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Constraint Reduction Reformulations for Projection Algorithms with Applications to Wavelet Construction

Author

Listed:
  • Minh N. Dao

    (Federation University Australia)

  • Neil D. Dizon

    (University of Newcastle)

  • Jeffrey A. Hogan

    (University of Newcastle)

  • Matthew K. Tam

    (The University of Melbourne)

Abstract

We introduce a reformulation technique that converts a many-set feasibility problem into an equivalent two-set problem. This technique involves reformulating the original feasibility problem by replacing a pair of its constraint sets with their intersection, before applying Pierra’s classical product space reformulation. The step of combining the two constraint sets reduces the dimension of the product spaces. We refer to this technique as the constraint reduction reformulation and use it to obtain constraint-reduced variants of well-known projection algorithms such as the Douglas–Rachford algorithm and the method of alternating projections, among others. We prove global convergence of constraint-reduced algorithms in the presence of convexity and local convergence in a nonconvex setting. In order to analyze convergence of the constraint-reduced Douglas–Rachford method, we generalize a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. Finally, we apply the constraint-reduced versions of Douglas–Rachford and alternating projections to solve the wavelet feasibility problems and then compare their performance with their usual product variants.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:joptap:v:190:y:2021:i:1:d:10.1007_s10957-021-01878-z
    DOI: 10.1007/s10957-021-01878-z
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    References listed on IDEAS

    as
    1. Minh N. Dao, & Hung M. Phan, 2019. "Linear Convergence of Projection Algorithms," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 715-738, May.
    2. Minh N. Dao & Hung M. Phan, 2018. "Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems," Journal of Global Optimization, Springer, vol. 72(3), pages 443-474, November.
    3. 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.
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    Cited by:

    1. Rubén Campoy, 2022. "A product space reformulation with reduced dimension for splitting algorithms," Computational Optimization and Applications, Springer, vol. 83(1), pages 319-348, September.

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