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Block-proximal methods with spatially adapted acceleration. (English) Zbl 1420.49034
Summary: We study and develop (stochastic) primal-dual block-coordinate descent methods for convex problems based on the method due to Chambolle and Pock. Our methods have known convergence rates for the iterates and the ergodic gap of \(O(1/N^2)\) if each block is strongly convex, \(O(1/N)\) if no convexity is present, and more generally a mixed rate \(O(1/N^2)+O(1/N)\) for strongly convex blocks if only some blocks are strongly convex. Additional novelties of our methods include blockwise-adapted step lengths and acceleration as well as the ability to update both the primal and dual variables randomly in blocks under a very light compatibility condition. In other words, these variants of our methods are doubly-stochastic. We test the proposed methods on various image processing problems, where we employ pixelwise-adapted acceleration.

49M29 Numerical methods involving duality
65K10 Numerical optimization and variational techniques
65K15 Numerical methods for variational inequalities and related problems
90C30 Nonlinear programming
90C47 Minimax problems in mathematical programming
ARock; iPiano
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