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

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.

Differential Calculus of Set-Valued Maps. An Update

IIASA has played a crucial role in the development of the "graphical approach" to the differential calculus of set-valued maps, around J.-P. Aubin, H. Frankowska, R.T. Rockafellar and allowed to make contacts with Soviet and eastern European mathematicians (C. Olech, B. Pschenichnyiy, E. Polovinkin, V. Tihomirov, etc.) who were following analogous approaches. Since 1981, they and their collaborators developed this calculus and applied it to a variety of problems, in mathematical programming (Kuhn-Tucker rules, sensitivity of solutions and Lagrange multipliers), in nonsmooth analysis (Inverse Functions Theorems, local uniqueness), in control theory (controllability of systems with feedbacks, Pontryagin's Maximum Principle, Hamilton-Jacobi-Bellman equations, observability and other issues), in viability theory (regulation of systems, heavy trajectories), etc. The first version of this survey appeared at IIASA in 1982, and constituted the seventh chapter of the book "Applied Nonlinear Analysis" published in 1984 by I. Ekeland and the author. Since then, many other results have been motivated by the successful applications of this calculus, and, maybe unfortunately, other concepts (such as the concept of intermediate tangent cone and derivatives introduced and used by H. Frankowska). Infinite-dimensional problems such as control problems or the more classical problems of calculus of variations require the use of adequate adaptations of the same main idea, as well as more technical assumptions. The time and the place (IIASA) were ripe to update the exposition of this differential calculus. The Russian translation of "Applied Nonlinear Analysis" triggered this revised version