An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Dao, Minh N.; Phan, Hung M.

doi:10.1186/s13663-021-00701-8

Cited by 3 publications

(10 citation statements)

References 17 publications

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“…x ∈ Ω γ using the notion of conically averaged operators recently introduced in [5], not only for a fixed relaxation parameter λ k = λ, as it was done in [17,Corollary 4.2]. Indeed, by [17,Theorem 4.1], the operator DY γ in (2) is conically (2 − γ /(2β)) −1 -averaged, so [5, Proposition 2.9] can be applied to deduce the convergence of the Krasnosel'ski ȋ-Mann iteration (3) to a fixed point of DY γ , which belongs to Ω γ by Lemma 3.1.…”

Section: Remark 34 (I)mentioning

confidence: 99%

“…The authors would like to thank Patrick Combettes for making us aware of [17] right before submitting this work. We thank the referees for their careful reading and their constructive comments which helped improve our manuscript.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…In addition, we derive in Theorem 3.7 a strengthened version of Davis-Yin splitting algorithm which permits computing the resolvent of A + B + T . Right before submitting this manuscript, we learnt about a recent preprint by Dao-Phan, which now has been published [17]. Using the notion of conically averaged operators introduced in [5], the authors prove in [17,Corollary 4.2] that the operator (1 − λ) Id +λDY γ is 2λβ/(4β − γ )-averaged when γ ∈ ]0, 4β[, from which the convergence of (3) for a fixed λ k = λ follows.…”

Section: Introductionmentioning

confidence: 99%

“…Right before submitting this manuscript, we learnt about a recent preprint by Dao-Phan, which now has been published [17]. Using the notion of conically averaged operators introduced in [5], the authors prove in [17,Corollary 4.2] that the operator (1 − λ) Id +λDY γ is 2λβ/(4β − γ )-averaged when γ ∈ ]0, 4β[, from which the convergence of (3) for a fixed λ k = λ follows. As a simple motivating example of the importance of the algorithm parameters, consider the problem of finding the minimum norm point in the intersection of two balls A and B in the Euclidean space whose intersection has nonempty interior.…”

Section: Introductionmentioning

confidence: 99%

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A Direct Proof of Convergence of Davis–Yin Splitting Algorithm Allowing Larger Stepsizes

Artacho

David

2022

Set-Valued Var. Anal

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This note is devoted to the splitting algorithm proposed by Davis and Yin (Set-valued Var. Anal. 25(4), 829–858, 2017) for computing a zero of the sum of three maximally monotone operators, with one of them being cocoercive. We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin. As a by-product, the same conclusion applies to the forward-backward splitting algorithm. Further, we use the notion of “strengthening” of a set-valued operator to derive a new splitting algorithm for computing the resolvent of the sum. Last but not least, we provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms.

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Section: Remark 34 (I)mentioning

confidence: 99%

Section: Acknowledgementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Direct Proof of Convergence of Davis–Yin Splitting Algorithm Allowing Larger Stepsizes

Artacho

David

2022

Set-Valued Var. Anal

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“…When n = 2 , there are also many methods. For instance, if F is cocoercive, Davis-Yin splitting [4][5][6] which takes the form (4)…”

Section: Splitting Algorithmsmentioning

confidence: 99%

Distributed forward-backward methods for ring networks

et al. 2022

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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 the product space. 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. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.

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An Adaptive Alternating Direction Method of Multipliers

Bartz

Campoy

Phan

2022

J Optim Theory Appl

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The alternating direction method of multipliers (ADMM) is a powerful splitting algorithm for linearly constrained convex optimization problems. In view of its popularity and applicability, a growing attention is drawn toward the ADMM in nonconvex settings. Recent studies of minimization problems for nonconvex functions include various combinations of assumptions on the objective function including, in particular, a Lipschitz gradient assumption. We consider the case where the objective is the sum of a strongly convex function and a weakly convex function. To this end, we present and study an adaptive version of the ADMM which incorporates generalized notions of convexity and penalty parameters adapted to the convexity constants of the functions. We prove convergence of the scheme under natural assumptions. To this end, we employ the recent adaptive Douglas–Rachford algorithm by revisiting the well-known duality relation between the classical ADMM and the Douglas–Rachford splitting algorithm, generalizing this connection to our setting. We illustrate our approach by relating and comparing to alternatives, and by numerical experiments on a signal denoising problem.

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An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Cited by 3 publications

References 17 publications

A Direct Proof of Convergence of Davis–Yin Splitting Algorithm Allowing Larger Stepsizes

A Direct Proof of Convergence of Davis–Yin Splitting Algorithm Allowing Larger Stepsizes

Distributed forward-backward methods for ring networks

An Adaptive Alternating Direction Method of Multipliers

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