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 In addition, unpublished results and new proofs are provided.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

Convex functions. --- Convex sets. --- Duality theory (Mathematics) --- Approximation theory. --- Convex domains. --- Convexity spaces. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Spaces, Convexity --- Convex sets --- Vector spaces --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Algebra --- Mathematical analysis --- Topology --- Sets, Convex --- Convex domains --- Set theory --- Functions, Convex --- Functions of real variables --- Operator theory. --- Functional analysis. --- Mathematical optimization. --- Mathematics. --- Operator Theory. --- Functional Analysis. --- Optimization. --- Approximations and Expansions. --- Math --- Science --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Functional calculus --- Functional equations --- Integral equations

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Analytical spaces