Convergence rates for an inertial algorithm of gradient type associated to a smooth non-convex minimization

• Published in: Mathematical programming Vol. 190; no. 1-2; pp. 285 - 329
• Main Author: László, Szilárd Csaba
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2021; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Convergence; Critical point; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Regularization; Sequences; Theoretical; Łojasiewicz exponent; Non-convex optimization; 90C30; Convergence rate; 65K10; inertial algorithm; 90C26; Kurdyka–Łojasiewicz inequality
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-020-01534-w

We investigate an inertial algorithm of gradient type in connection with the minimization of a non-convex differentiable function. The algorithm is formulated in the spirit of Nesterov’s accelerated convex gradient method. We prove some abstract convergence results which applied to our numerical scheme allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. Further, we obtain convergence rates for the generated sequences and the objective function values formulated in terms of the Łojasiewicz exponent of a regularization of the objective function. Finally, some numerical experiments are presented in order to compare our numerical scheme and some algorithms well known in the literature.