Solutions In Vector Optimization Research Articles

Unboundedness of the images of set-valued mappings having closed graphs: application to vector optimization

AbstractIn this paper, we propose criteria for unboundedness of the images of set-valued mappings having closed graphs in Euclidean spaces. We focus on mappings whose domains are non-closed or whose values are connected. These criteria allow us to see structural properties of solutions in vector optimization, where solution sets can be considered as the images of solution mappings associated to specific scalarization methods. In particular, we prove that if the domain of a certain solution mapping is non-closed, then the weak Pareto solution set is unbounded. Furthermore, for a quasi-convex problem, we demonstrate two criteria to ensure that if the weak Pareto solution set is disconnected then each connected component is unbounded.

Set-valued optimization in variable preference structures with new variants of generalized convexity

ABSTRACT A set-valued optimization problem with variable preferences is considered. Relations between local and global solutions, optimality conditions, and Wolfe and Mond-Weir duality properties are studied. Both minimal and nondominated solutions are discussed with general variable preferences. The results are proved for three main types of solutions in vector optimization: weak, Pareto, and strong solutions. New variants of generalized derivatives and convexity are proposed and used in all the results.

Necessary Conditions for Nondominated Solutions in Vector Optimization

In this paper, we study characterizations and necessary conditions for nondominated points of sets and nondominated solutions of vector-valued functions in vector optimization with variable domination structure. We study not only the case, where the intersection of all the involved domination sets has a nonzero element, but also the case, where it might be the singleton. While the first case has been studied earlier, the second case has not, to the best of our knowledge, done yet. Our results extend and improve the existing results in vector optimization with a fixed ordering cone and with a variable ordering structure.

Existence of efficient and properly efficient solutions to problems of constrained vector optimization

The paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint set. The main attention is paid to the two major notions of optimality in vector problems: Pareto efficiency and proper efficiency in the sense of Geoffrion. Employing adequate tools of variational analysis and generalized differentiation, we first establish relationships between the notions of properness, M-tameness, and the Palais–Smale conditions formulated for the restriction of the vector cost mapping on the constraint set. These results are instrumental to derive verifiable necessary and sufficient conditions for the existence of Pareto efficient solutions in vector optimization. Furthermore, the developed approach allows us to obtain new sufficient conditions for the existence of Geoffrion-properly efficient solutions to such constrained vector problems.

Characterization of Norm-Based Robust Solutions in Vector Optimization

In this paper, we study the norm-based robust (efficient) solutions of a vector optimization problem. We define two kinds of non-ascent directions in terms of Clarke’s generalized gradient and characterize norm-based robustness by means of the newly defined directions. This is done under a basic constraint qualification. We extend the provided characterization to vector optimization problems with conic constraints and semi-infinite ones. Moreover, we derive a necessary condition for norm-based robustness utilizing a non-smooth gap function.

A hybrid approach for finding efficient solutions in vector optimization with SOS-convex polynomials

This paper aims to find efficient solutions to a vector optimization problem (VOP) with SOS-convex polynomials. A hybrid scalarization method is used to transform (VOP) into a scalar one. A strong duality result, between the proposed scalar problem and its relaxation dual problem, is established, under certain regularity condition. Then, an optimal solution to the proposed scalar problem can be found by solving its associated semidefinite programming problem. Consequently, we observe that finding efficient solutions to (VOP) can be achieved.

Higher-order Optimality Conditions and Duality for Approximate Solutions in Non-convex Set-valued Optimization

This paper deals with higher-order optimality conditions and duality theory for approximate solutions in vector optimization involving non-convex set-valued maps. Firstly, under the assumption of near cone-subconvexlikeness for set-valued maps, the higher necessary and sufficient optimality conditions in terms of Studniarski derivatives are derived for local weak approximate minimizers of a set-valued optimization problem. Then, applications to Mond-Weir type dual problem are presented.

On some generalized equations with metrically C-increasing mappings: solvability and error bounds with applications to optimization

ABSTRACTGeneralized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat issues such as constraint systems, optimality and equilibrium conditions, variational inequalities, differential inclusions. This paper contains a study on solvability and error bounds for generalized equations of the form , where F is a given set-valued mapping and C is a closed, convex cone. A property called metric C-increase, matching the metric behaviour of F with the partial order associated with C, is singled out, which ensures solution existence and error bound estimates in terms of problem data. Applications to the exact penalization of optimization problems with constraint systems, defined by the above class of generalized equations, and to the existence of ideal efficient solutions in vector optimization are proposed.

Limit Behavior of Approximate Proper Solutions in Vector Optimization

In the framework of a vector optimization problem, we provide conditions for approximate proper solutions to tend to exact weak/efficient/proper solutions when the error tends to zero. This limit behavior depends on an approximation set that is used to define the approximate proper efficient solutions. We also study the special case when the final space of the vector optimization problem is normed, and more particularly, when it is finite dimensional. In these specific frameworks, we provide several explicit constructions of dilating ordering cones and approximation sets that lead to the desired limit behavior. In proving our results, new relationships between different concepts of approximate proper efficiency are stated.

Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure

In this paper, we introduce Hartley properly and super nondominated solutions in vector optimization with a variable ordering structure. We prove the connections between Benson properly nondominated, Hartley properly nondominated, and super nondominated solutions under appropriate assumptions. Moreover, we establish some necessary and sufficient conditions for newly-defined solutions invoking an augmented dual cone approach, the linear scalarization, and variational analysis tools. In addition to the theoretical results, various clarifying examples are given.

Existence Theorems in Vector Optimization with Generalized Order

In the present paper, we establish some results for the existence of optimal solutions in vector optimization in infinite-dimensional spaces, where the optimality notion is understood in the sense of generalized order (may not be convex and/or conical). This notion is induced by the concept of set extremality and covers many of the conventional notions of optimality in vector optimization. Some sufficient optimality conditions for optimal solutions of a class of vector optimization problems, which satisfies the free disposal hypothesis, are also examined.

Approximate solutions for nonconvex set-valued optimization and vector variational inequality

This note deals with approximate solutions in vector optimization involving a generalized cone-invex set-valued mapping. First, a new class of generalized cone-invex set-valued maps, called cone-subinvex set-valued maps, is introduced. Then the sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity. Finally, it also reveals the closed relationships between a weakly approximate minimizer of a cone-subinvex set-valued optimization problem and a weakly approximate solution of a kind of vector variational inequality.

Robust counterparts and robust efficient solutions in vector optimization under uncertainty

Two robust counterparts and associated concepts of robust efficient solution are established for a vector optimization problem under uncertainty. First, we propose a robust counterpart in the classical sense by following the line for scalar optimization problems under uncertainty. Then, from a relaxed model we derive another robust counterpart, which is a bilevel optimization problem involving a set-valued optimization problem at the upper level and a vector optimization problem at the lower level.

Newton-like methods for efficient solutions in vector optimization

In this work we study the Newton-like methods for finding efficient solutions of the vector optimization problem for a map from a finite dimensional Hilbert space X to a Banach space Y, with respect to the partial order induced by a closed, convex and pointed cone C with a nonempty interior. We present both exact and inexact versions, in which the subproblems are solved approximately, within a tolerance. Furthermore, we prove that under reasonable hypotheses, the sequence generated by our method converges to an efficient solution of this problem.

Lagrange multiplier rules for approximate solutions in vector optimization

In Asplund space, Lagrange multiplier rules for approximate solutions of nonsmooth vector optimization problems are studied. The relationships between the vector and the scalar optimization problems are established. And the optimality conditions of approximatesolutions for vector optimization are obtained. Moreover, thevector variational inequalities are considered by applying thepartial results given in this paper.

Generalized Proximal Method for Efficient Solutions in Vector Optimization

This article is devoted to developing the generalized proximal algorithm of finding efficient solutions to the vector optimization problem for a mapping from a uniformly convex and uniformly smooth Banach space to a real Banach space with respect to the partial order induced by a pointed closed convex cone. In contrast to most published literature on this subject, our algorithm does not depend on the nonemptiness of ordering cone of the space under consideration and deals with finding efficient solutions of the vector optimization problem in question. We prove that under some suitable conditions the sequence generated by our method weakly converges to an efficient solution of this problem.

Optimality conditions and duality on approximate solutions in vector optimization with arcwise connectivity

In this paper a new class of generalized vector-valued arcwise connected functions, termed sub-arcwise connected functions, is introduced. The properties of sub-arcwise connected functions are derived. The approximate quasi efficient solutions of vector optimization problems are studied, and the necessary and sufficient optimality conditions are obtained under the assumption of arcwise connectivity. An approximate Mond-Weir type dual problem is formulated and the duality theorems are established.

Viscosity-type Approximation Method for Efficient Solutions in Vector Optimization

The paper is devoted to developing the viscosity-type approximation algorithm of finding efficient solutions to the vector optimization problem for a mapping between finite dimensional Hilbert spaces with respect to the partial order induced by a pointed closed convex cone. We prove that under some suitable conditions either the sequence generated by our method converges to an efficient solution or its cluster points belong to the set of all efficient solutions of this problem.

Tikhonov-type regularization method for efficient solutions in vector optimization

The paper is devoted to developing the Tikhonov-type regularization algorithm of finding efficient solutions to the vector optimization problem for a mapping between finite dimensional Hilbert spaces with respect to the partial order induced by a pointed closed convex cone. We prove that under some suitable conditions either the sequence generated by our method converges to an efficient solution or all of its cluster points belong to the set of all efficient solutions of this problem.

Lagrange Multipliers for ε-Pareto Solutions in Vector Optimization with Nonsolid Cones in Banach Spaces

This paper presents some results concerning the existence of the Lagrange multipliers for vector optimization problems in the case where the ordering cone in the codomain has an empty interior. The main tool for deriving our assertions is a scalarization by means of a functional introduced by Hiriart-Urruty (Math. Oper. Res. 4:79–97, 1979) (the so-called oriented distance function). Moreover, we explain some applications of our results to a vector equilibrium problem, to a vector control-approximation problem and to an unconstrainted vector fractional programming problem.