Duality for Nonconvex Approximation and Optimization: CMS Books in Mathematics

Preliminaries.- Worst Approximation.- Duality for Quasi-convex Supremization.- Optimal Solutions for Quasi-convex Maximization.- Reverse Convex Best Approximation.- Unperturbational Duality for Reverse Convex Infimization.- Optimal Solutions for Reverse Convex Infimization.- Duality for D.C. Optimization Problems.- Duality for Optimization in the Framework of Abstract Convexity.- Notes and Remarks.

From the reviews:
"Being the first monograph devoted to nonconvex duality, this book is going to become a fundamental source for researchers in the field. An important feature of the book is that it is also accessible to nonspecialists, since, in spite of dealing with a rather specialized topic, it is essentially self-contained. … this monograph is a very useful addition to the existing literature on optimization and approximation and is undoubtedly going to constitute a major reference on nonconvex duality." (Juan-Enrique Martinez-Legaz, Mathematical Reviews, Issue 2006 k)
"This monograph, being the first book of this kind in the literature, covers a wide range of optimization and approximation problems. It provides an excellent overview over the literature and, moreover, it contains a lot of new results and new proofs of known results. The results and the choice of the classes of problems are well motivated. … The monograph is appropriate for graduate students and advanced readers." (Andreas Löhne, Mathematical Methods of Operations Research, Vol. 66, 2007)
"In this monograph the author presents some approaches to duality in nonconvex approximation in normed linear spaces and to duality in nonconvex global optimization in locally convex spaces. … It is my belief that the monograph under review will become a fundamental reference on nonconvex duality for researchers in the field, and, although the topics are very specialized, the monograph is also accessible to nonspecialists … . is strongly recommended to researchers, postgraduate and graduate students interested in nonconvex duality theory." (Fabián Flores Bazán, Zentralblatt MATH, Vol. 1119 (21), 2007)
"This is a nice addition to the literature on nonconvex optimization in locally convex spaces, devoted primarily to nonconvex duality. Most of the material appears for the first time in book form and examples are abundant. … The style is friendly. I stronglyrecommend this book to graduate students studying nonconvex optimization theory." (Constantin P. Niculescu, Revue Roumaine de Mathématique Pures et Appliquées, Vol. LII (5), 2007)

In this monograph the author presents the theory of duality for
nonconvex approximation in normed linear spaces and nonconvex global
optimization in locally convex spaces. Key topics include:
* duality for worst approximation (i.e., the maximization of the
distance of an element to a convex set)
* duality for reverse convex best approximation (i.e., the minimization of
the distance of an element to the complement of a convex set)
* duality for convex maximization (i.e., the maximization of a convex
function on a convex set)
* duality for reverse convex minimization (i.e., the minimization of a
convex function on the complement of a convex set)
* duality for d.c. optimization (i.e., optimization problems involving
differences of convex functions).
Detailed proofs of results are given, along with varied illustrations.
While many of the results have been published in mathematical journals,
this is the first time these results appear in book form. In
addition, unpublished results and new proofs are provided. This
monograph should be of great interest to experts in this and related
fields.
Ivan Singer is a Research Professor at the Simion Stoilow Institute of
Mathematics in Bucharest, and a Member of the Romanian Academy. He is
one of the pioneers of approximation theory in normed linear spaces, and
of generalizations of approximation theory to optimization theory. He
has been a Visiting Professor at several universities in the U.S.A.,
Great Britain, Germany, Holland, Italy, and other countries, and was the
principal speaker at an N. S. F. Regional Conference at Kent State
University. He is one of the editors of the journals Numerical
Functional Analysis and Optimization (since its inception in 1979),
Optimization, and Revue d'analyse num\'erique et de th\'eorie de
l'approximation. His previous books include Best Approximation in
Normed Linear Spaces by Elements of Linear Subspaces (Springer 1970),
The Theory of Best Approximation and Functional Analysis (SIAM 1974), Bases
in Banach Spaces I, II (Springer, 1970, 1981), and Abstract Convex Analysis
(Wiley-Interscience, 1997).

Contains the latest results (most recent related book was published almost 5 years ago) Reviews new and sophisticated variations capping 3 decades of progress in the field Provides a comprehensive analysis of so-called "anticonvex" and "convex-anticonvex" optimization problems An exhaustive presentation of the duality theory for these classes of problems and their generalizations