4 research outputs found
Cone-Henig Subdifferentials of Set-Valued Maps in Locally Convex Spaces.
In locally convex spaces, the concepts of cone-Henig subgradient and cone-Henig subdifferential for the set-valued mapping are introduced through the linear functionals. The theorems of existence for Henig efficient point and cone-Henig subdifferential are proposed, and the sufficient and necessary condition for a linear functional being a cone-Henig subgradient is established
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.
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
Quelques thĂšmes en l'analyse variationnelle et optimisation
In this thesis, we first study the theory of [gamma]-limits. Besides some basic properties of [gamma]-limits,expressions of sequential [gamma]-limits generalizing classical results of Greco are presented. These limits also give us a clue to a unified classification of derivatives and tangent cones. Next, we develop an approach to generalized differentiation theory. This allows us to deal with several generalized derivatives of set-valued maps defined directly in primal spaces, such as variational sets, radial sets, radial derivatives, Studniarski derivatives. Finally, we study calculus rules of these derivatives and applications related to optimality conditions and sensitivity analysis.Dans cette thĂšse, jâĂ©tudie dâabord la thĂ©orie des [gamma]-limites. En dehors de quelques propriĂ©tĂ©s fondamentales des [gamma]-limites, les expressions de [gamma]-limites sĂ©quentielles gĂ©nĂ©ralisant des rĂ©sultats de Greco sont prĂ©sentĂ©es. En outre, ces limites nous donnent aussi une idĂ©e dâune classification unifiĂ©e de la tangence et la diffĂ©rentiation gĂ©nĂ©ralisĂ©e. Ensuite, je dĂ©veloppe une approche des thĂ©ories de la diffĂ©rentiation gĂ©nĂ©ralisĂ©e. Cela permet de traiter plusieurs dĂ©rivĂ©es gĂ©nĂ©ralisĂ©es des multi-applications dĂ©finies directement dans lâespace primal, tels que des ensembles variationnels,des ensembles radiaux, des dĂ©rivĂ©es radiales, des dĂ©rivĂ©es de Studniarski. Finalement, jâĂ©tudie les rĂšgles de calcul de ces dĂ©rivĂ©es et les applications liĂ©es aux conditions dâoptimalitĂ© et Ă lâanalyse de sensibilitĂ©