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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. Adrian S. Lewis & Jérôme Malick, 2008. "Alternating Projections on Manifolds," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 216-234, February.
    4. 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.
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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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