Polyak Well-posedness Research Articles

Characterizing Levitin–Polyak well-posedness of split equilibrium problems

Characterizing Levitin–Polyak well-posedness of split equilibrium problems

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.

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

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 Parametric Set Optimization Problem

The aim of this paper is to introduce two notions of Levitin$-$Polyak (LP in short) well-posedness for a parametric set optimization problem, a pointwise and a global notion. Necessary and sufficient conditions for a parametric set optimization problem to be LP well-posed are given. Characterizations of LP well-posedness for a parametric set optimization problem in terms of upper Hausdorff convergence and Painlev\'{e}$-$Kuratowski convergence of sequences of approximate solution sets are also established.

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

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 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.

Characterizations of generalized Levitin–Polyak well-posed set optimization problems

In this paper, we introduce the notion of generalized Levitin–Polyak (in short gLP) well-posedness for set optimization problems. We provide some characterizations of gLP well-posedness in terms of the upper Hausdorff convergence and Painleve–Kuratowski convergence of a sequence of sets of approximate solutions, and in terms of the upper semicontinuity and closedness of an approximate solution map. We obtain some equivalence relationships between the gLP well-posedness of a set optimization problem and the gLP well-posedness of two corresponding scalar optimization problems. Also, we give some other characterizations of gLP well-posedness by two extended forcing functions and the Kuratowski noncompactness measure of the set of approximate solutions. Finally we show that certain cone-semicontinuous and cone-quasiconvex set optimization problems are gLP well-posed.

Levitin–Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints

In this paper we consider strong bilevel vector equilibrium problems and introduce the concepts of Levitin–Polyak well-posedness and Levitin–Polyak well-posedness in the generalized sense for such problems. The notions of upper/lower semicontinuity involving variable cones for vector-valued mappings and their properties are proposed and studied. Using these generalized semicontinuity notions, we investigate sufficient and/or necessary conditions of the Levitin–Polyak well-posedness for the reference problems. Some metric characterizations of these Levitin–Polyak well-posedness concepts in the behavior of approximate solution sets are also discussed. As an application, we consider the special case of traffic network problems with equilibrium constraints.

Levitin–Polyak well-posedness by perturbations of split minimization problems

In this paper, we extend well-posedness notions to the split minimization problem which entails finding a solution of one minimization problem such that its image under a given bounded linear transformation is a solution of another minimization problem. We prove that the split minimization problem in the setting of finite-dimensional spaces is Levitin–Polyak well-posed by perturbations provided that its solution set is nonempty and bounded. We also extend well-posedness notions to the split inclusion problem. We show that the well-posedness of the split convex minimization problem is equivalent to the well-posedness of the equivalent split inclusion problem.