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Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings

Author

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  • D. Russell Luke

    (Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083 Göttingen, Germany)

  • Nguyen H. Thao

    (Delft Center for Systems and Control, Delft University of Technology, 2628 CD Delft, Netherlands)

  • Matthew K. Tam

    (Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083 Göttingen, Germany)

Abstract

We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged mappings to expansive set-valued mappings that characterizes a type of strong calmness of the fixed point mapping. The second component to this analysis is an extension of the well-established notion of metric subregularity—or inverse calmness—of the mapping at fixed points. Convergence of expansive fixed point iterations is proved using these two properties, and quantitative estimates are a natural by-product of the framework. To demonstrate the application of the theory, we prove, for the first time, a number of results showing local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems, local linear convergence of the forward-backward algorithm for structured optimization without convexity, strong or otherwise, and local linear convergence of the Douglas-Rachford algorithm for structured nonconvex minimization. This theory includes earlier approaches for known results, convex and nonconvex, as special cases.

Suggested Citation

  • D. Russell Luke & Nguyen H. Thao & Matthew K. Tam, 2018. "Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1143-1176, November.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:4:p:1143-1176
    DOI: 10.1287/moor.2017.0898
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    References listed on IDEAS

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    1. Alexander Y. Kruger & Nguyen H. Thao, 2015. "Quantitative Characterizations of Regularity Properties of Collections of Sets," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 41-67, January.
    2. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Teemu Pennanen, 2002. "Local Convergence of the Proximal Point Algorithm and Multiplier Methods Without Monotonicity," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 170-191, February.
    4. Adrian S. Lewis & Jérôme Malick, 2008. "Alternating Projections on Manifolds," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 216-234, February.
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    Cited by:

    1. Minh N. Dao & Neil D. Dizon & Jeffrey A. Hogan & Matthew K. Tam, 2021. "Constraint Reduction Reformulations for Projection Algorithms with Applications to Wavelet Construction," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 201-233, July.
    2. Minh N. Dao, & Hung M. Phan, 2019. "Linear Convergence of Projection Algorithms," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 715-738, May.

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