A nested primal–dual iterated Tikhonov method for regularized convex optimization.
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| Title: | A nested primal–dual iterated Tikhonov method for regularized convex optimization. |
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| Authors: | Aleotti, Stefano1 (AUTHOR) saleotti@studenti.uninsubria.it, Bonettini, Silvia2 (AUTHOR) silvia.bonettini@unimore.it, Donatelli, Marco1 (AUTHOR) marco.donatelli@uninsubria.it, Prato, Marco2 (AUTHOR) marco.prato@unimore.it, Rebegoldi, Simone2 (AUTHOR) simone.rebegoldi@unimore.it |
| Source: | Computational Optimization & Applications. Jun2025, Vol. 91 Issue 2, p357-395. 39p. |
| Subjects: | Computational mathematics, Extrapolation, Algorithms |
| 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| 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. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 09266003 |
| DOI: | 10.1007/s10589-024-00613-4 |
