Nonlinear Forward-Backward Splitting with Momentum Correction

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The nonlinear, or warped, resolvent recently explored by Giselsson and Bùi-Combettes has been used to model a large set of existing and new monotone inclusion algorithms. To establish convergent algorithms based on these resolvents, corrective projection steps are utilized in both works. We present a different way of ensuring convergence by means of a nonlinear momentum term, which in many cases leads to cheaper per-iteration cost. The expressiveness of our method is demonstrated by deriving a wide range of special cases. These cases cover and expand on the forward-reflected-backward method of Malitsky-Tam, the primal-dual methods of Vũ-Condat and Chambolle-Pock, and the forward-reflected-Douglas-Rachford method of Ryu-Vũ. A new primal-dual method that uses an extra resolvent step is also presented as well as a general approach for adding momentum to any special case of our nonlinear forward-backward method, in particular all the algorithms listed above.

TidskriftSet-Valued and Variational Analysis
StatusPublished - 2023

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