2020/09/16 by Hédy Attouch, Attouch, Hedy, Aïcha Balhag +5 · 1 citation
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.2009.07620
openalex publication_date 2020/09/16 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28
In a Hilbert setting, we develop fast methods for convex unconstrained\noptimization. We rely on the asymptotic behavior of an inertial system\ncombining geometric damping with temporal scaling. The convex function to\nminimize enters the dynamic via its gradient. The dynamic includes three\ncoefficients varying with time, one is a viscous damping coefficient, the\nsecond is attached to the Hessian-driven damping, the third is a time scaling\ncoefficient. We study the convergence rate of the values under general\nconditions involving the damping and the time scale coefficients. The obtained\nresults are based on a new Lyapunov analysis and they encompass known results\non the subject. We pay particular attention to the case of an asymptotically\nvanishing viscous damping, which is directly related to the accelerated\ngradient method of Nesterov. The Hessian-driven damping significantly reduces\nthe oscillatory aspects. As a main result, we obtain an exponential rate of\nconvergence of values without assuming the strong convexity of the objective\nfunction. The temporal discretization of these dynamics opens the gate to a\nlarge class of inertial optimization algorithms.\n