On the convergence of an inertial proximal algorithm with a Tikhonov regularization term

TL;DR

This paper analyzes an inertial proximal algorithm with Tikhonov regularization, demonstrating fast convergence of the objective function to the global minimum and establishing conditions for strong convergence of the generated sequences.

Contribution

It provides new convergence results for an inertial proximal algorithm with Tikhonov regularization, including conditions for strong convergence and detailed parameter analysis.

Findings

Objective function values converge rapidly to the global minimum.

Weak convergence of the sequences to a minimizer is established.

Strong convergence depends on a critical relationship between parameters.

Abstract

This paper deals with an inertial proximal algorithm that contains a Tikhonov regularization term, in connection to the minimization problem of a convex lower semicontinuous function $f$. We show that for appropriate Tikhonov regularization parameters the value of the objective function in the sequences generated by our algorithm converges fast (with arbitrary rate) to the global minimum of the objective function and the generated sequences converges weakly to a minimizer of the objective function. We also obtain the fast convergence of the discrete velocities towards zero and some sum estimates. Nevertheless, our main goal is to obtain strong convergence results and also pointwise and sum estimates for the same constellation of the parameters involved. Our analysis reveals that the extrapolation coefficient and the Tikhonov regularization coefficient are strongly correlated and there is a critical setting of the parameters that separates the cases when strong respective weak convergence results can be obtained.