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The Proximal Alternating Minimization Algorithm for Two-Block Separable Convex Optimization Problems with Linear Constraints

Author

Listed:
  • Sandy Bitterlich

    (Chemnitz University of Technology)

  • Radu Ioan Boţ

    (University of Vienna)

  • Ernö Robert Csetnek

    (University of Vienna
    University of Göttingen)

  • Gert Wanka

    (Chemnitz University of Technology)

Abstract

The Alternating Minimization Algorithm has been proposed by Paul Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of this method does not usually correspond to the calculation of a proximal operator through a closed formula affects the implementability of the algorithm. In this paper, we allow in each block of the objective a further smooth convex function and propose a proximal version of the algorithm, which is achieved by equipping the algorithm with proximal terms induced by variable metrics. For suitable choices of the latter, the solving of the two subproblems in the iterative scheme can be reduced to the computation of proximal operators. We investigate the convergence of the proposed algorithm in a real Hilbert space setting and illustrate its numerical performances on two applications in image processing and machine learning.

Suggested Citation

  • Sandy Bitterlich & Radu Ioan Boţ & Ernö Robert Csetnek & Gert Wanka, 2019. "The Proximal Alternating Minimization Algorithm for Two-Block Separable Convex Optimization Problems with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 110-132, July.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-018-01454-y
    DOI: 10.1007/s10957-018-01454-y
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    References listed on IDEAS

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    1. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    2. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2016. "A block coordinate variable metric forward–backward algorithm," Journal of Global Optimization, Springer, vol. 66(3), pages 457-485, November.
    3. Kristian Bredies & Hongpeng Sun, 2017. "A Proximal Point Analysis of the Preconditioned Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 878-907, June.
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