Resolvent splitting for sums of monotone operators with minimal lifting

Malitsky, Yura; Tam, Matthew K.

doi:10.1007/s10107-022-01906-4

Cited by 9 publications

(2 citation statements)

References 22 publications

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“…In addition, the author showed that the minimal lifting for three operators is 2-fold. That result has been recently generalized in [23] for an arbitrary number of operators. Further, frugal splitting algorithms with minimal (r − 1)-lifting have been independently proposed in [12,23].…”

Section: Introductionmentioning

confidence: 82%

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Regularity of sets under a reformulation in a product space of reduced dimension

Campoy¹

2023

Preprint

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Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility problems defined by finitely many sets, some other require the use of a product space reformulation to construct equivalent problems with two sets. In this work we analyze how some regularity properties are preserved under a reformulation in a product space of reduced dimension. This allows us to establish local linear convergence of parallel projection methods which are constructed through this reformulation.

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Section: Introductionmentioning

confidence: 82%

“…That result has been recently generalized in [23] for an arbitrary number of operators. Further, frugal splitting algorithms with minimal (r − 1)-lifting have been independently proposed in [12,23]. By now, the analysis of splitting algorithms with reduced lifting has become a very active research topic; see, e.g.…”

Section: Introductionmentioning

confidence: 82%

Regularity of sets under a reformulation in a product space of reduced dimension

Campoy¹

2023

Preprint

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A product space reformulation with reduced dimension for splitting algorithms

Campoy

2022

Comput Optim Appl

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In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra’s classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments.

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A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

2022

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In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.

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Resolvent splitting for sums of monotone operators with minimal lifting

Cited by 9 publications

References 22 publications

Regularity of sets under a reformulation in a product space of reduced dimension

Regularity of sets under a reformulation in a product space of reduced dimension

A product space reformulation with reduced dimension for splitting algorithms

A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

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