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On the Convergence of Proximal Gradient Methods for Convex Simple Bilevel Optimization

Author

Listed:
  • Puya Latafat

    (IMT School for Advanced Studies Lucca)

  • Andreas Themelis

    (Kyushu University)

  • Silvia Villa

    (Università di Genova)

  • Panagiotis Patrinos

    (KU Leuven)

Abstract

This paper studies proximal gradient iterations for addressing simple bilevel optimization problems where both the upper and the lower level cost functions are split as the sum of differentiable and (possibly nonsmooth) prox-friendly functions. We develop a novel convergence recipe for iteration-varying stepsizes that relies on Barzilai-Borwein type local estimates for the differentiable terms. Leveraging the convergence recipe, under global Lipschitz gradient continuity, we establish convergence for a nonadaptive stepsize sequence, without requiring any strong convexity or linesearch. In the locally Lipschitz differentiable setting, we develop an adaptive linesearch method that introduces a systematic adaptive scheme enabling large and nonmonotonic stepsize sequences while being insensitive to the choice of hyperparameters and initialization. Numerical simulations are provided showcasing favorable convergence speed of our methods.

Suggested Citation

  • Puya Latafat & Andreas Themelis & Silvia Villa & Panagiotis Patrinos, 2025. "On the Convergence of Proximal Gradient Methods for Convex Simple Bilevel Optimization," Journal of Optimization Theory and Applications, Springer, vol. 204(3), pages 1-36, March.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:3:d:10.1007_s10957-024-02564-6
    DOI: 10.1007/s10957-024-02564-6
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Juan Peypouquet, 2012. "Coupling the Gradient Method with a General Exterior Penalization Scheme for Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 123-138, April.
    3. Lorenzo Lampariello & Gianluca Priori & Simone Sagratella, 2022. "On the solution of monotone nested variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 421-446, December.
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