Levitin-Polyak Well-posedness Research Articles

Levitin–Polyak well-posedness of split multivalued variational inequalities

ABSTRACT We introduce and study the split multivalued variational inequality problem (SMVIP) and the parametric SMVIP. We examine, in particular, Levitin–Polyak well-posedness of SMVIPs and parametric SMVIPs in Hilbert spaces. We provide several examples to illustrate our theoretical results. We also discuss several important special cases.

Well-posedness and global error bound for generalized mixed quasi-variational-hemivariational inequalities via regularized gap functions

The aim of this paper is to establish new results on a class of generalized mixed quasi-variational-hemivariational inequalities (GMQVHVI, for short) via regularized gap functions. First, we introduce the new regularized gap function of GMQVHVI. Then we establish the criterion of the Levitin-Polyak well-posedness for GMQVHVI under suitable conditions. Further, we provide the equivalence between the Levitin-Polyak well-posedness in the generalized sense for GMQVHVI and that for a quasi-optimization problem using the new regularized gap function. Finally, the global error bound for GMQVHVI in term of the regularized gap function is derived by employing some the properties of the Clarke generalized directional derivative and strong monotonicity conditions.

A note on “Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency” [Positivity.25(2021), 1923–1942

A note on “Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency” [Positivity.25(2021), 1923–1942

Well-posedness for bilevel vector equilibrium problems with variable domination structures

Abstract In this article, well-posedness for two types of bilevel vector equilibrium problems with variable domination structures are introduced and studied. With the help of cosmically upper continuity or Hausdorff upper semi-continuity for variable domination structures, sufficient and necessary conditions are given for such problems to be Levitin-Polyak (LP) well-posed and LP well-posedness in the generalized sense. As variable domination structure is a valid generalization of fixed one, the main results obtained in this article extend and develop some recent works in the literature.

Levitin–Polyak well-posedness for split equilibrium problems

Levitin–Polyak well-posedness for split equilibrium problems

Existence and well-posedness for excess demand equilibrium problems

<p style='text-indent:20px;'>In this paper, we study excess demand equilibrium problems in Euclidean spaces. Applying the Glicksberg's fixed point theorem, sufficient conditions for the existence of solutions for the reference problems are established. We introduce a concept of well-posedness, say Levitin–Polyak well-posedness in the sense of Painlevé–Kuratowski, and investigate sufficient conditions for such kind of well-posedness.</p>

Levitin-Polyak well-posedness for vector optimization problems in linear spaces

In this paper, we deal with vector optimization problems where the images of the objective maps are given in linear spaces not endowed with any topological structure. Firstly, we propose new concepts of semicontinuity of vector-valued maps acting from metric spaces into linear ones and discuss their properties and characterizations. Next, using Zorn's Lemma and these properties, we formulate existence conditions for such problems. Then, we introduce two notions of Levitin-Polyak well-posedness for the reference problems, and study sufficient conditions for their fulfillment, as well as relationships of these notions. Finally, using the Kuratowski measure of noncompactness, several metric characterizations of such well-posedness properties for vector optimization problems are investigated. Many examples are given to analyze our results.

Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds

Levitin–Polyak Well-Posedness by Perturbations for the Split Hemivariational Inequality Problem on Hadamard Manifolds

Existence and well-posedness of the α-core for generalized fuzzy games

The purpose of this paper is to study the existence and well-posedness of the α-core for generalized fuzzy games. First, we obtain α-core existence result with a weaker condition compared with Kajii's literature [7]. After that, we introduce a new concept of strong well-posedness which unified Tykhonov well-posedness and Levitin-Polyak well-posedness, and prove that generalized fuzzy games have generalized strong well-posedness. Finally, we present the characteristics of generalized strong well-posedness and strong well-posedness for generalized fuzzy games.

Levitin–Polyak well-posedness of variational–hemivariational inequalities

In this paper we consider the Levitin–Polyak well-posedness of a variational–hemivariational inequality in reflexive Banach spaces. First, we establish the equivalence between the Levitin–Polyak well-posedness of the considered inequality and the existence and uniqueness of its solution. Then, we discuss the relationships among the Levitin–Polyak well-posedness of the variational–hemivariational inequality, its corresponding inclusion problem and a constrained minimization problem. The proof of our results are based on arguments of monotonicity, convexity and the properties of the Clarke generalized gradient. Finally, we provide an application of our results in the study of a frictional contact problem with subdifferential boundary conditions.

On the Hadamard Well-Posedness of Generalized Mixed Variational Inequalities in Banach Spaces

We introduce a new concept of Hadamard well-posedness of a generalized mixed variational inequality in a Banach space. The relations between the Levitin–Polyak well-posedness and Hadamard well-posedness for a generalized mixed variational inequality are studied. The characterizations of Hadamard well-posedness for a generalized mixed variational inequality are established.

LP well-posed controlled systems for bounded quasi-equilibrium problems and their application to traffic networks

The aim of this paper is to establish the Levitin–Polyak well-posedness (LP well-posedness, for short) for two new classes of controlled systems of the bounded quasi-equilibrium problems, and two new classes of associated optimal control problems. First, we introduce two classes of the bounded quasi-equilibrium problems, and show, under suitable conditions, the equivalence between the LP well-posedness and existence of solution to these problems. Results on metric characterization of the LP well-posedness and LP well-posedness in the generalized sense for such problems in terms of the behavior of the approximate solution sets are provided. Second, we establish two classes of optimal control problems for systems described by the generalized bounded quasi-equilibrium problems. We also studied the LP well-posedness and LP well-posedness in the generalized sense for these problems. Finally, as a real-world application, we study the special case of controlled systems of the bounded traffic network problems.

Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency

Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency

On Levitin–Polyak well-posedness and stability in set optimization

On Levitin–Polyak well-posedness and stability in set optimization

Levitin–Polyak well-posedness for equilibrium problems with the lexicographic order

The aim of this work is to investigate optimization-related problems with the objective spaces ordered by the lexicographic cones, including parametric lexicographic equilibrium problems and optimization problems with lexicographic equilibrium constraints. We introduce concepts of Levitin–Polyak well-posedness for these problems and establish a number of sufficient conditions for such properties. The assumptions are imposed directly on the data of the problems and really verifiable. We do not need to suppose the existence (and/or convexity, compactness) of the solution set because it is proved using the mentioned assumptions on the data. Moreover, our assumptions are more relaxed than those which are usually imposed.

Generalized Levitin–Polyak well-posedness for controlled systems of FMQHI-fuzzy mixed quasi-hemivariational inequalities of Minty type

The aim of this paper is to establish some new results on the Levitin–Polyak well-posedness (in short, LP well-posedness) to a new class of the controlled systems of fuzzy mixed quasi-hemivariational inequalities of the Minty type (in short, FMQHI). First, we introduce a new class of the controlled systems for FMQHI. Then, we show, under suitable conditions, the equivalence between the LP well-posedness and existence of solution to this problem. Further, results on metric characterization of the LP well-posedness and the LP well-posedness in the generalized sense for such problem in terms of the behavior of the approximate solution set are provided. Finally, we introduce a new class of the gap function for controlled systems of FMQHI and use it to study criterion of the LP well-posedness for controlled systems of FMQHI. The results presented in the paper are new.

Levitin–Polyak well-posedness of generalized bilevel equilibrium problem with perturbations

In this paper, a new class of problem known as the generalized bilevel mixed equilibrium problem with perturbations (GBMEPP) has been introduced on a real Banach space. Based on the diameter and the measure of non-compactness of the approximate solution set of the above stated problem, criteria and characterizations have been derived for Levitin–Polyak (LP) well-posedness. A sufficient condition for the LP well-posedness has also been derived assuming the boundedness of the approximate solution set.

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.

Levitin-Polyak well-posedness for bilevel vector variational inequalities

Levitin-Polyak well-posedness for bilevel vector variational inequalities

Levitin–Polyak well-posedness for set optimization problems involving set order relations

In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set optimization problems and survey their relationships. After that, sufficient and necessary conditions for the reference problems to be LP well-posed are given. Furthermore, using the Kuratowski measure of noncompactness, we study characterizations of well-posedness for set optimization problems. Moreover, the links between stability and LP well-posedness of such problems are established via the study on approximating solution mappings. Tools and techniques used in this study and our results are different from existing ones in the literature.