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 $(\alpha_{ n})_{ n}$
Categories: math.OC