The contingent epiderivative and the calculus of variations on time scales

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists in applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler-Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results.Comment: Submitted 06/March/2010; revised 12/May/2010; accepted 03/July/2010; for publication in "Optimization---A Journal of Mathematical Programming and Operations Research

Higher order weak epiderivatives and applications to duality andoptimality conditions

AbstractIn 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)

First order optimality conditions in set-valued optimization

A a set-valued optimization problem minC F(x), x 2 X0, is considered, where X0 X, X and Y are Banach spaces, F : X0 Y is a set-valued function and C Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x0, y0), y0 2 F(x0), and are called minimizers. In particular the notions of w-minimizer (weakly efficient points), p-minimizer (properly efficient points) and i-minimizer (isolated minimizers) are introduced and their characterization in terms of the so called oriented distance is given. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive first order conditions, that is conditions in terms of suitable first order derivatives of F, for a pair (x0, y0), where x0 2 X0, y0 2 F(x0), to be a solution of this problem. We define and apply for this purpose the directional Dini derivative. Necessary conditions and sufficient conditions a pair (x0, y0) to be a w-minimizer, and similarly to be a i-minimizer are obtained. The role of the i-minimizers, which seems to be a new concept in set-valued optimization, is underlined. For the case of w-minimizers some comparison with existing results is done. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.

Higher-Order Weakly Generalized Epiderivatives and Applications to Optimality Conditions

The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results

The existence of contingent epiderivatives for set-valued maps

AbstractThis short note deals with the issue of existence of contingent epiderivatives for set-valued maps defined from a real normed space to the real line. A theorem of Jahn-Rauh [1], given for the existence of contingent epiderivatives, is used to obtain more general existence results. The strength and the limitations of the main result are discussed by means of some examples

On constrained set-valued optimization

The set-valued optimization problem minC F(x), G(x)\(-K) 6= ; is considered, where F : Rn Rm and G : Rn Rp are set-valued functions, and C Rm and K Rp are closed convex cones. Two type of solutions, called w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers), are treated. In terms of the Dini set-valued directional derivative first-order necessary conditions for a point to be a w-minimizer, and first-order sufficient conditions for a point to be an i-minimizer are established, both in primal and dual form. Key words: Set-valued optimization, First-order optimality conditions, Dini derivatives.

Higher-Order Weakly Generalized Epiderivatives and Applications to Optimality Conditions

The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent adjacent epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results

First order optimality condition for constrained set-valued optimization

A constrained optimization problem with set-valued data is considered. Different kind of solutions are defined for such a problem. We recall weak minimizer, efficient minimizer and proper minimizer. The latter are defined in a way that embrace also the case when the ordering cone is not pointed. Moreover we present the new concept of isolated minimizer for set-valued optimization. These notions are investigated and appear when establishing first-order necessary and sufficient optimality conditions derived in terms of a Dini type derivative for set-valued maps. The case of convex (along rays) data is considered when studying sufficient optimality conditions for weak minimizers. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.

Higher-Order optimality conditions for set-valued optimization

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