Characterizations of efficient sets by constrained objectives (Q795746)

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.