Weak Minimal Solutions Research Articles

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.

Continuity of a scalarization in vector optimization with variable ordering structures and application to convergence of minimal solutions

We consider a scalarization function, which was introduced by Eichfelder [Variable ordering structures in vector optimization. Berlin: Springer-Verlag; 2014 (Series in vector optimization)], based on the oriented distance of Hiriart–Urruty with respect to a general variable ordering structure (VOS). We first study the continuity of the composition of a set-valued map with the oriented distance. Then, using the obtained results, we study the continuity of the scalarization function by extending some concepts of continuity for cone-valued maps. As an application, convergence in the sense of Painlevé–Kuratowski of sets of weak minimal solutions is provided, with the vector criterion and a VOS. Illustrative examples are also given.

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

Semicontinuity of the minimal solution mappings to parametric set optimization problems on Banach lattices

This paper aims at investigating the stability of minimal solutions to set optimization problems on Banach lattices. We introduce the concepts of strictly quasi cone-convexlikeness and supremum mapping of the set-valued mappings and obtain an existence theorem of minimal solutions to set optimization problems by using Zorn's lemma. We first derive the continuity of supremum mapping of the set-valued mapping. Then, we establish the upper semicontinuity and lower semicontinuity of the minimal solution mapping and weak minimal solution mapping to parametric set optimization problems by utilizing strictly quasi cone-convexlikeness and the continuity of supremum mapping. Finally, our main results are applied to semicontinuity of the solution mappings to parametric vector optimization problems.

Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications

Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications

Six set scalarizations based on the oriented distance: continuity, convexity and application to convex set optimization

In the setting of normed spaces ordered by a convex cone not necessarily solid, we use six set scalarization functions, which are extensions of the oriented distance of Hiriart-Urruty, and we discuss convexity and continuity properties of their composition with two set-valued maps. Furthermore, as an application, we derive a multiplier rule for weak minimal solutions of a convex set optimization problem, with respect to the lower set less preorder of Kuroiwa. Some illustrative examples are also given.

Connectedness of weak minimal solution set for set optimization problems

This paper aims at investigating connectedness of weak p-minimal solution set for set optimization problems by using the scalarization method. We make a new attempt to derive a scalarization result for weak p-minimal solution set for set optimization problems. By using the scalarization result, we establish connectedness of weak p-minimal solution set for set optimization problems.

Approximate Solutions and Levitin–Polyak Well-Posedness for Set Optimization Using Weak Efficiency

The present study is devoted to define a new notion of approximate weak minimal solution based on a set order relation introduced by Karaman et al. (Positivity 22(3):783–802, 2018) for a constrained set optimization problem. Sufficient conditions have been found for the closedness of minimal solution sets. Using the Painleve–Kuratowski convergence, the stability aspects of the approximate weak minimal solution sets are discussed. Further, a notion of Levitin–Polyak well-posedness for the set optimization problem is introduced. Sufficiency criteria and some characterizations of the above defined well-posedness are established. An alternative approach to obtain robust solutions for uncertain vector optimization problems is discussed as an application.

Weak minimal elements and weak minimal solutions of a nonconvex set-valued optimization problem

In this paper, we characterize the nonemptiness of the set of weak minimal elements for a nonempty subset of a linear space. Moreover, we obtain some existence results for a nonconvex set-valued optimization problem under weaker topological conditions.

Characterization of set relations through extensions of the oriented distance

In the framework of normed spaces ordered by a convex cone not necessarily solid, we consider two set scalarization functions of type sup-inf, which are extensions of the oriented distance of Hiriart-Urruty. We investigate some of their properties and, moreover, we use these functions to characterize the lower and upper set less preorders of Kuroiwa and the strict lower and strict upper set relations. Finally, we apply the obtained results to characterize several concepts of minimal solution to a set optimization problem defined by a set-valued map. Minimal and weak minimal solutions with respect to the lower and upper set less relations are between the concepts considered. Illustrative examples are also given.

Scalarizations for a set optimization problem using generalized oriented distance function

The aim of this paper is to establish scalarizations for minimal and weak minimal solutions of a set optimization problem using generalized oriented distance function introduced by Crespi et al. (Math Methods Oper Res 63:87–106, 2006). The solution concepts are based on a partial set order relation on the family of nonempty bounded sets proposed by Karaman et al. (Positivity 22:783–802, 2018). Finally, we also provide existence results for minimal solutions and sufficient conditions for the solution sets to be closed.

External and internal stability in set optimization

The main aim of this paper is to establish stability in set optimization in terms of convergence of a sequence of solution sets of perturbed set optimization problems to the solution set of the original set optimization problem both in the image space and the given space. The perturbed problems are obtained by perturbing the feasible set without changing the objective map. Formulations of external stability and internal stability are considered in the image space. External stability, which pertains to complete convergence of a subsequence of weak minimal solution sets, both in the sense of Hausdorff and convergence of sets, is established under certain compactness and continuity assumptions. This leads to the upper convergence of the sequence of solution sets in the given space. Internal stability is established for minimal solution sets under certain continuity, compactness and domination assumptions which leads to the lower convergence of the sequence of solution sets in the given space. External stability for minimal solution sets and internal stability for weak minimal solution sets are deduced under the strict quasiconvexity assumption. In particular, the results are also deduced for a vector optimization problem.

Continuity of the solution set mappings to a parametric set optimization problem

In this paper, we establish the upper, lower semicontinuity and closedness of the minimal solution and weak minimal solution set mappings to a parametric set-valued vector optimization problem with set optimization criterion under some suitable assumptions.

Optimality conditions in non-convex set-valued optimization

The notion of radial epiderivative is introduced and then a necessary and sufficient condition for a point to be a weak minimal solution (weak-efficient solution) for a non-convex set-valued optimization problem is derived. Such a condition subsumes various necessary and/or sufficient conditions found in the literature for single-valued convex/non-convex mappings.

∈-Weak Minimal Solutions of Vector Optimization Problems with Set-Valued Maps

In this paper, we consider cone-subconvexlike vector optimization problems with set-valued maps in general spaces and derive scalarization results, ∈-saddle point theorems, and ∈-duality assertions using ∈-Lagrangian multipliers.

LEGENDRE CONDITIONS FOR A VARIATIONAL PROBLEM WITH ONE-SIDED PHASE CONSTRAINTS

This paper is concerned with a variational problem with inequality phase constraints. We present second-order necessary conditions (Legendre condition) for weak minimal solutions of the variational problem. Our optimality conditions give an information about the Hessian of the integrand at not only inactive points but also active points. Since the present constraints do not include x, our conditions differ from the Legendre-Clebsch condition.