Stochastic Forward Douglas-Rachford Splitting Method for Monotone Inclusions

Cevher, Volkan; Vu, Cong Bang; Yurtsever, Alp

doi:10.1007/978-3-319-97478-1_7

Cited by 13 publications

(21 citation statements)

References 31 publications

(68 reference statements)

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“…Moreover,task (4), in the case where J = 2, X := X 0 = X 1 = X 2 , H 1 = H 2 = Id, r 1 = r 2 = 0, and A := X , i.e., min x∈X [f(x) + g 1 (x) + g 2 (x)], has also attracted attention in the context of the "three-term operator splitting" framework [18,19]. As in [14,15,16], ∇f, Prox g 1 and Prox g 2 are employed via computationally efficient recursions in [18,19] to generate a sequence which converges weakly (and under certain hypotheses, strongly) to a solution of the minimization task at hand. All studies in [14,15,16,18,19] (1) in the case where X is a Euclidean space and A := {x ∈ X | a x = 0}, for some a ∈ X \ {0}, was treated, within a stochastic setting, in [20].…”

Section: Prior Artmentioning

confidence: 99%

“…Theorems 3.1 and 3.6. Fixed-point theory, variational inequalities and affine-nonexpansive mappings are utilized to accommodate the affine constraint A in a more flexible way (see, e.g., Proposition 2.10 and Example A.4) than the usage of the indicator function and its associated metric-projection mapping that methods [15,16,18,19] promote. Such flexibility is combined with the first-order information of f and the proximal mapping of g to build recursions of tunable complexity that can score low-computational-complexity footprints, wellsuited for large-scale minimization tasks.…”

Section: Contributionsmentioning

confidence: 99%

“…where (21) is used in (37a), the convexity of g, (15) and the self adjointness of U in (37b), and finally (19) and (36) in (37c). Since lim k→∞ (x n k − x n k +1 ) = 0 by (29), the continuity of Q α implies lim k→∞ Q α (x n k +1 − x n k ) = 0, and (34) yields lim k→∞ (T − Id)x n k +1 = 0.…”

Section: Algorithm and Convergence Analysismentioning

confidence: 99%

“…where (19) is used in (40). Since (x n k +1 − x) k 0 and v n k +1 k v, and due to (29), (34) and (38), as well as the continuity of the linear mapping Q α , it turns out by [3, Lem.…”

Section: Algorithm and Convergence Analysismentioning

confidence: 99%

“…where the definition of Υ, given after (52), is used in (59a), (19) in (59b), (21) in (59c), (24) with η := ρ/(ρ − 1), a := U v n+1 + λ[∇f (x n ) + ξ n+1 ], b := (1 − 2α)(Id −T )x n+1 and Π := Q −1 α , as well as ρ > 1 in (59d), and…”

Section: Algorithm and Convergence Analysismentioning

confidence: 99%

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Fejér-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks

Slavakis

Yamada

2018

Optimization

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This paper introduces the Fejér-monotone hybrid steepest descent method (FM-HSDM), a new member to the HSDM family of algorithms, for solving affinely constrained minimization tasks in real Hilbert spaces, where convex smooth and non-smooth losses compose the objective function. FM-HSDM offers sequences of estimates which converge weakly and, under certain hypotheses, strongly to solutions of the task at hand. In contrast to its HSDM's precursors, FM-HSDM enjoys Fejér monotonicity, the step-size parameter stays constant across iterations to promote convergence speed-ups of the sequence of estimates to a minimizer, while only Lipschitzian continuity, and not strong monotonicity, of the derivative of the smooth-loss function is needed to ensure convergence. FM-HSDM utilizes fixed-point theory, variational inequalities and affine-nonexpansive mappings to accommodate affine constraints in a more versatile way than state-of-the-art primal-dual techniques and the alternating direction method of multipliers do. Recursions can be tuned to score low computational footprints, well-suited for large-scale optimization tasks, without compromising convergence guarantees. Results on the rate of convergence to an optimal point are also presented. Finally, numerical tests on synthetic data are used to validate the theoretical findings.Proof. The proof of Proposition 2.10(i) follows easily from Example 2.6(iv) and Proposition 2.9. The formula appearing in Proposition 2.10(ii) can be deduced by mathematical induction on J. Further, Q J Q J−1 · · · Q 1 Q 0 Q 1 · · · Q J−1 Q J is self adjoint, and its positivity follows from the fundamental observation that ∀x ∈ X , Q J Q J−1 · · · Q 1 Q 0 Q 1 · · · Q J−1 Q J x | x = Q 0 (Q 1 · · · Q J−1 Q J x) | Q 1 · · · Q J−1 Q J x ≥ 0, due to the positivity of Q 0 . Finally, the claim of Proposition 2.10(ii) is established by Q J · · · Q 1 Q 0 Q 1 · · · Q J ≤ Q 0 J j=1 Q j 2 ≤ 1.

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Section: Prior Artmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Algorithm and Convergence Analysismentioning

confidence: 99%

“…where (19) is used in (40). Since (x n k +1 − x) k 0 and v n k +1 k v, and due to (29), (34) and (38), as well as the continuity of the linear mapping Q α , it turns out by [3, Lem.…”

Section: Algorithm and Convergence Analysismentioning

confidence: 99%

Section: Algorithm and Convergence Analysismentioning

confidence: 99%

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Fejér-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks

Slavakis

Yamada

2018

Optimization

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An inertial three-operator splitting algorithm with applications to image inpainting

2019

Appl. Set-Valued Anal. Optim.

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The three-operators splitting algorithm is a popular operator splitting method for finding the zeros of the sum of three maximally monotone operators, with one of which is cocoercive operator. In this paper, we propose a class of inertial threeoperator splitting algorithm. The convergence of the proposed algorithm is proved by applying the inertial Krasnoselskii-Mann iteration under certain conditions on the iterative parameters in real Hilbert spaces. As applications, we develop an inertial three-operator splitting algorithm to solve the convex minimization problem of the sum of three convex functions, where one of them is differentiable with Lipschitz continuous gradient. Finally, we conduct numerical experiments on a constrained image inpainting problem with nuclear norm regularization. Numerical results demonstrate the advantage of the proposed inertial three-operator splitting algorithms.

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Convergence Analysis of an Inexact Three-Operator Splitting Algorithm

2018

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The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature.

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Stochastic Forward Douglas-Rachford Splitting Method for Monotone Inclusions

Cited by 13 publications

References 31 publications

Fejér-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks

Fejér-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks

An inertial three-operator splitting algorithm with applications to image inpainting

Convergence Analysis of an Inexact Three-Operator Splitting Algorithm

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