Generalized Convexity and Vector Optimization
Springer Science & Business Media, Dec 19, 2008 - Mathematics - 294 pages
The present lecture note is dedicated to the study of the optimality conditions and the duality results for nonlinear vector optimization problems, in ?nite and in?nite dimensions. The problems include are nonlinear vector optimization problems, s- metric dual problems, continuous-time vector optimization problems, relationships between vector optimization and variational inequality problems. Nonlinear vector optimization problems arise in several contexts such as in the building and interpretation of economic models; the study of various technolo- cal processes; the development of optimal choices in ?nance; management science; production processes; transportation problems and statistical decisions, etc. In preparing this lecture note a special effort has been made to obtain a se- contained treatment of the subjects; so we hope that this may be a suitable source for a beginner in this fast growing area of research, a semester graduate course in nonlinear programing, and a good reference book. This book may be useful to theoretical economists, engineers, and applied researchers involved in this area of active research. The lecture note is divided into eight chapters: Chapter 1 brie?y deals with the notion of nonlinear programing problems with basic notations and preliminaries. Chapter 2 deals with various concepts of convex sets, convex functions, invex set, invex functions, quasiinvex functions, pseudoinvex functions, type I and generalized type I functions, V-invex functions, and univex functions.
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Assume assumptions Banach spaces completes the proof consider the following constraint functions contradicts Converse Duality convex functions convex sets convex subsets defined Definition differentiable function dual problem duality results exists a function feasible solution function f function with respect G Xq global optimal GMWD Hanson Hessian matrix higher-order i=l i=l implies invex set invex with respect Lemma linearly independent locally Lipschitz minimax Mishra Mond Mond-Weir type n-set nonempty nonlinear programming objective and constraint order dual Pareto efficient solution piecewise smooth problem VP programming problems pseudo-invex pseudo-quasi pseudo-quasi-type quasi-invex respect to TJ satisfied second order strictly pseudo type sublinear sublinear function Suppose symmetric duality TJ x,x type I univex V-invex variational-like inequality problem vector function vector optimization problem vector variational-like inequality Weak Duality weak Pareto efficient weak strictly pseudo weakly efficient solution