Titlebook: Convex Analysis and Monotone Operator Theory in Hilbert Spaces; Heinz H. Bauschke,Patrick L. Combettes Book 2017Latest edition Springer In

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Introduction: Video Games and Storytelling related to each other. In this chapter, we provide fundamental results on these relationships, as well as basic results on the steepest descent direction, the Chebyshev center, and the max formula that relates the directional derivative to the support function of the subdifferential at a given poin

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https://doi.org/10.1057/9781137525055is chapter, we study the interplay between primal and dual problems in the context of Fenchel–Rockafellar duality and, more generally, for bivariate functions. The latter approach leads naturally to saddle points and Lagrangians. Special attention is given to minimization under equality constraints

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Convex Sets,asserts that every nonempty closed convex subset . of . is a Chebyshev set, i.e., that every point in . possesses a unique best approximation from ., and which provides a characterization of this best approximation.

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,Fejér Monotonicity and Fixed Point Iterations,quences possess attractive properties that simplify the analysis of their asymptotic behavior. In this chapter, we provide the basic theory for Fejér monotone sequences and apply it to obtain in a systematic fashion convergence results for various classical iterations involving (quasi)nonexpansive operators.

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Convex Cones and Generalized Interiors,omposition based on closed linear subspaces. They also arise naturally in convex analysis in the local study of a convex set via the tangent cone and the normal cone operators, and they are central in the analysis of various extensions of the notion of an interior that will be required in later chapters.

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Conjugation,detail in this chapter. In particular, it is shown that the conjugate of an infimal convolution is the sum of the conjugates. The key result of this chapter is the Fenchel–Moreau theorem, which states that the proper convex lower semicontinuous functions are precisely those functions that coincide with their biconjugates.

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