Variational Principle For Set-valued Mappings Research Articles

A limiting subdifferential version of Ekeland’s variational principle in set optimization

The paper is devoted to a new subdifferential version of Ekeland’s variational principle for set-valued maps in terms of Mordukhovich’s limiting differentiation, where Kuroiwa’s lower set-less preorder is used to compare images of set-valued maps. As a consequence, we study necessary conditions for strict positive minimizers of set-valued maps.

Ekeland’s variational principle with weighted set order relations

The main results of the paper are a minimal element theorem and an Ekeland-type variational principle for set-valued maps whose values are compared by means of a weighted set order relation. This relation is a mixture of a lower and an upper set relation which form the building block for modern approaches to set-valued optimization. The proofs rely on nonlinear scalarization functions which admit to apply the extended Brezis–Browder theorem. Moreover, Caristi’s fixed point theorem and Takahashi’s minimization theorem for set-valued maps based on the weighted set order relation are obtained and the equivalences among all these results is verified. An application to generalized intervals is given which leads to a clear interpretation of the weighted set order relation and versions of Ekeland’s principle which might be useful in (computational) interval mathematics.

Fixed Points and Variational Principles with Applications to Capability Theory of Wellbeing via Variational Rationality

In this paper we first develop two new results of variational analysis. One is a fixed point theorem for parametric dynamic systems in quasimetric spaces, which can also be interpreted as an existence theorem of minimal points with respect to reflexive and transitive preferences for sets in products spaces. The other one is a variational principle for set-valued mappings acting from quasimetric spaces to vector spaces with variable ordering structures, which can be treated as a far-going extension of Ekeland’s variational principle to this setting. Both of these results are motivated by applications to adaptive dynamical aspects of Sen’s capability theory of wellbeing. As consequence of our mathematical results, we develop new applications to such human behavioral models by using a recent variational rationality approach to behavioral sciences.

On Ha’s version of set-valued Ekeland’s variational principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.

An Ekeland’s variational principle for set-valued mappings with applications

In this paper, we obtain a general Ekeland’s variational principle for set-valued mappings in complete metric space, which is different from those in [G.Y. Chen, X.X. Huang, Ekeland’s ε -variational principle for set-valued mapping, Mathematical Methods of Operations Research 48 (1998) 181–186; G.Y. Chen, X.X. Huang, S.H. Hou, General Ekeland’s Variational Principle for Set-Valued Mappings, Journal of Optimization Theory and Applications 106 (2000) 151–164; S.J. Li, W.Y. Zhang, On Ekeland’s variational Principle for set-valued mappings, Acta Mathematicae Application Sinica, English Series 23 (2007) 141–148]. By the result, we prove some existence results for a general vector equilibrium problem under nonconvex and compact or noncompact assumptions of its domain, respectively. Moreover, we give some equivalent results to the variational principle.

On the [formula omitted]-variational principle for set-valued mappings

In this paper, the well-known Ekeland variational principle is generalized to the case where set-valued mappings are involved. More specifically, the ϵ -efficient points of a vector optimization problem for set-valued mappings are investigated via a vector Ekeland variational principle for set-valued mappings.

On Ekeland's Variational Principle for Set-Valued Mappings

In this paper, we derive a general vector Ekeland variational principle for set-valued mappings, which has a closed relation to ek 0-efficient points of set-valued optimization problems. The main result presented in this paper is a generalization of the corresponding result in [3].

The Ekeland variational principle for set-valued maps involving coderivatives

In this paper we use the Fréchet, Clarke, and Mordukhovich coderivatives to obtain variants of the Ekeland variational principle for a set-valued map F and establish optimality conditions for set-valued optimization problems. Our technique is based on scalarization with the help of a marginal function associated with F and estimates of subdifferentials of this function in terms of coderivatives of F.

Errata Corrige: General Ekeland's Variational Principle for Set-Valued Mappings

Errata Corrige: General Ekeland's Variational Principle for Set-Valued Mappings

A New Variant of Ekeland's Variational Principle for Set-Valued Maps

In this article, following the idea used by Göpfert et al . [A. Göpfert, Chr. Tammer and C. Zalinescu (2000). On the vectorial Ekeland's variational principle and minimal points in product spaces. Nonlinear Analysis, Theory, Methods & Applications , 39 , 909-922] to derive an Ekeland's variational principle for vector-valued functions, we derive a new variant of Ekeland's variational principle for set-valued maps. Finally, we apply this variational principle to obtain an approximate necessary optimality condition for a class of set-valued optimization problems.

Stability Results for Ekeland's ε Variational Principle for Set-Valued Mappings

In this paper, we define the Mosco convergence and Kuratowski-Painleve (P.K.) convergence for set-valued mapping sequence F n . Under some conditions, we derive the following result If a set-valued mapping sequence F n , which are nonempty, compact valued, upper semicontinuous and uniformly bounded below, Mosco (or P.K.) converges to a set-valued mapping F , which is upper semicontinuous, nonempty, compact valued, then Q l >0, u >0, $\\varepsilon / \\lambda - {\\rm ext}\\, F := \\{ \\bar x \\in X : (F(x) - \\bar y + \\varepsilon / \\lambda \\Vert x - \\bar x \\Vert e)$

Extended and strongly extended well-posedness of set-valued optimization problems

In this paper, we deal with the extended well-posedness and strongly extended well-posedness of set-valued optimization problems. These two concepts are generalizations of the extended well-posedness of real-valued optimization probems defined by Zolezzi. We obtain some criteria and characterizations of these two types of extended well-posedness, further generalizing most results obtained by Zolezzi for the extended well-posedness of scalar optimization problems. In the mean time, many results obtained by us for the extended well-posedness of vector optimization problems have been generalized to set-valued optimization. Finally, we present an approximate variational principle for set-valued maps, derive a necessary approximate optimality condition for set-valued optimization, based on which we introduce a condition, which is somewhat analogous to the Palais-Smale condition (C), and provide sufficient conditions for the extended and strongly extended well-posedness of set-valued optimization problems.

Equivalents of a general approximate variational principle for set-valued maps and application to efficiency

In this paper, three equivalents of a corollary of the general approximate variational principle for set-valued maps which we obtained in [5] are derived. Applying a corollary of the general approximate variational principle, we obtain existence theorems for efficient solutions to vector optimization problems.

General Ekeland's Variational Principle for Set-Valued Mappings

In this paper, we introduce the concept of approximate solutions for set-valued optimization problems. A sufficient condition for the existence of approximate solutions is obtained. A general Ekeland's variational principle for set-valued mappings in complete ordered metric spaces and complete metric spaces are derived. These results are generalizations of results for vector-valued functions in Refs. 1–4.

Ekeland's ε-variational principle for set-valued mappings

In this paper, we introduce the concept of approximate solutions for set-valued mappings and provide a sufficient condition for the existence of approximate solutions of set-valued mappings. We obtain an approximate variational principle for set-valued mappings.