A proximal-gradient inertial algorithm with Tikhonov regularization: strong convergence to the minimal norm solution
<|reference_start|>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 $(x_k)$ 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 $\mathcal{O}(k^{-2})$ for the potential energy $(f+g)(x_k)-\min(f+g)$ can be obtained.<|reference_end|>
@article{lászló2024a,
    title={A proximal-gradient inertial algorithm with Tikhonov regularization:
  strong convergence to the minimal norm solution},
    author={Szil'ard Csaba L'aszl'o},
    journal={arXiv preprint arXiv:2407.10350},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.10350},
    primaryClass={math.OC cs.NA math.NA}
}