Combining Strong Convergence, Values Fast Convergence and Vanishing of Gradients for a Proximal Point Algorithm Using Tikhonov Regularization in a Hilbert Space

TL;DR

This paper introduces a proximal point algorithm with Tikhonov regularization in Hilbert spaces, achieving strong convergence, fast value convergence, and vanishing gradients for convex functions, including non-smooth cases.

Contribution

It develops a novel proximal algorithm that guarantees strong convergence and optimal rates for convex and non-smooth functions using Tikhonov regularization.

Findings

Objective function values converge at rate O(1/β_k).

Generated sequences strongly converge to the minimum norm solution.

Gradient norms tend to zero, indicating optimality.

Abstract

In a real Hilbert space $\mathcal{H}$. Given any function $f$ convex differentiable whose solution set $\argmin_{\mathcal{H}}\,f$ is nonempty, by considering the Proximal Algorithm $x_{k+1}=\text{prox}_{\b_k f}(d x_k)$, where $0<d<1$ and $(\b_k)$ is nondecreasing function, and by assuming some assumptions on $(\b_k)$, we will show that the value of the objective function in the sequence generated by our algorithm converges in order $\mathcal{O} \left( \frac{1}{ \beta _k} \right)$ to the global minimum of the objective function, and that the generated sequence converges strongly to the minimum norm element of $\argmin_{\mathcal{H}}\,f$, we also obtain a convergence rate of gradient toward zero. Afterward, we extend these results to non-smooth convex functions with extended real values.