Generalized convexity and vector optimization
type: Item
instance of: scholarly article
title: Generalized convexity and vector optimization
zbMATH Open document ID: 1162.49001
DOI: 10.1007/978-3-540-85671-9
author: S. K. Mishra
author: Kin Keung Lai
author: Shouyang Wang
published in: Nonconvex Optimization and Its Applications
publication date: 2008-10-30T00:00:00Z
review text: Monograph devoted to various types of generalized convexity and their application to optimization and multi-objective programming problems. The text of the book is divided into 8 chapters. Chapter 1 contains the formulation of nonlinear programming problem and introduces the basic notions necessary in further parts of the book. Chapters 2 and 3 describe various approaches to generalization the notions of convex sets and functions. In this context invex sets, invex functions, quasi- and pseudoinvex functions, \(V\)-invex and univex functions as well as their modifications are defined. Both differentiable and non-differentiable generalizations are considered. Optimality conditions for multi-objective problems, fractional programming problems (both one-objective and multiobjective) and mathematical programming problems; under various types of generalized convexity are derived in Chapter 4. Chapter 5 deals with various approaches to duality theory for problems considered in Chapter 4. Especially Mond-Weir type duality and its generalization are considered. Chapter 6 investigates the second and higher-order duality results for minimax problems and some other non-differentiable optimization problems under generalized convexity assumptions. In Chapter 7 symmetric, and mixed-symmetric duality results for problems from previous chapters are presented. Especially mathematical programming problems, minimax mixed integer problems, and multi-objective non-differentiable fractional variational problems are investigated in this context. The last Chapter 8 investigates relationships between vector variational-like inequality problems and vector optimization problems under various generalizations of convexity whereas both smooth and non-smooth vector optimization problems are considered. The authors mention also motivations to generalizations of convexity. The motivation stemmed from the fact that the ``classical'' convexity assumptions in many cases do not satisfy the real-world re…
reviewed by: Karel Zimmermann
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 90Cxx
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 90C29
Mathematics Subject Classification ID: 52A01
zbMATH DE Number: 5358937
zbMATH Keywords: generalized convexity
zbMATH Keywords: vector optimization
zbMATH Keywords: generalized convex functions
zbMATH Keywords: optimality conditions
zbMATH Keywords: duality theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-85671-9
OpenAlex ID: W590002453
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description: scientific article; zbMATH DE number 5358937