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Journal of Convex Analysis 29 (2022), No. 3, 893--920
Copyright Heldermann Verlag 2022

Fast Convergence of Generalized Forward-Backward Algorithms for Structured Monotone Inclusions

Paul-Emile Maingé
Université des Antilles, Martinique, France


We develop rapidly convergent forward-backward algorithms for computing zeroes of the sum of finitely many maximally monotone operators. A modification of the classical forward-backward method for two general operators is first considered, by incorporating an inertial term (close to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates $(x_n)$, with worst-case rates of $o(n^{-1})$ in terms of both the discrete velocity and the fixed point residual, instead of the classical rates of ${\cal O}(n^{-1/2})$ established so far for related algorithms. Our procedure is then adapted to more general monotone inclusions and a fast primal-dual algorithm is proposed for solving convex-concave saddle point problems.

Keywords: Nesterov-type algorithm, inertial-type algorithm, global rate of convergence, fast first-order method, relaxation factors, correction term, accelerated proximal algorithm, fixed point problem.

MSC: 90C25, 90C30, 90C60, 68Q25, 49M25.

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