Set-valued Ekeland Variational Principle Research Articles

Equilibrium set-valued variational principles and the lower boundedness condition with application to psychology

We first give a pre-order principle whose form is very general. Combining the pre-order principle and generalized Gerstewitz functions, we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a subset of the ordering cone multiplied by the quasi-metric. From this, we obtain a number of new results which essentially improve the related results. Particularly, the earlier lower boundedness condition has been weakened. Finally, we apply the new EVPs to Psychology.

Equilibrium versions of set-valued variational principles and their applications to organizational behavior

ABSTRACT By using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of left-complete quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a cone-convex subset of the ordering cone multiplied by the quasi-metric. Moreover, we obtain an equilibrium EVP, where the perturbation contains a σ-convex subset and the quasi-metric. From the above two general EVPs, we deduce several interesting corollaries, which extend and improve the related known results. Several examples show that the obtained set-valued EVPs are new. Finally, applying the above EVPs to organizational behavior sciences, we obtain some interesting results on organizational change and development with leadership. In particular, we show that the existence of robust organizational traps.

Ekeland variational principles for set-valued functions with set perturbations

By using generalized nonconvex separation functionals and a pre-order principle in Qiu [J Math Anal Appl. 2014;419:904–937], we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective function is a set-valued map taking values in a real vector space quasi-ordered by a convex cone K and the perturbation consists of a cone-convex subset H of K multiplied by the distance function. Here, the assumption on lower semi-continuity of the objective function is replaced by a weaker one: sequentially lower monotony. And the assumption on lower boundedness of the objective function is taken to be the weakest of several different kinds. From the general set-valued EVP, we deduce a number of particular versions of set-valued EVP, which extend and improve the related results in the literature. In particular, we give several EVPs for approximately efficient solutions in set-valued optimization, which not only extend the related results from vector-valued objective functions into set-valued objective functions, but also improve the related results by removing a usual assumption for K-boundedness (by scalarization) of the objective function's range.

Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

In this paper, we first generalize Gerstewitz’s functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl., 419, 904–937 (2014)]. By using the generalized Gerstewitz’s functions and the partial order principle, we obtain a vector EVP for e-efficient solutions in the sense of Nemeth, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.

Equilibrium versions of variational principles in quasi-metric spaces and the robust trap problem

Using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective bimap is defined on the product of left-complete quasi-metric spaces and taking values in a quasi-order linear space, and the perturbation consists of the quasi-metric and a positive vector . Here, the ordering is only to be -closed, which is strictly weaker than to be topologically closed. From the general equilibrium version, we deduce a number of particular equilibrium versions of EVP with set-valued bimaps or with vector-valued bimap. As applications of the equilibrium versions of EVP, we present several interesting results on equilibrium problems, vector optimization and fixed point theory in the setting of quasi-metric spaces. These results extend and improve the related known results. Using the obtained EVPs, we further study the existence and the robustness of traps in Behavioural Sciences.

A revised pre-order principle and set-valued Ekeland variational principles with generalized distances

In my former paper “A pre-order principle and set-valued Ekeland variational principle” (see [J. Math. Anal. Appl., 419, 904–937 (2014)]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy’s EVP in [On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396 (2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy’s EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the objective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces (Z, d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain X ⊂ Z is countably compact in any Hausdorff topology weaker than that induced by d. When (Z, d) is a Fechet space (i.e., a complete metrizable locally convex space), our existence result only requires that the domain X ⊂ Z is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems, which extend and improve the related known results.

On extended versions of Dancs-Hegedüs-Medvegyev’s fixed-point theorem

In this article, we establish some fixed-point (known also as critical point, invariant point) theorems in quasi-metric spaces. Our results unify and further extend in some regards the fixed-point theorem proposed by Dancs, S.; Hegedüs, M.; Medvegyev, P. (A general ordering and fixed-point principle in complete metric space. Acta Sci. Math. 1983;46:381–388), the results given by Khanh, P.Q., Quy D.N. (A generalized distance and enhanced Ekeland?s variational principle for vector functions. Nonlinear Anal. 2010;73:2245–2259), the preorder principles established by Qiu, J.H. (A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl. 2014;419:904–937) and the results obtained by Bao, T.Q., Mordukhovich, B.S., Soubeyran, A. (Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality. Set-Valued Var. Anal. 2015;23:375–398). In addition, we provide examples to illustrate that the improvements of our results are significant.

A pre-order principle and set-valued Ekeland variational principle

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi-ordered linear space and the perturbation contains a family of set-valued maps satisfying certain property. From this general set-valued Ekeland variational principle, we deduce a number of particular versions of set-valued Ekeland variational principle, which include many known Ekeland variational principles, their improvements and some new results.

Set-valued Ekeland variational principles in fuzzy metric spaces

In this paper, we establish a general set-valued Ekeland's variational principle in fuzzy metric spaces, where the objective function is a set-valued map defined on a fuzzy metric space and taking values in a pre-ordered locally convex space, and the perturbation involves a quasi-metric family generating the fuzzy topology of the domain space. Moreover, the direction of the perturbation is a convex subset of the positive cone instead of a single positive vector. In our general version, the assumption that the objective function is lower semi-continuous and one that the range of the function is lower bounded are both weakened. From the general Ekeland's variational principle, we obtain several particular set-valued Ekeland's variational principles in fuzzy metric spaces, which generalize and improve some related known results. From these, we deduce the corresponding Caristi's fixed point theorems for set-valued maps and the corresponding Takahashi's non-convex minimization theorems in set-valued optimization. Finally, we extend the obtained results to F-type topological spaces.

Set-Valued Quasi-Metrics and a General Ekeland's Variational Principle in Vector Optimization

We introduce the notion of compatibility between a set-valued quasi-metric and the original metric. By using this notion we prove a general set-valued Ekeland's variational principle (EVP), where the perturbation contains a set-valued quasi-metric which is compatible with the original metric. Here, we need not assume that the order cone is $w$-normal. By using the unified approach for approximate solutions introduced by Gutierrez, Jimenez, and Novo, we deduce a general version of set-valued EVP based on $(C,\epsilon )$-efficient solutions, where $C$ is a coradiant set contained in the order cone. By choosing two specific versions of the coradiant set $C$ in the general version of EVP, we obtain several particular set-valued EVPs for $\epsilon$-efficient solutions in the sense of Nemeth and of Dentcheva and Helbig, respectively. These set-valued EVPs improve and generalize the related interesting results in [C. Gutierrez, B. Jimenez, and V. Novo, SIAM J. Control Optim., 47 (2008), pp. 883--903].

A Set-Valued Ekeland's Variational Principle in Vector Optimization

This paper deals with Ekeland's variational principle for vector optimization problems. By using a set-valued metric, a set-valued perturbed map, and a cone-boundedness concept based on scalarization, we introduce an original approach to extending the well-known scalar Ekeland's principle to vector-valued maps. As a consequence of this approach, we obtain an Ekeland's variational principle that does not depend on any approximate efficiency notion. This result is related to other Ekeland's principles proved in the literature, and the finite-dimensional case is developed via an $\varepsilon$-efficiency notion that we introduced in [Math. Methods Oper. Res., 64 (2006), pp. 165–185; SIAM J. Optim., 17 (2006), pp. 688–710].