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

Abstract

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 $(\alpha_{n})_{n}$