From differential equation solvers to accelerated first-order methods for convex optimization
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Publication:6324832
DOI10.1007/S10107-021-01713-3zbMATH Open1515.65157arXiv1909.03145WikidataQ115385301 ScholiaQ115385301MaRDI QIDQ6324832
Publication date: 6 September 2019
Abstract: Convergence analysis of accelerated first-order methods for convex optimization problems are presented from the point of view of ordinary differential equation solvers. A new dynamical system, called Nesterov accelerated gradient flow, has been derived from the connection between acceleration mechanism and -stability of ODE solvers, and the exponential decay of a tailored Lyapunov function along with the solution trajectory is proved. Numerical discretizations are then considered and convergence rates are established via a unified discrete Lyapunov function. The proposed differential equation solver approach can not only cover existing accelerated methods, such as FISTA, G"{u}ler's proximal algorithm and Nesterov's accelerated gradient method, but also produce new algorithms for composite convex optimization that possess accelerated convergence rates.
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