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

• Published in: Optimization methods & software Vol. 40; no. 4; pp. 947 - 976
• Main Author: László, Szilárd Csaba
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 04.07.2025
• Subjects: convex optimization; Inertial algorithm; optimal rate; proximal-gradient algorithm; strong convergence; Tikhonov regularization
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2025.2517172

We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection with 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 the strong convergence of the generated sequence $ (x_k){_{k\ge 0}} $ ( x k ) k ≥ 0 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 setting of the parameters the optimal rate of order $ \mathcal {O}(k^{-2}) $ O ( k − 2 ) for the potential energy $ (f+g)(x_k)-\min (f+g) $ ( f + g ) ( x k ) − min ( f + g ) can be obtained.