Ekeland variational principles involving set perturbations in vector equilibrium problems
- PDF / 470,252 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 74 Downloads / 218 Views
Ekeland variational principles involving set perturbations in vector equilibrium problems Le Phuoc Hai1,2 Received: 11 May 2020 / Accepted: 30 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract On the basis of the notion of approximating family of cones and a generalized type of Gerstewitz’s/Tammer’s nonlinear scalarization functional, we establish variants of the Ekeland variational principle (for short, EVP) involving set perturbations for a type of approximate proper solutions in the sense of Henig of a vector equilibrium problem. Initially, these results are obtained for both an unconstrained and a constrained vector equilibrium problem, where the objective function takes values in a real locally convex Hausdorff topological linear space. After that, we consider special cases when the objective function takes values in a normed space and in a finite-dimensional vector space. For the finite-dimensional objective space with a polyhedral ordering cone, we give the explicit representation of variants of EVP depending on matrices, and in such a way, some selected applications for multiobjective optimization problems and vector variational inequality problems are also derived. Keywords Ekeland variational principle · Set perturbations · Approximate proper efficiency · Approximating family of cones · Multiobjective programming · Variational inequalities Mathematics Subject Classification 90C33 · 90C26 · 90C29 · 49J52 · 49J53
1 Introduction From the very beginning [9], EVP proved in complete metric spaces has become quickly a fundamental tool with many applications in optimization and nonlinear analysis. Actually, EVP formulated in finite-dimensional spaces (see [40, Corollary 2.13]) is one of the first and most powerful results of modern variational analysis. Therefore, several formulations of EVP for vector-valued and set-valued mappings are proved, e.g., in [17,25,34,35,38,44,49]. Also, some works related to bifunctions can be found in [1,7,11,28,29,42], where the authors in [11] extended the Deville–Godefroy–Zizler variational principle to bifunctions and utilized
B
Le Phuoc Hai [email protected]
1
Department of Mathematics and Computing, University of Science, Ho Chi Minh City, Vietnam
2
Vietnam National University, Ho Chi Minh City, Vietnam
123
Journal of Global Optimization
it to obtain two variational principles, i.e., one of Borwein-Preiss type and another one of Ekeland type. Furthermore, regarding set perturbations, there have been also a lot of extensions of the EVP involving set perturbations in various types. For instance, several papers considered previously a set perturbation in the simple form d(x, y)D with a convex set D, e.g., [4,5,36,38], or the other more general form considered in [25,43], where the perturbation contains set-valued metrics or set-valued quasi-metrics. Quite recently, by using a new type of generalized nonconvex separation functionals and a pre-order principle introduced in [44], several versions of EVP involving set perturbat
Data Loading...
