A proximal-gradient inertial algorithm with Tikhonov regularization: strong convergence to the minimal norm solution

We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection to the minimization problem of the sum of a convex lower semi-continuous function f and a smooth convex function g. For the appropriate setting of the parameters we provide strong convergence of the generated sequence (xₖ) to the minimum norm minimizer of our objective function $f+g$. Further, we obtain fast convergence to zero of the objective function values in a generated sequence but also for the discrete velocity and the sub-gradient of the objective function. We also show that for another settings of the parameters the optimal rate of order O(k⁻²) for the potential energy (f+g)(xₖ)-min(f+g) can be obtained.