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

Puya Latafat (), Andreas Themelis (), Silvia Villa () and Panagiotis Patrinos ()
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Puya Latafat: IMT School for Advanced Studies Lucca
Andreas Themelis: Kyushu University
Silvia Villa: Università di Genova
Panagiotis Patrinos: KU Leuven

Journal of Optimization Theory and Applications, 2025, vol. 204, issue 3, No 16, 36 pages

Abstract: 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.

Keywords: Convex optimization; Bilevel programming; Adaptive proximal gradient methods; Locally Lipschitz gradient; 65K05; 90C06; 90C25; 90C30 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-024-02564-6

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