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

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.