Viscosity approximation method for a variational problem

TL;DR

This paper introduces a viscosity approximation iterative method for solving variational problems involving nonexpansive and inverse strongly monotone operators in Hilbert spaces, proving strong convergence under certain conditions.

Contribution

The paper develops a new iterative algorithm combining viscosity approximation with projection methods and proves its strong convergence for variational problems in Hilbert spaces.

Findings

The iterative sequence converges strongly to a solution of the variational problem.

The convergence is maintained under perturbations of the algorithm.

Numerical experiments show the impact of parameter choices on convergence rate.

Abstract

Let $Q$ be a nonempty closed and convex subset of a real Hilbert space $% \mathcal{H}$, $S:Q\rightarrow Q$ a nonexpansive mapping, $A:Q\rightarrow Q$ an inverse strongly monotone operator, and $f:Q\rightarrow Q$ a contraction mapping. We prove, under appropriate conditions on the real sequences $% \{\alpha_{n}\}$ and $\{\lambda_{n}\},$ that for any starting point $x_{1}$ in $Q,$ the sequence $\{x_{n}\}$ generated by the iterative process \begin{equation} x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})SP_{Q}(x_{n}-\lambda_{n}Ax_{n}) \label{Alg} \end{equation} converges strongly to a particular element of the set $F_{ix}(S)\cap S_{VI(A,Q)}$ which we suppose that it is nonempty, where $F_{ix}(S)$ is the set of fixed point of the mapping $% S$ and $S_{VI(A,Q)}$ is the set of $q\in Q$ such that $\langle Aq,x-q\rangle\geq0$ for every $x\in Q.$ Moreover, we study the strong convergence of a perturbed version of the algorithm generated by the above process. Finally, we apply the main result to construct an algorithm associated to a constrained convex optimization problem and we provide a numerical experiment to emphasize the effect of the parameter $\{\alpha_{n}\}$ on the convergence rate of this algorithm.