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

Abstract

In a real Hilbert space $\mathcal{H}$. Given any function $f$ convex differentiable whose solution set ${\mathrm{argmin}}_{\mathcal{H}}f$ 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 ${\mathrm{argmin}}_{\mathcal{H}}f$, we also obtain a convergence rate of gradient toward zero. Afterward, we extend these results to non-smooth convex functions with extended real values.