Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces

TL;DR

This paper introduces modified proximal splitting algorithms with Tikhonov regularization that ensure strong convergence to minimal norm solutions in Hilbert spaces, improving upon traditional weak convergence guarantees.

Contribution

It proposes new strongly convergent algorithms for monotone inclusions and convex optimization, utilizing Tikhonov regularization and fixed point methods.

Findings

Algorithms achieve strong convergence to minimal norm solutions.

Derived primal-dual algorithms for structured monotone problems.

Applicable to convex minimization with enhanced convergence guarantees.

Abstract

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\u{\i}--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.