Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping

• Published in: Optimization Vol. 72; no. 5; pp. 1199 - 1238
• Main Authors: Attouch, Hedy, Chbani, Zaki, Fadili, Jalal, Riahi, Hassan
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04.05.2023; Taylor & Francis LLC
• Subjects: Computational geometry; Convergence; Convergence of iterates; Convexity; First order algorithms; Hessian driven damping; Hilbert space; inertial optimization algorithms; Mathematics; Nesterov accelerated gradient method; Numerical Analysis; Optimization; Optimization and Control; time rescaling; Viscous damping; inertial optimization algorithms; Convergence of iterates; Nesterov accelerated gradient method; Hessian driven damping; time rescaling
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2021.2009828

In a Hilbert space setting, for convex optimization, we show the convergence of the iterates to optimal solutions for a class of accelerated first-order algorithms. They can be interpreted as discrete temporal versions of an inertial dynamic involving both viscous damping and Hessian-driven damping. The asymptotically vanishing viscous damping is linked to the accelerated gradient method of Nesterov while the Hessian driven damping makes it possible to significantly attenuate the oscillations. By treating the Hessian-driven damping as the time derivative of the gradient term, this gives, in discretized form, first-order algorithms. These results complement the previous work of the authors where it was shown the fast convergence of the values, and the fast convergence towards zero of the gradients.