Generalized concavity
type: Item
instance of: scholarly article
title: Generalized concavity
zbMATH Open document ID: 0679.90029
published in: Mathematical Concepts and Methods in Science and Engineering
publication date: 1992-09-17T00:00:00Z
review text: The book represents a detailed study of various approaches to the generalization of concave functions, which occupy a significant position in economics, engineering, management science and so on, because of the following two important properties they have: 1) a local maximizer is also a global minimizer, 2) the usual first order necessary conditions over an open set (i.e. the vanishing of the gradient) are also sufficient. The book is divided into nine chapters. The first chapter has an introductory character. The second chapter reviews the well-kown properties of the concave functions and concave mathematical programming problems. Chapter 3 deals with various types of generalized concave functions. The properties of the following cases of the generalized concave functions are investigated: quasiconcave, semistrictly quasiconcave and strictly quasiconcave functions, further pseudoconcave, strictly pseudoconcave functions. Necessary and sufficient optimality conditions for mathematical programming problems with these types of generalized concave functions are investigated. Chapter 4 considers four models of economic behaviour, in which each of the types of generalized concavity studied in the preceding chapters arise in their natural application. Some economic duality theorems are proved and their applications are discussed. Chapter 5 and 6 provide some hints how to recognize whether a given function of n variables has a generalized concavity property. The characterization of the generalized concavity properties of quadratic functions are studied. Chapter 7 investigates the generalized concavity concepts in the framework of the fractional programming theory and gives some applications. Chapter 8 is devoted to the investigation of properties of transconcave functions and (h,\(\phi)\)-concave functions. A function f defined over a convex set C, \(C\subseteq E^ n\), is transconcave if there exists a monotonically increasing function G of one variable such that the funct…
Mathematics Subject Classification ID: 90-01
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 49-01
Mathematics Subject Classification ID: 49M37
Mathematics Subject Classification ID: 49K10
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 90C32
zbMATH DE Number: 43677
zbMATH Keywords: generalization of concave functions
zbMATH Keywords: pseudoconcave functions
zbMATH Keywords: Necessary and sufficient optimality conditions
zbMATH Keywords: arcwise connected functions
author: Mordecai Avriel
author: W. Erwin Diewert
author: S. Schaible
author: Israel Zang
reviewed by: Karel Zimmermann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 43677