We propose a level proximal subdifferential for a proper lower semicontinuous function. Level proximal subdifferential is a uniform refinement of the well-known proximal subdifferential, and has the pleasant feature t...
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We propose a level proximal subdifferential for a proper lower semicontinuous function. Level proximal subdifferential is a uniform refinement of the well-known proximal subdifferential, and has the pleasant feature that its resolvent always coincides with the proximal mapping of a function. It turns out that the resolvent representation of proximal mapping in terms of Mordukhovich limiting subdifferential is only valid for hypoconvex functions. We also provide properties of level proximal subdifferential and numerous examples to illustrate our results.
The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly no...
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The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly nonexpansive operators form a proper subclass of the more general-but still pleasant from an algorithmic perspective-class of averaged operators. In this paper, we introduce the new notion of conically nonexpansive operators which generalize nonexpansive mappings. We characterize averaged operators as being resolvents of comonotone operators under appropriate scaling. As a consequence, we characterize the proximal point mappings associated with hypoconvex functions as cocoercive operators, or equivalently;as displacement mappings of conically nonexpansive operators. Several examples illustrate our analysis and demonstrate tightness of our results.
Proximal mappings are essential in splitting algorithms for both convex and nonconvex optimization. In this paper, we show that proximal mappings of every prox-bounded function are nonexpansive if and only if they are...
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Proximal mappings are essential in splitting algorithms for both convex and nonconvex optimization. In this paper, we show that proximal mappings of every prox-bounded function are nonexpansive if and only if they are firmly nonexpansive if and only if they are averaged if and only if the function is convex. Lipschitz proximal mappings of prox-bounded functions are also characterized via hypoconvex or strongly convex functions. Our results generalize a recent result due to
Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. ...
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Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1/lambda-hypoconvex if its proximal mapping P(lambda)f is single-valued. When the function f is bounded below, and P(lambda)f is single-valued for every lambda > 0, the function must be convex. Similarly, we show that the function f is 1/mu-strongly convex if the farthest mapping Q(mu)f is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in R(n). We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin. (C) 2010 Elsevier Inc. All rights reserved.
Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the com...
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Many iterative optimization algorithms involve compositions of special cases of Lipschitz continuous operators, namely firmly nonexpansive, averaged, and nonexpansive operators. The structure and properties of the compositions are of particular importance in the proofs of convergence of such algorithms. In this paper, we systematically study the compositions of further special cases of Lipschitz continuous operators. Applications of our results include compositions of scaled conically nonexpansive mappings, as well as the Douglas-Rachford and forward-backward operators, when applied to solve certain structured monotone inclusion and optimization problems. Several examples illustrate and tighten our conclusions.
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