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Envelope Functions: Unifications and Further Properties

Author

Listed:
  • Pontus Giselsson

    (Lund University)

  • Mattias Fält

    (Lund University)

Abstract

Forward–backward and Douglas–Rachford splitting are methods for structured nonsmooth optimization. With the aim to use smooth optimization techniques for nonsmooth problems, the forward–backward and Douglas–Rachford envelopes where recently proposed. Under specific problem assumptions, these envelope functions have favorable smoothness and convexity properties and their stationary points coincide with the fixed-points of the underlying algorithm operators. This allows for solving such nonsmooth optimization problems by minimizing the corresponding smooth convex envelope function. In this paper, we present a general envelope function that unifies and generalizes existing ones. We provide properties of the general envelope function that sharpen corresponding known results for the special cases. We also present a new interpretation of the underlying methods as being majorization–minimization algorithms applied to their respective envelope functions.

Suggested Citation

  • Pontus Giselsson & Mattias Fält, 2018. "Envelope Functions: Unifications and Further Properties," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 673-698, September.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:3:d:10.1007_s10957-018-1328-z
    DOI: 10.1007/s10957-018-1328-z
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    References listed on IDEAS

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    1. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
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    Cited by:

    1. Yanli Liu & Wotao Yin, 2019. "An Envelope for Davis–Yin Splitting and Strict Saddle-Point Avoidance," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 567-587, May.
    2. Ghaderi, Susan & Ahookhosh, Masoud & Arany, Adam & Skupin, Alexander & Patrinos, Panagiotis & Moreau, Yves, 2024. "Smoothing unadjusted Langevin algorithms for nonsmooth composite potential functions," Applied Mathematics and Computation, Elsevier, vol. 464(C).
    3. Aviad Aberdam & Amir Beck, 2022. "An Accelerated Coordinate Gradient Descent Algorithm for Non-separable Composite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 219-246, June.

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