A nested primal–dual iterated Tikhonov method for regularized convex optimization

Stefano Aleotti (), Silvia Bonettini (), Marco Donatelli (), Marco Prato () and Simone Rebegoldi ()
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Stefano Aleotti: Università dell’Insubria
Silvia Bonettini: Università di Modena e Reggio Emilia
Marco Donatelli: Università dell’Insubria
Marco Prato: Università di Modena e Reggio Emilia
Simone Rebegoldi: Università di Modena e Reggio Emilia

Computational Optimization and Applications, 2025, vol. 91, issue 2, No 2, 357-395

Abstract: Abstract Proximal–gradient methods are widely employed tools in imaging that can be accelerated by adopting variable metrics and/or extrapolation steps. One crucial issue is the inexact computation of the proximal operator, often implemented through a nested primal–dual solver, which represents the main computational bottleneck whenever an increasing accuracy in the computation is required. In this paper, we propose a nested primal–dual method for the efficient solution of regularized convex optimization problems. Our proposed method approximates a variable metric proximal–gradient step with extrapolation by performing a prefixed number of primal–dual iterates, while adjusting the steplength parameter through an appropriate backtracking procedure. Choosing a prefixed number of inner iterations allows the algorithm to keep the computational cost per iteration low. We prove the convergence of the iterates sequence towards a solution of the problem, under a relaxed monotonicity assumption on the scaling matrices and a shrinking condition on the extrapolation parameters. Furthermore, we investigate the numerical performance of our proposed method by equipping it with a scaling matrix inspired by the Iterated Tikhonov method. The numerical results show that the combination of such scaling matrices and Nesterov-like extrapolation parameters yields an effective acceleration towards the solution of the problem.

Keywords: Primal–dual methods; Iterated Tikhonov; Convex optimization; Image deblurring (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-024-00613-4

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