• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Primal-dual block-proximal splitting for a class of non-convex problems

    Stanislav Mazurenko, Jyrki Jauhiainen, Tuomo Valkonen

ETNA - Electronic Transactions on Numerical Analysis, pp. 509-552, 2020/09/23

doi: 10.1553/etna_vol52s509

doi: 10.1553/etna_vol52s509


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doi:10.1553/etna_vol52s509



doi:10.1553/etna_vol52s509

Abstract

We develop block structure-adapted primal-dual algorithms for non-convexnon-smooth optimisation problems, whose objectives can be written as compositions$G(x)+F(K(x))$ of non-smooth block-separable convex functions $G$ and $F$ with anonlinear Lipschitz-differentiable operator $K$. Our methods are refinements ofthe nonlinear primal-dual proximal splitting method for such problems withoutthe block structure, which itself is based on the primal-dual proximal splittingmethod of Chambolle and Pock for convex problems. We propose individual steplength parameters and acceleration rules for each of the primal and dual blocksof the problem. This allows them to convergence faster by adapting to thestructure of the problem. For the squared distance of the iterates to a criticalpoint, we show local $O(1/N)$, $O(1/N^2)$, and linear rates under varyingconditions and choices of the step length parameters. Finally, we demonstrate the performance of the methods for the practical inverseproblems of diffusion tensor imaging and electrical impedance tomography.

Keywords: primal-dual algorithms, convex optimization, non-smooth optimization, step length

Published Online: 2020/09/23 09:26:01

Object Identifier: 0xc1aa5572 0x003bd91d

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