Painleve Kuratowski Convergence Research Articles

Painlevé–Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with application to controlling traffic networks under uncertainty

The purpose of this article is to establish some new results on the Painleve–Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with a sequence of mappings $$\varGamma _C$$ -converging. First, we introduce a new class of controlled systems for fuzzy vector quasi-optimization problems and establish some conditions for the existence of approximate solutions to these problems using the Kakutani–Fan–Glicksberg fixed-point theorem. Then, we study the Painleve–Kuratowski lower convergence, Painleve–Kuratowski upper convergence and Painleve–Kuratowski convergence of the solution sets for such problems. Finally, as a real-world application, we consider the special case of controlled systems of fuzzy traffic network problems. Existence conditions and the Painleve–Kuratowski convergence of the solution sets for these problems are also investigated and studied. The results presented in the paper are new and extend the main results given by some authors in the literature.

Metric Regularity and Lyusternik-Graves Theorem via Approximate Fixed Points of Set-Valued Maps in Noncomplete Metric Spaces

This paper considers global metric regularity and approximate fixed points of set-valued mappings. We establish a very general Theorem extending to noncomplete metric spaces a recent result by A.L. Dontchev and R.T. Rockafellar on sharp estimates of the distance from a point to the set of exact fixed points of composition set-valued mappings. In this way, we find again the famous Nadler’s Theorem, and mainly, we accordingly come up with new conclusions in this research field concerning approximate versions of Lim’s Lemma as well as the celebrated global Lyusternik-Graves Theorem. The presented results are accompanied with examples and counter-examples when it is needed. Our approach follows up numerical procedures without recourse to convergence of Cauchy sequences. Moreover, we connect metric regularity to set-convergence in metric spaces such as Painleve-Kuratowski convergence and Pompeiu-Hausdorff convergence for sets of approximate fixed points of set-valued maps. In the same context, we analyse the possibilities of the passage of regularity estimates from approximate fixed points to exact ones under the motivation of some Beer’s observations related to Wijsman convergence. As a by-product, we obtain the approximative counterpart of a recent result by A. Arutyunov on coincidence points of set-valued maps besides a new characterization of globally metrically regular set-valued maps, wherein completeness and closedness conditions are not needed.

Stability of efficient solutions to set optimization problems

This article deals with considering stability properties of Pareto minimal solutions to set optimization problems with the set less order relation in real topological Hausdorff vector spaces. We focus on studying the Painleve–Kuratowski convergence of Pareto minimal elements in the image space. Employing convexity properties, we study the external stability of Pareto minimal solutions via weak ones. Then, we use converse properties to investigate external stability conditions to such problems where Pareto minimal solution sets and weak/ideal ones are distinct. For the internal stability, we propose a concept of compact convergence in the sense of Painleve–Kuratowski and use it together with a domination property to analyze stability conditions for the reference 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.

Stability and Scalarization for a Unified Vector Optimization Problem

This paper aims at investigating the Painleve–Kuratowski convergence of solution sets of a sequence of perturbed vector problems, obtained by perturbing the feasible set and the objective function of a unified vector optimization problem, in real normed linear spaces. We establish convergence results, both in the image and given spaces, under the assumptions of domination and strict domination properties. Moreover, scalarization techniques are employed to establish the Painleve–Kuratowski convergence in terms of the solution sets of a sequence of scalarized problems.

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.

Painlevé-Kuratowski Stability of the Solution Sets to Perturbed Vector Generalized Systems

In this paper, stability results of solution mappings to perturbed vector generalized system are studied. Firstly, without the assumption of monotonicity, the Painleve-Kuratowski convergence of global efficient solution sets of a family of perturbed problems to the corresponding global efficient solution set of the generalized system is obtained, where the perturbations are performed on both the objective function and the feasible set. Then, the density and Painleve-Kuratowski convergence results of efficient solution sets are established by using gamma convergence, which is weaker than the assumption of continuous convergence. These results extend and improve the recent ones in the literature.

Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

In this paper, we consider vector quasi-equilibrium problems under perturbation in terms of suitable asymptotically solving sequences, not embedding given problems into a parameterized family. By employing some types of convergences for mapping and set sequences, we obtain the Painleve–Kuratowski upper convergence of solution sets for the reference problems. Then, using nonlinear scalarization functions, we propose gap functions for such problems, and later employing these functions, we study necessary and sufficient conditions for Painleve–Kuratowski lower convergence and Painleve–Kuratowski convergence. As an application, we discuss the special case of vector quasi-variational inequality.

Painlevé–Kuratowski stability of approximate efficient solutions for perturbed semi-infinite vector optimization problems

This paper is concerned with the stability of semi-infinite vector optimization problems (SIVOP) under functional perturbations of both objective functions and constraint sets. First, we establish the Berge-lower semicontinuity and Painleve–Kuratowski convergence of the constraint set mapping. Then, using the obtained results, we obtain sufficient conditions of Painleve–Kuratowski stability for approximate efficient solution mapping and approximate weakly efficient solution mapping to the (SIVOP). Furthermore, an application to the traffic network equilibrium problems is also given.

Stability of approximate solution mappings for generalized Ky Fan inequality

This paper is concerned with the stability for a generalized Ky Fan inequality when it is perturbed by vector-valued bifunction sequence and set sequence. By continuous convergence of the bifunction sequence and Painleve–Kuratowski convergence of the set sequence, we establish the Painleve–Kuratowski convergence of the approximate solution mappings of a family of perturbed problems to the corresponding solution mapping of the original problem. Our main results are new and different from the ones in the literature.

Painlevé-Kuratowski convergences of the solution sets for perturbed vector equilibrium problems without monotonicity

In this paper, we obtain some stability results for perturbed vector equilibrium problems. Under new assumptions, which are weaker than the assumption of C-strict monotonicity, we provide sufficient conditions for the Painleve-Kuratowski Convergence of the weak efficient solution sets and efficient solution sets for the perturbed vector equilibrium problems with a sequence of mappings converging in real linear metric spaces. These results extend and improve some known results in the literature.

Painlevé-Kuratowski Convergences of the solution sets to perturbed generalized systems

In this paper, we obtain the Painleve-Kuratowski Convergence of the efficient solution sets, the weak efficient solution sets and various proper efficient solution sets for the perturbed generalized system with a sequence of mappings converging in a real locally convex Hausdorff topological vector spaces.

Painleve-Kuratowski Convergences for the Solution Sets of Set-Valued Weak Vector Variational Inequalities

Painleve-Kuratowski convergence of the solution sets is investigated for the perturbed set-valued weak vector variational inequalities with a sequence of mappings converging continuously. The closedness and Painleve-Kuratowski upper convergence of the solution sets are obtained. We also obtain Painleve-Kuratowski upper convergence when the sequence of mappings converges graphically. By virtue of a sequence of gap functions and a key assumption, Painleve-Kuratowski lower convergence of the solution sets is established. Some examples are given for the illustration of our results.

A Lévy Type Martingale Convergence Theorem for Random Sets with Unbounded Values

Given a nondecreasing sequence (bernou n ) of sub-sgr-fields and a real or vector valued random variable f, the Levy Martingale convergence Theorem (LMCT) asserts that E(f/bernou n ) converges to E(f/bernouinfin) almost surely and in L 1, where bernouinfin stands for the sgr-field generated by the bernou n . In the present paper, we study the validity of the multivalued analog this theorem for a random set F whose values are members of Fscr(X), the space of nonempty closed sets of a Banach space X, when Fscr(X) is endowed either with the Painleve–Kuratowski convergence or its infinite dimensional extensions. We deduce epi-convergence results for integrands via the epigraphical multifunctions. As it is known, these results are useful for approximating optimization problems. The method relies on countability supportness hypotheses which are shown to hold when the values of the random set E(F/bernou n ) do not contain any line. On the other hand, since the values of F are not assumed to be bounded, conditions involving barrier and asymptotic cones are shown to be necessary. Moreover, we discuss the relations with other multivalued martingale convergence theorems and provide examples showing the role of the hypotheses. Even in the finite dimensional setting, our results are new or subsume already existing ones.