Foundations of Mathematical Optimization: Convex Analysis without Linearity
Springer Science & Business Media, Feb 28, 1997 - Mathematics - 585 pages
Many books on optimization consider only finite dimensional spaces. This volume is unique in its emphasis: the first three chapters develop optimization in spaces without linear structure, and the analog of convex analysis is constructed for this case. Many new results have been proved specially for this publication. In the following chapters optimization in infinite topological and normed vector spaces is considered. The novelty consists in using the drop property for weak well-posedness of linear problems in Banach spaces and in a unified approach (by means of the Dolecki approximation) to necessary conditions of optimality. The method of reduction of constraints for sufficient conditions of optimality is presented. The book contains an introduction to non-differentiable and vector optimization.
Audience: This volume will be of interest to mathematicians, engineers, and economists working in mathematical optimization.
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algebra approximation arbitrary assume Banach space belonging bounded called Clarke compact condition consequence consider constant constraints construct contains continuous function convergent convex function Corollary definition denote derivative differentiable directional Dolecki easy element equal Example exists formula Frechet function f(x given globalization gradient hand Hence holds immediately implies inequality interior Let f(x Let X linear linear functionals linear space Lipschitz functions Lipschitzian locally loss lower semi-continuous mapping marginal function metric space minimal minimum monotone Moreover multifunction Namely neighbourhood normed space notion Observe obtain operator optimization point xo problem Proof Proposition real-valued functions defined recall relation resp respect satisfies separated sequence similar strict subgradient subset supergradient Suppose Take tangent cone tending Theorem topology uniformly convex unique upper semi-continuous values vector weakly well-posed