Duality for Nonconvex Approximation and Optimization

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addition, unpublished results and new proofs are provided.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).

Résumé
The theory of convex optimization has been constantly developing over the past 30 years. Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems. This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity. This manuscript will be of great interest for experts in this and related fields.

Contenu
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.