Set-valued Optimization Research Articles

A global optimality result using coupled proximal contractions with stability analysis

In this work a multi-valued coupled proximal contraction mapping is used for determining the distance between two sets through determination of two pairs of points simultaneously. It is a global optimization problem by its very nature which is converted here into a problem of determining an optimal approximation of the solution of a multi-valued coupled fixed point problem where the actual solution does not exist in the most general cases. The result is well demonstrated with an example. In another two sections we derive a data dependence result and a weak stability result respectively for the problem considered here. For our purpose we define a notion of weak stability of the solution sets. The study is in the framework of metric spaces and broadly falls within the domain of set-valued optimization.

Approximate weak efficiency of the set-valued optimization problem with variable ordering structures

Approximate weak efficiency of the set-valued optimization problem with variable ordering structures

Existence of weak efficient solutions of set-valued optimization problems

Existence of weak efficient solutions of set-valued optimization problems

Second-order optimality conditions for set-valued optimization problems under the set criterion

Second-order optimality conditions for set-valued optimization problems under the set criterion

Correction to: Existence of Pseudo-Relative Sharp Minimizers in Set-Valued Optimization

Correction to: Existence of Pseudo-Relative Sharp Minimizers in Set-Valued Optimization

Higher-order [formula omitted]-cone arcwisely connectedness in optimization problems associated with difference of set-valued maps

In this paper, an optimization problem (DP) is studied where the objective maps and the constraints are the difference of set-valued maps (abbreviated as SVMs). The higher-order σ-cone arcwise connectedness is described as an entirely new type of generalized higher-order arcwise connectedness for set-valued optimization problems. Under the higher-order contingent epiderivative and higher-order σ-cone arcwise connectedness suppositions, the higher-order sufficient Karush–Kuhn–Tucker (KKT) optimality requirements are demonstrated for the problem (DP). The higher-order Wolfe (WD) form of duality is investigated and the corresponding higher-order weak, strong, and converse theorems of duality are established between the primary (DP) and the corresponding dual problem by employing the higher-order σ-cone arcwise connectedness supposition. In order to demonstrate that higher-order σ-cone arcwise connectedness is more generalized than higher-order cone arcwise connectedness, an example is also constructed. As a special case, the results coincide with the existing ones available in the literature.

Higher-order optimality conditions with separated derivatives and sensitivity analysis for set-valued optimization

In this paper, we establish optimality conditions and sensitivity analysis of set-valued optimization problems in terms of higher-order radial derivatives. First, we obtain the optimality conditions with separated derivatives for a set-valued optimization problem, here separated derivatives means the derivatives of objective and constraint functions are different. Then, some duality theorems for a mixed type of primal-dual set-valued optimization problem are gained. Finally, several results concerning higher-order sensitivity analysis are presented. The main results of this paper are illustrated by some concrete examples.

Generalized set-valued Ekeland variational principles via the null set concept with applications to some set-valued optimization problems

In this paper, we establish two new versions of generalized set-valued Ekeland variational principles related to the so-called null set in a hyperspace ordered by a convex cone. Moreover, we propose some interval versions of Ekeland variational principles. As applications, the obtained theoretical results are applied to prove a set-valued fixed point theorem, to solve a set-valued optimization problem, and to explore a set-valued noncooperative game.

Higher-order mixed duality for set-valued optimization problems

Higher-order mixed duality for set-valued optimization problems

Relations Among Minimal Elements of a Family of Sets with Respect to Various Set Order Relations

In this article, some set order relations given for set-optimization criterion which is one of the solution concept of set-valued optimization problems are considered. Minimal elements of a family of sets with respect to these set order relations are compared in detail. For this comparison relations between set order relations mentioned in this article are used. Also, in cases where a family of minimal sets is not a subset of the other one, examples are given.

Optimality Conditions for Set-Valued Optimization Problems

In this paper, we first prove that the generalized subconvexlikeness introduced by Yang, Yang and Chen [1] and the presubconvelikeness introduced by Zeng [2] are equivalent. We discuss set-valued nonconvex optimization problems and obtain some optimality conditions.

Optimality Conditions for Directional Efficiency in Set-Valued Optimization

Optimality Conditions for Directional Efficiency in Set-Valued Optimization

Subdifferential calculus for ordered multifunctions with applications to set-valued optimization

Subdifferential calculus for ordered multifunctions with applications to set-valued optimization

Stable strong duality for cone-constrained set-valued optimization problems: A perturbation approach

In this paper we consider a general set-valued otimimization problem of the model Winf F(x), s.t. x in X where $F\colon X\rightrightarrows Y\cup\{+\infty_{Y}\}$ is a proper mapping. The problem is then embed into a parametric problem and can be express as (P) $\mathop{\winf}\bigcup\limits_{x\in X}\Phi(x,0_Z)$ where $\Phi\colon X\times Z\rightrightarrows Y^\bullet$ is a proper set-valued {\it perturbation mapping} such that $ \Phi(x,0_Z) = F(x) $. A representation of the epigraph of the conjugate mapping $\Phi^\ast$ is established (Theorem 1) and it is used as the basic tool for establishing a general stable strong duality for the problem (P) (Theorem 3). As applications, the mentioned general strong duality result is then applied to a cone-constrained set-valued optimization problem (CSP) to derive three dual problems for (CSP): The Lagrange dual problem and two forms of Fenchel-Lagrange dual problems for (CSP). Consequently, three stable strong duality results for the three primal-dual pairs of problems are derived (Theorems 4 and 5), among them, one is entirely new while the others extend some known ones in the literature.

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.

Optimality conditions in set-valued optimization using approximations as generalized derivatives

Abstract The set criterion is an appropriate defining approach regarding the solutions for the set-valued optimization problems. By using approximations as generalized derivatives of set-valued mappings, we establish necessary optimality conditions for a constrained set-valued optimization problem in the sense of set optimization in terms of asymptotical pointwise compact approximations. Sufficient optimality conditions are then obtained through first-order strong approximations of data set-valued mappings.

A New Topological Framework and Its Application to Well-Posedness in Set-Valued Optimization

In this paper, we introduce a topology on the power set of a partially ordered normed space Z from which we derive a topological convergence on along with new concepts of continuity and semicontinuity for set-valued mappings. Our goal is to propose an appropriate framework to address set optimization problems involving set relations based on a cone ordering. Taking advantage of this new setting, we establish several results regarding the well-posedness of set-valued optimization problems that are consistent with the state-of-the-art.

Optimality Conditions and Dualities for Robust Efficient Solutions of Uncertain Set-Valued Optimization with Set-Order Relations

In this paper, we introduce a second-order strong subdifferential of set-valued maps, and discuss some properties, such as convexity, sum rule and so on. By the new subdifferential and its properties, we establish a necessary and sufficient optimality condition of set-based robust efficient solutions for the uncertain set-valued optimization problem. We also introduce a Wolfe type dual problem of the uncertain set-valued optimization problem. Finally, we establish the robust weak duality theorem and the robust strong duality theorem between the uncertain set-valued optimization problem and its robust dual problem. Several main results extend to the corresponding ones in the literature.

Higher-order generalized tangent epiderivatives and applications to set-valued optimization

Higher-order generalized tangent epiderivatives and applications to set-valued optimization

Global well-posedness of set-valued optimization with application to uncertain problems

Global well-posedness of set-valued optimization with application to uncertain problems