Convergence of a splitting inertial proximal method for monotone operators

By Abdellatif Moudafi, M. Oliny. 2003

319 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5dd3f20236c3a0e1b6f1cf7bbf6c3914b1f23aff

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In Computational Theory and Mathematics

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In Numerical Analysis

Citations in the corpus

A Hybrid Proximal-Extragradient Algorithm with Inertial Effects (2014, background): “(iii) The inertial forward-backward algorithm for finding the zeros of T := A+B, where A : H → H is a γ-cocoercive operator with γ > 0 and B : H⇒ H is a maximally monotone operator reads in the error-free case (see [20]):”
A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions (2017, methodology): “In [41], Moudafi and Oliny introduced an additional single-valued, cocoercive and Lipschitz continuous operator A into the inertial proximal point algorithm: { yn = xn + αn(xn − xn−1) xn+1 = (I + rnB)−1(yn − rnAxn), n ≥ 1.”
An Inertial Forward-Backward Algorithm for Monotone Inclusions (2014, background): “In subsequent work [33], Moudafi and Oliny introduced an additional single-valued and Lipschitz continuous operator B into the inertial proximal point algorithm: { yk = xk + αk(x − xk−1) xk+1 = (Id +λk A)−1(yk − λk B(xk)) , (6)”
On the convergence of the iterates of "FISTA" (2015, methodology): “This part is strongly inspired from a recent paper of Pock and Lorenz [9], inspired by works of Alvarez and Attouch [1] and Moufadi and Oliny [10].”
The Rate of Convergence of Nesterov's Accelerated Forward-Backward Method is Actually Faster Than 1/k2 (2015, methodology): “This algorithm is closely connected with proximal-based inertial algorithms [1, 14, 23].”
Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity (2018, methodology): “This algorithm is within the scope of the proximal-based inertial algorithms [3], [24], [33], and forward-backward methods.”
An introduction to continuous optimization for imaging (2016, unknown): “The heavy ball method has been generalized to monotone operators by Alvarez and Attouch (2001) (see also Alvarez 2003, Moudafi and Oliny 2003), so there exist general convergence results that allow for non-smooth terms.”
Convergence theorems for inertial KM-type algorithms (2008, methodology): “In particular, an inertial type extrapolation was recently proposed as an acceleration process [1–3,12,13,15,16].”
iPiano: Inertial Proximal Algorithm for Nonconvex Optimization (2014, methodology): “In a subsequent work [34], it has been applied to a forward-backward splitting algorithm, again in the general framework of maximal monotone operators.”
A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces (2019, background): “To satisfy Condition (A2) of the sequence {xn}, it is necessary to first calculate θn at each step (see [32]).”
An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces (2018, unknown): “In [31], Moudafi and Oliny proposed the following inertial proximal point algorithm for solving the zero-finding problem of the sum of two monotone operators: { yn = xn + θn(xn − xn−1) xn+1 = (I + rnB)−1(yn − rnAxn), n ≥ 1, (1.”
On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm” (2015, methodology): “This part is strongly inspired from a recent paper of Pock and Lorenz [11], inspired by works of Alvarez and Attouch [12] and Moufadi and Oliny [13].”
Inertial Douglas-Rachford splitting for monotone inclusion problems (2014, methodology): “roximal algorithm is that the next iterate is dened by making use of the last two iterates. Since its introduction, one can notice an increasing interest in algorithms having this particularity, see [2{4,11,17,27,28, 31]. In order to prove the convergence of the proposed inertial Douglas-Rachford al”