Proper Efficiency In Vector Optimization Research Articles

On Proper Separation Theorems by Means of the Quasi-Relative Interior with Applications

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results.

Henig proper efficiency in vector optimization with variable ordering structure

In this paper we introduce a notion of Henig proper efficiency for constrained vector optimization problems in the setting of variable ordering structure. In order to get an appropriate concept, we have to explore firstly the case of fixed ordering structure and to observe that, in certain situations, the well-known Henig proper efficiency can be expressed in a simpler way. Then, we observe that the newly introduced notion can be reduced, by a Clarke-type penalization result, to the notion of unconstrained robust efficiency. We show that this penalization technique, coupled with sufficient conditions for weak openness, serves as a basis for developing necessary optimality conditions for our Henig proper efficiency in terms of generalized differentiation objects lying in both primal and dual spaces.

A Kind of Unified Proper Efficiency in Vector Optimization

Based on the ideas of the classical Benson proper efficiency, a new kind of unified proper efficiency namedS-Benson proper efficiency is introduced by using Assumption (B) proposed by Flores-Bazán and Hernández, which unifies some known exact and approximate proper efficiency including(C,ε)-proper efficiency andE-Benson proper efficiency in vector optimization. Furthermore, a characterization ofS-Benson proper efficiency is established via a kind of nonlinear scalarization functions introduced by Göpfert et al.

E-Benson proper efficiency in vector optimization

Starting from the innovative ideas of Chicco et al. (Vector optimization problems via improvement sets. J. Optim. Theory Appl. 2011;150:516–529), in this paper, the concepts of improvement set and -efficiency are introduced in a real locally convex Hausdorff topological vector space. Furthermore, some properties of the improvement sets are given and a kind of proper efficiency, named as -Benson proper efficiency, which unifies some proper efficiency and approximate proper efficiency, is proposed via the improvement sets in vector optimization. Moreover, the concept of -subconvexlikeness of set-valued maps is introduced via the improvement sets and an alternative theorem is proved. In the end, some scalarization theorems and Lagrange multiplier theorems of -Benson proper efficiency are established for a vector optimization problem with set-valued maps.

Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike Set-Valued Maps

A scalarization theorem and two Lagrange multiplier theorems are established for tightly proper efficiency in vector optimization involving nearly cone-subconvexlike set-valued maps. A dual is proposed, and some duality results are obtained in terms of tightly properly efficient solutions. A new type of saddle point, which is called tightly proper saddle point of an appropriate set-valued Lagrange map, is introduced and is used to characterize tightly proper efficiency.

Optimality for Henig Proper Efficiency in Vector Optimization Involving Dini Set-Valued Directional Derivatives

This note studies the optimality conditions of vector optimization problems involving generalized convexity in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary and sufficient optimality conditions are established for Henig proper and strong minimal solutions respectively in generalized preinvex vector optimization problems.

Duality Results for Generalized Vector Variational Inequalities with Set-Valued Maps

In this paper, we introduce new dual problems of generalized vector variational inequality problems with set-valued maps and we discuss a link between the solution sets of the primal and dual problems. The notion of solutions in each of these problems is introduced via the concepts of efficiency, weak efficiency or Benson proper efficiency in vector optimization. We provide also examples showing that some earlier duality results for vector variational inequality may not be true.

Strong Duality for Proper Efficiency in Vector Optimization

In this paper, we give counterexamples showing that the strong duality results obtained in Refs. 1–5 for several dual problems of multiobjective mathematical programs are false. We provide also the conditions under which correct results can be established.

Proper Efficiency in Vector Optimization on Real Linear Spaces

In this paper, we use an algebraic type of closure, which is called vector closure, and through it we introduce some adaptations to the proper efficiency in the sense of Hurwicz, Benson, and Borwein in real linear spaces without any particular topology. Scalarization, multiplier rules, and saddle-point theorems are obtained in order to characterize the proper efficiency in vector optimization with and without constraints. The usual convexlikeness concepts used in such theorems are weakened through the vector closure.

Proper Efficiency in Vector Optimization on Real Linear Spaces

Proper Efficiency in Vector Optimization on Real Linear Spaces

On the notion of proper efficiency in vector optimization

In this paper, we consider the main definitions of proper efficiency for a vector optimization problem in topological linear spaces. The implications among these definitions generalize the inclusion structure holding in Euclidean spaces with componentwise ordering.

Density Results for Proper Efficiencies

Several density results are established for different notions of proper efficiency in vector optimization without the requirement of ordering cones being boundedly based. For a compact set, the set of Henig proper efficient points is shown to be dense in the set of all efficient points. Density theorems for Borwein's proper efficiency and for positive scalarizable efficiency follow immediately with appropriate assumptions.

Approximating families of cones and proper efficiency in vector optimization

In the paper we prove that a closed cone C in a normed space Y admits an approximating family of cones if and only if a C-strictly positive functional exists. In the sequel, we use this result to show that in this case, every cone-efficient point of a weakly compact subset can be approximated by properly efficient points