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A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity [article]

Yura Malitsky, Matthew K. Tam
2020 arXiv   pre-print
In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators.  ...  Our method converges under the same assumptions as Tseng's forward-backward-forward method, namely, it does not require cocoercivity of the single-valued operator.  ...  The second author's research was supported in part by a Fellowship from the Alexander von Humboldt Foundation and in part by a Discovery Early Career Research Award from the Australian Research Council  ... 
arXiv:1808.04162v4 fatcat:dwzgevhe4fevzlzplv2h6sxheq

Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions [article]

Janosch Rieger, Matthew K. Tam
2020 arXiv   pre-print
By specialising to two operator inclusions, we recover the forward-reflected-backward and the reflected-forward-backward splitting methods as particular cases.  ...  In this work, we propose and analyse two splitting algorithms for finding a zero of the sum of three monotone operators, one of which is assumed to be Lipschitz continuous.  ...  MKT is the recipient of a Discovery Early Career Research Award (DE200100063) from the Australian Research Council.  ... 
arXiv:2001.07327v1 fatcat:anomtifuabesvh6p5i4ye7zda4

On the Weak and Strong Convergence of a Conceptual Algorithm for Solving Three Operator Monotone Inclusions [article]

Yunier Bello-Cruz, Oday Hazaimah
2021 arXiv   pre-print
In this paper, a conceptual algorithm modifying the forward-backward-half-forward (FBHF) splitting method for solving three operator monotone inclusion problems is investigated.  ...  The FBHF splitting method adjusts and improves Tseng's forward-backward-forward (FBF) splitting method when the inclusion problem has a third-part operator that is cocoercive.  ...  In order to relax the cocoercivity assumption, Tseng [17] proposed a modification of the FB splitting method, known as the forward-backward-forward (FBF) splitting method, which usually requires L-Lipschitz  ... 
arXiv:2003.05604v2 fatcat:uw4kbtugpzhrvhwyuu2hrtf2gy

Nonlinear Forward-Backward Splitting with Projection Correction [article]

Pontus Giselsson
2021 arXiv   pre-print
The key of the method is the nonlinearity in the forward-backward step, where the backward part is based on a nonlinear resolvent construction that allows for the kernel in the resolvent to be a nonlinear  ...  In particular, we present a four-operator splitting method as a special case of NOFOB that relies nonlinearity and nonsymmetry in the forward-backward kernel.  ...  We have presented the versatile and conceptually simple algorithm NOFOB (nonlinear forward-backward splitting) for solving monotone inclusion problems.  ... 
arXiv:1908.07449v3 fatcat:z56skcydg5cxvlokqx3oud6fai

Dual three-operator splitting algorithms for solving composite monotone inclusion with applications to convex minimization

2021 Journal of Applied and Numerical Optimization  
In this paper, we study a monotone inclusion problem involving the mixtures of composite and parallel-sum type monotone operators with one of them being a cocoercive operator.  ...  Furthermore, we present two iterative algorithms to solve the composite monotone inclusion problem, and prove their convergence based on the inexact three-operator splitting algorithm.  ...  As a result, Vu [5] showed a new splitting algorithm for solving the composite monotone inclusions involving cocoercive operators, which depends on forward-backward splitting algorithms.  ... 
doi:10.23952/jano.3.2021.3.08 fatcat:zidkhcgxvrgllm7cmmmi4t7h2m

Distributed Forward-Backward Methods for Ring Networks [article]

Francisco J. Aragón-Artacho, Yura Malitsky, Matthew K. Tam, David Torregrosa-Belén
2022 arXiv   pre-print
In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in  ...  Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator.  ...  The authors are thankful to the anonymous referees for their careful reading and for providing very helpful comments.  ... 
arXiv:2112.00274v2 fatcat:dswvtqzqgffj3ehl7exgrssfgi

Nonlinear Forward-Backward Splitting with Momentum Correction [article]

Martin Morin, Sebastian Banert, Pontus Giselsson
2022 arXiv   pre-print
A new primal-dual method that uses an extra resolvent step is also presented as well as a general approach for adding Polyak momentum to any special case of our nonlinear forward-backward method, in particular  ...  The nonlinear, or warped, resolvent recently explored by Giselsson and B\'ui-Combettes has been used to model a large set of existing and new monotone inclusion algorithms.  ...  When C = 0 this is the same method as [22, Equation 4 .1] without relaxation and when D = 0 it is forward-backward splitting with Polyak momentum.  ... 
arXiv:2112.00481v3 fatcat:ma6vpct6qfgsfpz36gtyg2kqyq

Finding the forward-Douglas-Rachford-forward method [article]

Ernest K. Ryu, Bang Cong Vu
2019 arXiv   pre-print
The classical Douglas--Rachford and Forward-backward-forward methods respectively solve the monotone inclusion problem with a sum of 2 monotone operators and a sum of 1 monotone and 1 monotone-Lipschitz  ...  We then present a method that naturally combines Douglas--Rachford and forward-reflected-backward, a recently proposed alternative to Forward-backward-forward by Malitsky and Tam [arXiv:1808.04162, 2018  ...  Recently, Malitsky and Tam [12] have proposed forward-reflected-backward (FRB) splitting, another method for the case B = 0: x n+1 = J γA (x n − γ(2Cx n − Cx n−1 )), where the step size parameter satisfies  ... 
arXiv:1909.09747v2 fatcat:i37grjnjrje2tnl3sb2dcnm6dy

Projected-gradient algorithms for generalized equilibrium seeking in Aggregative Games are preconditioned Forward-Backward methods [article]

Giuseppe Belgioioso, Sergio Grammatico
2018 arXiv   pre-print
We show that projected-gradient methods for the distributed computation of generalized Nash equilibria in aggregative games are preconditioned forward-backward splitting methods applied to the KKT operator  ...  Specifically, we adopt the preconditioned forward-backward design, recently conceived by Yi and Pavel in the manuscript "A distributed primal-dual algorithm for computation of generalized Nash equilibria  ...  The main technical contribution of the paper is to conceive a design procedure for the preconditioned forward-backward splitting method.  ... 
arXiv:1803.10441v1 fatcat:kzpw6lfn3raobimf3sfyi24sti

Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions [article]

Luis M. Briceño-Arias, Jinjian Chen, Fernando Roldán, Yuchao Tang
2022 arXiv   pre-print
In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian  ...  The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forward-backward-half forward splitting, among other methods.  ...  In Section 4 we derive a method for solving a composite monotone primal-dual inclusion, including monotone, Lipschitzian, cocoercive, and bounded linear operators.  ... 
arXiv:2104.01516v3 fatcat:4wems6lz3rdyhcvucelomrfhcy

Stochastic inertial primal-dual algorithms [article]

Lorenzo Rosasco, Silvia Villa, Bang Cong Vu
2015 arXiv   pre-print
Key in our analysis is considering the framework of splitting algorithm for solving a monotone inclusions in suitable product spaces and for a specific choice of preconditioning operators.  ...  Our analysis provide convergence results in a general setting, that allows to analyze in a unified framework a variety of special cases of interest.  ...  Stochastic Inertial Forward-backward splitting method for solving monotone inclusions While stochastic proximal gradient methods have been studied in several papers (see e.g.  ... 
arXiv:1507.00852v1 fatcat:v5haldhk2zesbptjudtd3nmqoe

A class of Fejer convergent algorithms, approximate resolvents and the Hybrid Proximal-Extragradient method [article]

B. F. Svaiter
2012 arXiv   pre-print
method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method.  ...  We also introduce a new definition of approximations of resolvents which preserve some useful features of the exact resolvent, and use this concept to present an unifying view of the Forward-Backward splitting  ...  The Forward-Backward Splitting method solves the inclusion problem 0 ∈ (A + B)x where f1) A : X → X is α-cocoercive, α > 0; f2) B : X ⇉ X is maximal monotone.  ... 
arXiv:1204.1353v3 fatcat:r7z33o5bcfcljjhts2dzcaw5mq

A Class of Fejér Convergent Algorithms, Approximate Resolvents and the Hybrid Proximal-Extragradient Method

Benar F. Svaiter
2013 Journal of Optimization Theory and Applications  
method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method.  ...  We also introduce a new definition of approximations of resolvents which preserve some useful features of the exact resolvent, and use this concept to present an unifying view of the Forward-Backward splitting  ...  The Forward-Backward Splitting method solves the inclusion problem 0 ∈ (A + B)x where f1) A : X → X is α-cocoercive, α > 0; f2) B : X ⇒ X is maximal monotone.  ... 
doi:10.1007/s10957-013-0449-7 fatcat:hsxjnmc5nrd5dhhpia5k2wmoma

A Note on the Forward-Douglas--Rachford Splitting for Monotone Inclusion and Convex Optimization [article]

Hugo Raguet
2018 arXiv   pre-print
We shed light on the structure of the "three-operator" version of the forward-Douglas--Rachford splitting algorithm for finding a zero of a sum of maximally monotone operators A + B + C, where B is cocoercive  ...  We show that it is a straightforward extension of a fixed-point algorithm proposed by us as a generalization of the forward-backward splitting algorithm, initially designed for finding a zero of a sum  ...  Forward-Backward and Douglas-Rachford Algorithms viewed as Compositions of Averaged Operators e forward-backward splitting algorithm is a well-studied method for solving monotone inclusion problem P when  ... 
arXiv:1704.06948v3 fatcat:uxfyyhnusnd3plrdowmgbhuova

Stochastic Forward–Backward Splitting for Monotone Inclusions

Lorenzo Rosasco, Silvia Villa, Bang Công Vũ
2016 Journal of Optimization Theory and Applications  
The algorithm we propose is a natural stochastic extension of the classical forward-backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as  ...  We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator.  ...  (ii) A stochastic forward-backward splitting algorithm for monotone inclusions has been recently analyzed in [25, 26] , under rather different assumptions.  ... 
doi:10.1007/s10957-016-0893-2 fatcat:ix6l5fv3xrhgvnlekp3v7toh7q
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