On The Strong Convergence of The Gradient Projection Algorithm with Tikhonov regularizing term

Ramzi May

Published 2019-10-17Version 1

We investigate the strong and the weak convergence properties of the following gradient projection algorithm with Tikhonov regularizing term \[ x_{n+1}=P_{Q}(x_{n}-\gamma_{n}\nabla f(x_{n})-\gamma_{n}\alpha_{n}\nabla \phi (x_{n})), \] where $P_{Q}$ is the projection operator from a Hilbert space $\mathcal{H}$ onto a given nonempty, closed and convex subset $Q,$ $f:\mathcal{H}% \rightarrow \mathbb{R}$ a regular convex function, $\phi :\mathcal{H}% \rightarrow \mathbb{R}$ a regular strongly convex function, and $\gamma_{n}$ and $\alpha_{n}$ are positive real numbers. Following a Lyuapunov approach inspired essentially from the paper [Comminetti R, Peypouquet J Sorin S. Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization. J. Differential Equations. (2001); 245:3753-3763], we establish the strong convergence of $(x_{n})_{n}$ to a particular minimizer $x^{\ast }$ of $f$ on $Q$ under some simple and natural conditions on the objective function $f$\ and the sequences $(\gamma_{n})_{n}$ and $(\alpha_{n})_{n}$