Strengthened splitting methods for computing resolvents

Artacho, Francisco J. Aragón; Campoy, Rubén; Tam, Matthew K.

doi:10.1007/s10589-021-00291-6

Cited by 12 publications

(6 citation statements)

References 43 publications

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“…The following result records the basic convergence properties by Ryu [25], and recently improved by Arag ón-Artacho, Campoy, and Tam [2].…”

Section: Ryu Splittingsupporting

confidence: 56%

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The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

Bauschke¹,

Singh²,

Wang³

2022

Preprint

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Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolvents of the operators are available, this problem can be tackled with the Douglas-Rachford algorithm. However, when dealing with three or more operators, one must work in a product space with as many factors as there are operators. In groundbreaking recent work by Ryu and by Malitsky and Tam, it was shown that the number of factors can be reduced by one. A similar reduction was achieved recently by Campoy through a clever reformulation originally proposed by Kruger. All three splitting methods guarantee weak convergence to some solution of the underlying sum problem; strong convergence holds in the presence of uniform monotonicity.In this paper, we provide a case study when the operators involved are normal cone operators of subspaces and the solution set is thus the intersection of the subspaces. Even though these operators lack strict convexity, we show that striking conclusions are available in this case: strong (instead of weak) convergence and the solution obtained is (not arbitrary but) the projection onto the intersection. Numerical experiments to illustrate our results are also provided.

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“…The following result records the basic convergence properties by Ryu [25], and recently improved by Arag ón-Artacho, Campoy, and Tam [2].…”

Section: Ryu Splittingsupporting

confidence: 56%

Section: Introductionmentioning

confidence: 99%

The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

Bauschke¹,

Singh²,

Wang³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The following result records the basic convergence properties by Ryu [20], and recently improved by Arag ón-Artacho, Campoy, and Tam [2].…”

Section: Ryu Splittingsupporting

confidence: 56%

“…In general, solving (2) may be quite hard. Luckily, in many interesting cases, we have access to the firmly nonexpansive resolvents J A i := (Id +A i ) −1 which opens the door to employ splitting algorithms to solve (2). The most famous instance is the Douglas-Rachford algorithm [15] whose importance for this problem was brought to light in the seminal paper by Lions and Mercier [17].…”

Section: Introductionmentioning

confidence: 99%

The splitting algorithms by Ryu and by Malitsky-Tam applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

Bauschke,

Singh,

Wang

2021

Preprint

View full text Add to dashboard Cite

Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that resolvents of the operators are available, this problem can be tackled with the Douglas-Rachford algorithm. However, when dealing with three or more operators, one must work in a product space with as many factors as there are operators. In groundbreaking recent work by Ryu and by Malitsky and Tam, it was shown that the number of factors can be reduced by one. These splitting methods guarantee weak convergence to some solution of the underlying sum problem; strong convergence holds in the presence of uniform monotonicity.In this paper, we provide a case study when the operators involved are normal cone operators of subspaces and the solution set is thus the intersection of the subspaces. Even though these operators lack strict convexity, we show that striking conclusions are available in this case: strong (instead of weak) convergence and the solution obtained is (not arbitrary but) the projection onto the intersection. Numerical experiments to illustrate our results are also provided.

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“…Remark 3.8. For some applications of Example 3.7, we refer the reader to [8] and also to the recent preprint [2].…”

Section: Rectangular Matricesmentioning

confidence: 99%

Projecting onto rectangular matrices with prescribed row and column sums

Bauschke¹,

Singh²,

Wang³

2021

Preprint

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In 1998, Khoury as well as Glunt, Hayden, and Reams, discovered independently a beautiful formula for the projection onto the set of generalized bistochastic (square) matrices. Their result has since found various generalizations and applications.In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed (scaled) row and column sums. Our approach is based on computing the Moore-Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert-Schmidt operators. We also perform numerical experiments featuring the new projection operator.

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Strengthened splitting methods for computing resolvents

Cited by 12 publications

References 43 publications

The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

The splitting algorithms by Ryu and by Malitsky-Tam applied to normal cones of linear subspaces converge strongly to the projection onto the intersection

Projecting onto rectangular matrices with prescribed row and column sums

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