Directional Pareto Efficiency: Concepts and Optimality Conditions

Abstract

We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations concerning the objective mapping and the constraints are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation and the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pareto minima are considered in our specific situation.1 means of an inverse image of a cone through another set-valued map. For the study of this general case, we introduce an adapted tangent cone, along with several directional regularity properties of the involved maps, and this approach allows us to derive necessary optimality conditions that, in turn, generalize the prototype of Fermat Theorem at an endpoint of an interval. Furthermore, we present as well optimality conditions in terms of tangent limiting cones and coderivatives. Both on primal and dual spaces we have under consideration several situations concerning the objective and constraint mappings with their specific techniques of study, among which we mention generalized constraint qualification conditions, Gerstewitz scalarization, openness vs. minimality paradigm, Clarke penalization, extremal principle. Some results are dedicated to the sufficient optimality conditions under convexity assumptions. Finally, we consider as well the situation of minimality for sets and a brief discussion of this concept reveals the similarities and the differences with respect to the known situation of Pareto efficiency.The paper is organized as follows. First of all, we introduce the notation we use and then we present the concepts of directional minimality we study in this work. The definitions of these notions along with some comparisons and examples are the subjects of the second section. The main section of the paper is the third one, and it deals with optimality conditions for the above introduced concepts, being, in turn, divided into two subsections. Firstly, we derive optimality conditions using tangent cones and to this aim we adapt a classical concept of the Bouligand tangent cone and Bouligand derivative of a set-valued map. Using some directional metric regularities, we get several assertions concerning these objects and this allows us to present necessary optimality conditions for a wide range of situations going from problems governed by set-valued mappings having generalized inequalities constraints to fully smooth constrained problems. Secondly, we deal with optimality conditions using normal limiting cones and, again, we consider several types of problems. In this process of getting necessary optimality conditions we adapt several techniques from classical vector optimization. Moreover, some generalized convex cases are considered in order to obtain sufficient opti...

Cone enlargements and applications to vector optimization

“…Consider a nonempty set L ⊂ S X and take K as a pointed closed cone in X. The following notion of directional efficiency is introduced and briefly studied in [14]. Definition 5.1.…”

Exact Penalization and Optimality Conditions for Approximate Directional Minima

“…Remark 2.5. Note that the "one point" regularity properties defined above are particular cases of the notions of directional calmness and directional metric subregularity, respectively, defined and studied in [3]. Namely, the neighborhood of the element x is an open ball centered in x and of radius ε.…”

“…Besides considering the directional setting, the minimum concept studied in this paper contains two constants that, in some sense, measure the approximate character of efficiency. By this paper, we had aimed to prove that some results from the papers [2,3,5] still hold for approximate directional Pareto minima. The novelty here is the fact that it's required a careful tracking (see Lemma 3.2) of the aforementioned constants that are involved in the definition of the minimality concepts.…”

“…The novelty we propose here is related to the extension of the concept of directional Pareto minimality, introduced in [3], to the case of approximate directional solutions. Therefore the penalization results and the necessary optimality conditions we derive here, even if it follows some known methods of investigation (see, for instance, [2]), it brings some new technical issues that we have to overcome.…”