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In a Hilbert setting, we develop a gradient-based dynamic approach for fastsolving convex optimization problems. By applying time scaling, averaging, andperturbation techniques to the continuous steepest descent (SD), we obtainhigh-resolution ODEs of the Nesterov and Ravine methods. These dynamics involveasymptotically vanishing viscous damping and Hessian driven damping (either inexplicit or implicit form). Mathematical analysis does not require developing aLyapunov analysis for inertial systems. We simply exploit classical convergenceresults for (SD) and its external perturbation version, then use tools ofdifferential and integral calculus, including Jensen's inequality. The methodis flexible and by way of illustration we show how it applies starting fromother important dynamics in optimization. We consider the case where theinitial dynamics is the regularized Newton method, then the case where thestarting dynamics is the differential inclusion associated with a convex lowersemicontinuous potential, and finally we show that the technique can benaturally extended to the case of a monotone cocoercive operator. Our approachleads to parallel algorithmic results, which we study in the case of fastgradient and proximal algorithms. Our averaging technique shows new linksbetween the Nesterov and Ravine methods.

Fast convex optimization via time scale and averaging of the steepest descent