Theory of Vector Optimization

by Dinh The Luc

Overview

These notes grew out of a series of lectures given by the author at the University of Budapest during 1985-1986. Additional results have been included which were obtained while the author was at the University of Erlangen-Niirnberg under a grant of the Alexander von Humboldt Foundation. Vector optimization has two main sources coming from economic equilibrium and welfare theories of Edgeworth (1881) and Pareto (1906) and from mathematical backgrounds of ordered spaces of Cantor (1897) and Hausdorff (1906). Later, game theory of Borel (1921) and von Neumann (1926) and production theory of Koopmans (1951) have also contributed to this area. However, only in the fifties, after the publication of Kuhn-Tucker's paper (1951) on the necessary and sufficient conditions for efficiency, and of Deubreu's paper (1954) on valuation equilibrium and Pareto optimum, has vector optimization been recognized as a mathematical discipline. The stretching development of this field began later in the seventies and eighties. Today there are a number of books on vector optimization. Most of them are concerned with the methodology and the applications. Few of them offer a systematic study of the theoretical aspects. The aim of these notes is to provide a unified background of vector optimization,with the emphasis on nonconvex problems in infinite dimensional spaces ordered by convex cones. The notes are arranged into six chapters. The first chapter presents preliminary material.

Product Details

ISBN-13:	9783540505419
Publisher:	Springer Berlin Heidelberg
Publication date:	12/20/1988
Series:	Lecture Notes in Economics and Mathematical Systems , #319
Edition description:	Softcover reprint of the original 1st ed. 1989
Pages:	176
Product dimensions:	7.01(w) x 10.00(h) x 0.02(d)

Table of Contents

1: Analysis over Cones.- 1.Convex cones.- 2.Recession cones.- 3.Cone closed sets.- 4.Cone monotonie functions.- 5.Cone continuous functions.- 6.Cone convex functions.- 7.Set-valued maps.- 2: Efficient Points and Vector Optimization Problems.- 1.Binary relations and partial orders.- 2.Efficient points.- 3.Existence of efficient points.- 4.Domination property.- 5.Vector optimization problems.- 3: Nonsmooth Vector Optimization Problems.- 1 .Contingent derivatives.- 2.Unconstrained problems.- 3.Constrained problems.- 4.Differentiable case.- 5.Convex case.- 4: Scalarization and Stability.- 1.Separation by monotonic functions.- 2.Scalar representations.- 3.Completeness of scalarization.- 4.Stability.- 5: Duality.- l.Lagrangean duality.- 2.Conjugate duality.- 3.Axiomatic duality.- 4.Duality and alternative.- 6: Structure of Optimal Solution Sets.- 1.General case.- 2.Linear case.- 3.Convex case.- 4.Quasiconvex case.- Comments.- References.