A Nesterov Type Algorithm with Double Tikhonov Regularization: Fast Convergence of the Function Values and Strong Convergence to the Minimal Norm Solution

Abstract

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function f. We show that the generated sequences converge strongly to the minimal norm element from \(\text {argmin}f\). We also show fast convergence for the potential energies \(f(x_n)-\text {min}f\) and \(f(y_n)-\text {min}f\), where \((x_n),\,(y_n)\) are the sequences generated by our algorithm. Further we obtain fast convergence to zero of the discrete velocity and some estimates concerning the value of the gradient of the objective function in the generated sequences. Via some numerical experiments we show that we need both Tikhonov regularization terms in our algorithm in order to obtain the strong convergence of the generated sequences to the minimum norm minimizer of our objective function.