Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces
For researchers in optimization and monotone operator theory, this provides a general framework to obtain strong convergence without restrictive assumptions like strong monotonicity, addressing a known limitation of proximal splitting algorithms.
This paper proposes modified proximal splitting algorithms (Krasnosel'skiĭ-Mann, forward-backward, Douglas-Rachford) with Tikhonov regularization that achieve strong convergence to the minimal norm solution in Hilbert spaces, overcoming the typical weak convergence guarantee. The methods are extended to primal-dual settings for structured monotone inclusions and convex optimization.
Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity (respectively, strong convexity for optimization problems). In this paper, we propose a modified Krasnosel'ski\uı--Mann algorithm in connection with the determination of a fixed point of a nonexpansive mapping and show strong convergence of the iteratively generated sequence to the minimal norm solution of the problem. Relying on this, we derive a forward-backward and a Douglas-Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. Furthermore, we formulate strong convergent primal-dual algorithms of forward-backward and Douglas-Rachford-type for highly structured monotone inclusion problems involving parallel-sums and compositions with linear operators. The resulting iterative schemes are particularized to the solving of convex minimization problems.