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A Relative Inexact Proximal Gradient Method with an Explicit Linesearch

Author

Listed:
  • Yunier Bello-Cruz

    (Northern Illinois University)

  • Max L. N. Gonçalves

    (Federal University of Goias)

  • Jefferson G. Melo

    (Federal University of Goias)

  • Cassandra Mohr

    (Northern Illinois University)

Abstract

This paper presents and investigates an inexact proximal gradient method for solving composite convex optimization problems characterized by an objective function composed of a sum of a full-domain differentiable convex function and a non-differentiable convex function. We introduce an explicit line search applied specifically to the differentiable component of the objective function, requiring only a relative inexact solution of the proximal subproblem per iteration. We prove the convergence of the sequence generated by our scheme and establish its iteration complexity, considering both the functional values and a residual associated with first-order stationary solutions. Additionally, we provide numerical experiments to illustrate the practical efficacy of our method.

Suggested Citation

  • Yunier Bello-Cruz & Max L. N. Gonçalves & Jefferson G. Melo & Cassandra Mohr, 2025. "A Relative Inexact Proximal Gradient Method with an Explicit Linesearch," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-32, July.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:1:d:10.1007_s10957-025-02684-7
    DOI: 10.1007/s10957-025-02684-7
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    References listed on IDEAS

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    1. M. Marques Alves & Jonathan Eckstein & Marina Geremia & Jefferson G. Melo, 2020. "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," Computational Optimization and Applications, Springer, vol. 75(2), pages 389-422, March.
    2. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    3. Jonathan Eckstein & Wang Yao, 2017. "Approximate ADMM algorithms derived from Lagrangian splitting," Computational Optimization and Applications, Springer, vol. 68(2), pages 363-405, November.
    4. Vando A. Adona & Max L. N. Gonçalves & Jefferson G. Melo, 2019. "A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 640-666, August.
    5. Yunier Bello-Cruz & Max L. N. Gonçalves & Nathan Krislock, 2023. "On FISTA with a relative error rule," Computational Optimization and Applications, Springer, vol. 84(2), pages 295-318, March.
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