Quasiconvex Optimization and Location Theory / Edition 1

by J.A. dos Santos Gromicho

Overview

grams of which the objective is given by the ratio of a convex by a positive (over a convex domain) concave function. As observed by Sniedovich (Ref. [102, 103]) most of the properties of fractional pro­ grams could be found in other programs, given that the objective function could be written as a particular composition of functions. He called this new field C­ programming, standing for composite concave programming. In his seminal book on dynamic programming (Ref. [104]), Sniedovich shows how the study of such com­ positions can help tackling non-separable dynamic programs that otherwise would defeat solution. Barros and Frenk (Ref. [9]) developed a cutting plane algorithm capable of optimizing C-programs. More recently, this algorithm has been used by Carrizosa and Plastria to solve a global optimization problem in facility location (Ref. [16]). The distinction between global optimization problems (Ref. [54]) and generalized convex problems can sometimes be hard to establish. That is exactly the reason why so much effort has been placed into finding an exhaustive classification of the different weak forms of convexity, establishing a new definition just to satisfy some desirable property in the most general way possible. This book does not aim at all the subtleties of the different generalizations of convexity, but concentrates on the most general of them all, quasiconvex programming. Chapter 5 shows clearly where the real difficulties appear.

Product Details

ISBN-13:	9780792346944
Publisher:	Springer US
Publication date:	01/31/1998
Series:	Applied Optimization , #9
Edition description:	1998
Pages:	219
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1 Introduction.- 2 Elements of Convexity.- 2.1 Generalities.- 2.2 Convex sets.- 2.3 Convex functions.- 2.4 Quasiconvex functions.- 2.5 Other directional derivatives.- 3 Convex Programming.- 3.1 Introduction.- 3.2 The ellipsoid method.- 3.3 Stopping criteria.- 3.4 Computational experience.- 4 Convexity in Location.- 4.1 Introduction.- 4.2 Measuring convex distances.- 4.3 A general model.- 4.4 A convex location model.- 4.5 Characterizing optimality.- 4.6 Checking optimality in the planar case.- 4.7 Computational results.- 5 Quasiconvex Programming.- 5.1 Introduction.- 5.2 A separation oracle for quasiconvex functions.- 5.3 Easy cases.- 5.4 When we meet a “bad” point.- 5.5 Convergence proof.- 5.6 An ellipsoid algorithm for quasiconvex programming.- 5.7 Improving the stopping criteria.- 6 Quasiconvexity in Location.- 6.1 Introduction.- 6.2 A quasiconvex location model.- 6.3 Computational results.- 7 Conclusions.

Editorial Reviews

For graduate students and researchers working in continuous optimization using separation techniques or in general continuous location models, describes variants of the ellipsoid method for convex and quasi-convex problems and applies them to very general convex and quasi-convex models in location theory. Describes the adopted notation and reviews the basics of convexity and convex optimization, then shows how the new concepts diverge from rather than replace the classical ones, and uses the ellipsoid method to illustrate methods that rely on separation hyperplanes. Explains the new algorithm from its derivation to its convergence analysis, using a simpler approach than is found in previous literature. Slightly enhanced from a 1995 Ph. D. dissertation for the Erasmus University in Rotterdam. Annotation c. by Book News, Inc., Portland, Or.

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