Titlebook: Convex Analysis and Nonlinear Optimization; Theory and Examples Jonathan M. Borwein,Adrian S. Lewis Textbook 20001st edition Springer Scien

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Fenchel Duality,We have already seen, in the First order sufficient condition (2.1.2), one benefit of convexity in optimization: critical points of convex functions are global minimizers. In this section we extend the types of functions we consider in two important ways:

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Special Cases,In our earlier section on theorems of the alternative (Section 2.2), we observed that finitely generated cones are closed. Remarkably, a finite linear-algebraic assumption leads to a topological conclusion. In this section we pursue the consequences of this type of assumption in convex analysis.

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Fixed Points,Many questions in optimization and analysis reduce to solving a nonlinear equation . (.) = 0, for some function .: .. Equivalently, if we define another map . = . − . (where . is the identity map), we seek a point . in . satisfying . (.) = .; we call . a . of ..

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https://doi.org/10.1007/978-1-4757-9859-3Mathematica; Microsoft Access; PostScript; boundary element method; constraint; convex analysis; duality; e

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Springer Science+Business Media New York 2000

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Jonathan M. Borwein,Adrian S. LewisReviews the increasingly sophisticated state of computational optimization techniques.Provides an accessible account of convex analysis and its applications and extensions.New Edition adds material on

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Background,onal vector space over the reals ., equipped with an inner product ‹·,·›. We would lose no generality if we considered only the space .. of real (column) .-vectors (with its standard inner product), but a more abstract, coordinate-free notation is often more flexible and elegant.

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Convex Analysis,ontinuous on the interior of their domains. Part of the surprise is that an . assumption (convexity) leads to a . conclusion (continuity). It is this powerful fact that guarantees the usefulness of regularity conditions like Adom . ∩ cont . ≠ ∅ (3.3.9), which we studied in the previous section.

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