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First-order methods with momentum such as Nesterov's fast gradient method arevery useful for convex optimization problems, but can exhibit undesirableoscillations yielding slow convergence rates for some applications. An adaptiverestarting scheme can improve the convergence rate of the fast gradient method,when the parameter of a strongly convex cost function is unknown or when theiterates of the algorithm enter a locally strongly convex region. Recently, weintroduced the optimized gradient method, a first-order algorithm that has aninexpensive per-iteration computational cost similar to that of the fastgradient method, yet has a worst-case cost function rate that is twice fasterthan that of the fast gradient method and that is optimal for large-dimensionalsmooth convex problems. Building upon the success of accelerating the fastgradient method using adaptive restart, this paper investigates similarheuristic acceleration of the optimized gradient method. We first derive a newfirst-order method that resembles the optimized gradient method for stronglyconvex quadratic problems with known function parameters, yielding a linearconvergence rate that is faster than that of the analogous version of the fastgradient method. We then provide a heuristic analysis and numerical experimentsthat illustrate that adaptive restart can accelerate the convergence of theoptimized gradient method. Numerical results also illustrate that adaptiverestart is helpful for a proximal version of the optimized gradient method fornonsmooth composite convex functions.

Adaptive Restart of the Optimized Gradient Method for Convex Optimization