Nuclear and full nuclear cones in product spaces: Pareto efficiency and an Ekeland type variational principle
type: Item
instance of: scholarly article
title: Nuclear and full nuclear cones in product spaces: Pareto efficiency and an Ekeland type variational principle
zbMATH Open document ID: 1110.49018
author: G. Isac
author: Chr. Tammer
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: Ekeland's principle is a maximal point theorem in a product space, the second component of which is partially ordered by a cone, that has to satisfy special conditions (f.i. bounded base, nuclearity, Daniel's property, inner points). The aim of the paper is to find a variational principle of Ekeland's type, where that cone is only convex, pointed, and closed. The authors succeed: For the case, that the mentioned product space is the product of two Banach spaces (that means, that the objective function has values in a Banach space), and the cone in the second space has only the permitted properties, they prove a theorem of Ekeland's type (their theorem 9), where only the third condition (the condition for approximate efficiency) has a slightly changed form (compared with the up to now known results). The proof uses essentially a helping cone, which is well-based. The new results are not only an essential contribution to make the connection between Ekeland's principle, Pareto efficiency, and nuclearity of cones clearer, but there are interesting applications, f.i. for vector valued approximation problems (generalized Kolmogorov conditions for approximate solutions) and, using a nonconvex separation theorem, for optimality conditions of approximate solutions, if the objective function satisfies differentiability conditions. Finally hints are given, that the results can be extended to Hausdorff locally convex spaces.
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49K27
zbMATH DE Number: 5070762
zbMATH Keywords: Nuclear Cones
zbMATH Keywords: Ekeland's Variational Principle
zbMATH Keywords: Pareto Efficiency
zbMATH Keywords: Maximal Point Theorem
reviewed by: Alfred Göpfert
MaRDI profile type: Publication
full work available at URL: s11117-004-2770-8
OpenAlex ID: W2163613677
cites work: Stability Results for Ekeland's ε-Variational Principle and Cone Extremal Solutions
cites work: Continuity and Differentiability Properties of Convex Operators
cites work: A remark on Ekeland's principle in locally convex topological vector spaces.
cites work: A vector variational inequality and optimization over an efficient set
cites work: A unified approach to the existing three types of variational principles for vector valued functions
cites work: General Ekeland's variational principle for set-valued mappings.
cites work: On the variational principle
cites work: Nonconvex minimization problems
cites work: The strong Ekeland variational principle, the strong drop theorem and applications
cites work: Nonconvex separation theorems and some applications in vector optimization
cites work: Q3944358
cites work: On the vectorial Ekeland's variational principle and minimal points in product spaces
cites work: Q4341807
cites work: Supernormal cones and fixed point theory
cites work: Q4887340
cites work: Ekeland's principle and nuclear cones: a geometrical aspect
cites work: Full nuclear cones associated to a normal cone. Application to Pareto efficiency.
cites work: Q4014092
cites work: Convex programming and variational inequalities
cites work: A nonconvex vector minimization problem
cites work: Q4174553
cites work: Equivalents of Ekeland's principle
cites work: Convex functions, monotone operators and differentiability.
cites work: Inclusion theorems for non-explosive and strongly exposed cones in normed spaces
cites work: Q4279633
cites work: A generalization of ekellandz’s variational principle
cites work: On the existence of efficient points in locally convex spaces
cites work: A note on a class of cones ensuring the existence of efficient points in bounded complete sets1
cites work: Vector variational inequality and its duality
cites work: On vector variational inequalities: Application to vector equilibria
cites work: Pareto optimization in topological vector spaces
DOI: 10.1007/S11117-004-2770-8
description: scientific article; zbMATH DE number 5070762