Set-valued Optimization Problems Research Articles

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.

KÜME DEĞERLİ OPTİMİZASYON PROBLEMLERİNİN ZAYIF MİNİMAL ÇÖZÜMLERİ VE BAZI KÜME SIRALAMALARININ KARŞILAŞTIRILMASI ÜZERİNE

In this work, some set order relations are compared with each other. In addition, it is shown that every weak minimal solution of a set valued optimization problem with respect to vector optimization criterion, is also a weak minimal solution with respect to set optimization criterion considering some special set orders.

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.

Pointwise well-posedness of a set-valued optimization problem at a weak solution

Pointwise well-posedness of a set-valued optimization problem at a weak solution

Stability in Set-Valued Optimization Problems Using Asymptotic Analysis and Epi-Convergence

Stability in Set-Valued Optimization Problems Using Asymptotic Analysis and Epi-Convergence

Using ρ-cone arcwise connectedness on parametric set-valued optimization problems

Within the inquiry about work, we explore a parametric set-valued optimization problem, where the objective as well as constraint maps are set-valued. A generalization of cone arcwise associated set-valued maps is presented named ρ-cone arcwise connectedness of set-valued maps. We set up adequate Karush–Kuhn–Tucker optimality conditions for the problem beneath contingent epiderivative and ρ-cone arcwise connectedness presumptions. Assist, Mond–Weir, Wolfe, and blended sorts duality models are examined. We demonstrate the related theorems between the primal and the comparing dual problems beneath the presumption.

Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets

Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

On constrained set-valued optimization problems with $$\rho $$-cone arcwise connectedness

On constrained set-valued optimization problems with $$\rho $$-cone arcwise connectedness

Sharp Lagrange multipliers for set-valued optimization problems

In this paper, we give a comparison among some notions of weak sharp minima introduced in Amahroq et al. [Le matematiche J. 73 (2018) 99–114], Durea and Strugariu [Nonlinear Anal. 73 (2010) 2148–2157] and Zhu et al. [Set-Valued Var. Anal. 20 (2012) 637–666] for set-valued optimization problems. Besides, we establish sharp Lagrange multiplier rules for general constrained set-valued optimization problems involving new scalarization functionals based on the oriented distance function. Moreover, we provide sufficient optimality conditions for the considered problems without any convexity assumptions.

Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

A Note on Pointwise Well-Posedness of Set-Valued Optimization Problems

A Note on Pointwise Well-Posedness of Set-Valued Optimization Problems