker type necessary and sufficient optimality conditions for (SOP).

Higher-order optimality conditions for set-valued optimization with ordering cones having empty interior using variational sets

“…Thus, (x 0 , y 0 ) is not a proper Henig solution of (P), but other results do not work since int D = ∅; e.g., Theorem 3.2 in [4], Lemma 4.1, Theorems 4.2, 6.1 in [11], Theorems 3.4, 3.5 in [19], Theorems 4.1, 4.2, 5.1, 5.2 in [20], Theorems 4.1, 5.2 in [30]. …”

Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization

“…Therefore, epiderivatives based on epigraphs, in a similar manner as the contingent derivative is based on graphs, have certain advantages over other kinds of derivatives, see [3] for epiderivatives of extended-real valued functions, and [4,8] for that of set-valued maps. For higher-order epiderivatives and applications to set-valued optimization, the reader is referred to [6,12,16,17]. All these higher-order epiderivatives are defined upon informations of lower-order approximating directions.…”

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

“…The theory of duality and optimality conditions for optimization problems has received considerable attention (see [1][2][3][4][5][6][7][8][9][10]). The derivative (epiderivative) plays an important role in studying duality and optimality conditions for set-valued optimization problems.…”

“…Later, the second-order epiderivatives [13], higher-order generalized contingent (adjacent) epiderivatives [14] and generalized higher-order contingent (adjacent) derivatives [15] for set-valued maps are used to study the second (or high) order necessary or/and sufficient optimality conditions for set-valued optimization problems. Chen et al [2] utilized the weak efficiency to introduce higher-order weak adjacent (contingent) epiderivative for a set-valued map, they then investigate higher-order Mond-Weir (Wolfe) type duality and higher-order Kuhn-Tucker type optimality conditions for constrained set-valued optimization problems. Li et al [3] used the higher-order contingent derivatives to discuss the weak duality, strong duality and converse duality of a higher-order Mond-Weir type dual for a set-valued optimization problem.…”