Tikhonov Regularized Iterative Methods for Nonlinear Problems

TL;DR

This paper introduces new Tikhonov regularized iterative algorithms for solving nonlinear monotone inclusion problems, achieving strong convergence without requiring strong convexity or monotonicity assumptions.

Contribution

It proposes a unified fixed point algorithm based on normal S-iteration with Tikhonov regularization, along with new primal-dual algorithms for complex monotone problems, ensuring strong convergence.

Findings

Proposed algorithms achieve strong convergence without strong assumptions.

Numerical experiments demonstrate effectiveness in image deblurring.

New methods extend applicability to complex structured problems.

Abstract

We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of nonexpansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward-backward-type algorithm and a Douglas-Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward-backward-type primal-dual algorithm and a Douglas-Rachford-type primal-dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.