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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,Fenchel–Rockafellar Duality,Of central importance in convex analysis are conditions guaranteeing that the conjugate of a sum is the infimal convolution of the conjugates. The main result in this direction is a theorem due to Attouch and Brézis. In turn, it gives rise to the Fenchel–Rockafellar duality framework for convex optimization problems.

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Convex Analysis and Monotone Operator Theory in Hilbert Spaces978-3-319-48311-5Series ISSN 1613-5237 Series E-ISSN 2197-4152

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https://doi.org/10.1057/9781137508416asserts 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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https://doi.org/10.1057/9781137508416lems in nonlinear analysis reduce to finding fixed points of nonexpansive operators. In this chapter, we discuss nonexpansiveness and several variants. The properties of the fixed point sets of nonexpansive operators are investigated, in particular in terms of convexity.

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https://doi.org/10.1057/9781137508416quences 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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