Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling
type: Item
instance of: scholarly article
title: Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling
zbMATH Open document ID: 1255.49026
author: Boris S. Mordukhovich
author: Truong Quang Bao
published in: Discrete and Continuous Dynamical Systems
publication date: 2011-12-20T00:00:00Z
review text: The paper represents new results on first order necessary optimality conditions for local and global minima in set-valued optimization, multiplier rules included (Theorem 4.3). The application to economics with \(n\in\mathbb N\) customers and \(m\in \mathbb N\) producers in normed commodity spaces yields a generalized second fundamental theorem of welfare economics (multiplier rule, Theorem 5.2). The used main tools are variational analysis, generalized differentiation for multivalued mappings (generalized normal cones) and very general preference mappings. The transitivity of the ordering is replaced by a much more general local shift property, the so called asymptotic closedness of the level sets with respect to the generalized ordering. The paper starts with a very general extremal principle for partially sequentially normally compact finite set systems (cf. [\textit{B. S. Mordukhovich}, Variational analysis and generalized differentiation. I: Basic theory. II: Applications. Grundlehren der Mathematischen Wissenschaften 331/332. Berlin: Springer (2006; Zbl 1100.49002)]) (part II, Lemma 5.58) and apply this principle for deriving necessary conditions in set-valued optimization - vector optimization included (Theorem 3.3). They consider three kinds of local minimizers: (i) fully localized weak minimizers (generalization of weak minimizers in vector optimization), (ii) fully localized minimizers (similar to Pareto minimality) and (iii) fully localized strong minimizers which mainly base on different asymptotic closedness properties of the used level set mappings. Comparisons of the new concept to former results in set-valued and vector optimization are given. The proofs of the important results are given in detail. Unfortunately, not all used notations are introduced e. g. the locally closedness of sets (closedness with respect to the intersection with some closed ball). For global minima in this concept, the asymptotic closedness of the level sets is replaced weake…
reviewed by: Armin Hoffmann
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 91B15
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 90C29
zbMATH DE Number: 5990793
zbMATH Keywords: variational analysis
zbMATH Keywords: nonsmooth and multiobjective optimization
zbMATH Keywords: generalized differentiation
zbMATH Keywords: necessary optimality conditions
zbMATH Keywords: second fundamental theorem of welfare economics
zbMATH Keywords: extremal principle
zbMATH Keywords: asymptototical closedness
zbMATH Keywords: fully localized weak minimizers
zbMATH Keywords: fully localized minimizers
zbMATH Keywords: fully localized strong minimizers
zbMATH Keywords: M-normal cone
zbMATH Keywords: coderivative
zbMATH Keywords: set-valued optimization
zbMATH Keywords: intransitive preference relation
zbMATH Keywords: epi-Lipschitzian sets
zbMATH Keywords: refined net demand qualification condition
MaRDI profile type: Publication
full work available at URL: dcds.2011.31.1069
OpenAlex ID: W2316980383
DOI: 10.3934/DCDS.2011.31.1069
description: scientific article; zbMATH DE number 5990793