Accelerated Inertial Gradient Algorithms with Vanishing Tikhonov Regularization
par
XR203
XLIM
We study a first-order gradient algorithm to find the minimum-norm solution of a smooth convex minimization problem with Lipschitz continuous gradient. The algorithm is derived via an explicit time discretization of a damped inertial system with vanishing Tikhonov regularization. For general Tikhonov regularization parameters, we establish a Lyapunov-type analysis under appropriate control of the decay rate of the Tikhonov term. For polynomial choice of Tikhonov terms $\varepsilon_k = k^{−p}$ with $0 < p < 2$, we provide a fast convergence rate for the objective values and prove the strong convergence of the iterates to the minimum-norm minimizer. In the critical case $p = 2$, our analysis ensures the fast convergence of the objective values, but it does not guarantee the strong convergence to the minimum-norm minimizer.