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This paper provides a theoretical and numerical comparison of classicalfirst-order splitting methods for solving smooth convex optimization problemsand cocoercive equations. From a theoretical point of view, we compareconvergence rates of gradient descent, forward-backward, Peaceman-Rachford, andDouglas-Rachford algorithms for minimizing the sum of two smooth convexfunctions when one of them is strongly convex. A similar comparison is given inthe more general cocoercive setting under the presence of strong monotonicityand we observe that the convergence rates in optimization are strictly betterthan the corresponding rates for cocoercive equations for some algorithms. Weobtain improved rates with respect to the literature in several instances byexploiting the structure of our problems. Moreover, we indicate which algorithmhas the lowest convergence rate depending on strong convexity and cocoerciveparameters. From a numerical point of view, we verify our theoretical resultsby implementing and comparing previous algorithms in well-established signaland image inverse problems involving sparsity. We replace the widely used$\ell_1$ norm with the Huber loss and we observe that fully proximal-basedstrategies have numerical and theoretical advantages with respect to methodsusing gradient steps.

Theoretical and numerical comparison of first-order algorithms for cocoercive equations and smooth convex optimization