Characterizations of efficient sets by constrained objectives
type: Item
instance of: scholarly article
title: Characterizations of efficient sets by constrained objectives
zbMATH Open document ID: 0542.90090
DOI: 10.1007/BF00939137
published in: Journal of Optimization Theory and Applications
publication date: 1985-01-01T00:00:00Z
review text: This paper provides a complete or partial characterization of the efficient set of a set \(X\subseteq {\mathbb{R}}^ m\) with respect to m objective functions, \(\{f^ i\}\) in terms of constrained optimization problems of the equality or inequality types, e.g., maximize \(f^ i(x)\), subject to \(x\in X\), \(f^ j(x)\geq \alpha_ j\), \(\forall j\neq i\), \(\alpha \in {\mathbb{R}}^{m-1}.\) Related to these maximization problems are the lexicographic maximization problems, introduced since optimizations of the above kind do not always produce efficient solutions. With appropriate definitions of the lexicographic maximization problems, precisely the set of efficient solutions is obtained in the inequality form given above. The introduction of a special condition gives a related, but weaker, result for the equality constrained form. The introduction of efficient constraints also gives fairly strong characterization results. Finally, the lexicographic optimization problem is reduced to a uniform linear optimization problem, and conditions for the special conditions required to strengthen the characterizations are also given.
Mathematics Subject Classification ID: 90C31
zbMATH DE Number: 3862953
zbMATH Keywords: weighting factors
zbMATH Keywords: efficient set
zbMATH Keywords: constrained optimization
zbMATH Keywords: lexicographic maximization
zbMATH Keywords: efficient constraints
author: D. J. White
MaRDI profile type: Publication
cites work: Q5514145
cites work: Q4120209
cites work: Proper efficiency and the theory of vector maximization
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description: scientific article; zbMATH DE number 3862953