A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators

Boţ, Radu Ioan; Csetnek, Ernö Robert; Heinrich, André

doi:10.48550/arxiv.1206.5953

Cited by 2 publications

(6 citation statements)

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“…+ C is strongly monotone, as well, thus the monotone inclusion problem (5) has at most one solution. Hence, if (x, v 1 , ..., v m ) is a primal-dual solution to Problem 3, then x is the unique solution to (5). Notice that the problem (6) may not have an unique solution.…”

Section: Algorithmmentioning

confidence: 99%

“…The problem of finding the zeros of the sum of two (or more) maximally monotone operators in Hilbert spaces continues to be a very active research field, with applications in convex optimization, partial differential equations, signal and image processing, etc. (see [1,[5][6][7]9,12,13,21]). To the most prominent methods in this area belong the proximal point algorithm for finding the zeros of a maximally monotone operator (see [17]) and the Douglas-Rachford splitting algorithm for finding the zeros of the sum of two maximally monotone operators (see [14]).…”

Section: Introductionmentioning

confidence: 99%

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On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

et al. 2014

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We present two modified versions of the primal-dual splitting algorithm relying on forward-backward splitting proposed in [21] for solving monotone inclusion problems. Under strong monotonicity assumptions for some of the operators involved we obtain for the sequences of iterates that approach the solution orders of convergence of O( 1 n ) and O(ω n ), for ω ∈ (0, 1), respectively. The investigated primal-dual algorithms are fully decomposable, in the sense that the operators are processed individually at each iteration. We also discuss the modified algorithms in the context of convex optimization problems and present numerical experiments in image processing and support vector machines classification.

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Section: Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

et al. 2014

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“…(a) For every u ∈ zer(A + D + N C ) one can take w = 0 in (10) and the conclusion follows from Lemma 2.…”

Section: The General Casementioning

confidence: 93%

“…For example, when H and G are real Hilbert spaces and L : H → G is nonzero linear continuous, then (x, v) → (L * v, −Lx) is an example in this sense. This operator appears in a natural way when employing primal-dual approaches in the context of monotone inclusion problems as done in [12] (see also [10,11,15,24]).…”

Section: Relaxing Cocoercivity To Monotonicity and Lipschitz Continuitymentioning

confidence: 99%

“…The problem under consideration is the following. We use the product space approach (see [4,[10][11][12]15]) in order to show that the above problem can be reformulated as the monotone inclusion problem treated in Subsection 3.1 in an appropriate product space. To this end we consider the real Hilbert space H × G endowed with the inner product…”

Section: Monotone Inclusion Problems Involving Compositions With Line...mentioning

confidence: 99%

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Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems

Boţ

Csetnek

2014

Set-Valued Var. Anal

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We deal with monotone inclusion problems of the form 0 ∈ Ax + Dx + N C (x) in real Hilbert spaces, where A is a maximally monotone operator, D a cocoercive operator and C the nonempty set of zeros of another cocoercive operator. We propose a forwardbackward penalty algorithm for solving this problem which extends the one proposed by H. Attouch, M.-O. Czarnecki and J. Peypouquet in [3]. The condition which guarantees the weak ergodic convergence of the sequence of iterates generated by the proposed scheme is formulated by means of the Fitzpatrick function associated to the maximally monotone operator that describes the set C. In the second part we introduce a forward-backwardforward algorithm for monotone inclusion problems having the same structure, but this time by replacing the cocoercivity hypotheses with Lipschitz continuity conditions. The latter penalty type algorithm opens the gate to handle monotone inclusion problems with more complicated structures, for instance, involving compositions of maximally monotone operators with linear continuous ones.

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A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators

Cited by 2 publications

References 0 publications

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems

Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems

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