Accelerated forward–backward algorithms for structured monotone inclusions

Paul-Emile Maingé () and André Weng-Law ()
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Paul-Emile Maingé: F.W.I., MEMIAD, Université des Antilles
André Weng-Law: F.W.I., MEMIAD, Université des Antilles

Computational Optimization and Applications, 2024, vol. 88, issue 1, No 6, 167-215

Abstract: Abstract In this paper, we develop rapidly convergent forward–backward algorithms for computing zeroes of the sum of two maximally monotone operators. A modification of the classical forward–backward method is considered, by incorporating an inertial term (closed to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term, along with a preconditioning process. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates $$(x_n)$$ ( x n ) , with worst-case rates of $$ o(n^{-1})$$ o ( n - 1 ) in terms of both the discrete velocity and the fixed point residual, instead of the rates of $$\mathcal {O}(n^{-1/2})$$ O ( n - 1 / 2 ) classically established for related algorithms. Our procedure can be also adapted to more general monotone inclusions. In particular, we propose a fast primal-dual algorithmic solution to some class of convex-concave saddle point problems. In addition, we provide a well-adapted framework for solving this class of problems by means of standard proximal-like algorithms dedicated to structured monotone inclusions. Numerical experiments are also performed so as to enlighten the efficiency of the proposed strategy.

Keywords: Nesterov-type algorithm; Optimal gradient method; Inertial-type algorithm; Global rate of convergence; Fast first-order method; Restarting techniques; 90C25; 90C06; 65F22 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-023-00547-3

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