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Description
Originally published in 1988, this enduring text remains the most comprehensive book on generalized convexity and concavity. The authors present generalized concave functions in a unified framework, exploring them primarily from the domains of optimization and economics.
Concavity of a function is a common property used in most of the important theorems concerning properties of optimization problems in mathematical economics, operations research, mathematical programming, engineering, and management science. Generalized Concavity deals with the many nonconcave functions that have properties similar to those of concave functions.
Specific topics covered in this book include: a review of concavity and the basics of generalized concavity; applications of generalized concavity to economics; special function forms such as composite forms, products, ratios, and quadratic functions; fractional programming; and concave transformable functions.
Keywords: convexity/concavity, generalized convexity/generalized concavity, convex programming, fractional programming, convexity in economics
Table of Contents
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Front Matter
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1. Introduction
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3. Generalized Concavity
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7. Fractional Programming
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Back Matter
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Excerpt
With the republication of our book, first published in 1988, we would like to recall how it all started.
Mordecai Avriel worked on nonlinear programming, where convexity plays a major role. If the nonlinear program is minimizing a convex function (or maximizing a concave function) over a convex set, every local optimum is a global one. Consequently, searching for a global optimum of a convex nonlinear program is much easier than seeking the optimum in a general nonlinear program. In other words, convexity is a sufficient condition for local minima to be a global one, but it is definitely not a necessary one. Therefore, Mordecai began studying functions that are more general than convex and in special cases can be reduced to convex ones. He published his first results in 1972, in the paper “r-Convex Functions.”
Siegfried Schaible was working at about the same time on various forms of generalized convexity in his doctoral dissertation in Germany. Siegfried and Mordecai met at the Department of Operations Research of Stanford University.
At about the same time Israel Zang started his doctoral research at the Technion in Haifa, Israel, under the supervision of Mordecai. The dissertation dealt with convex transformable functions and programs. The concept of r-convex functions is concerned with a particular range transformation of the function under discussion. Israel's work created the framework to handle general range and/or domain transformations. In addition, this dissertation discusses the characterization of functions whose local minima are global through continuity properties of their level sets mappings.
The next phase of collaboration started when Mordecai spent a sabbatical at the University of British Columbia (UBC) in Vancouver in 1974–1975, where he met Erwin Diewert, who was interested in generalized convexity from an economist's viewpoint. It turns out that convexity (and concavity) have important implications in economic theory.
©2010 SIAM
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