# Shadow Douglas–Rachford Splitting for Monotone Inclusions

@article{Csetnek2019ShadowDS, title={Shadow Douglas–Rachford Splitting for Monotone Inclusions}, author={Ern{\"o} Robert Csetnek and Yura Malitsky and Matthew K. Tam}, journal={Applied Mathematics \& Optimization}, year={2019}, volume={80}, pages={665 - 678} }

In this work, we propose a new algorithm for finding a zero of the sum of two monotone operators where one is assumed to be single-valued and Lipschitz continuous. This algorithm naturally arises from a non-standard discretization of a continuous dynamical system associated with the Douglas–Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved. Each iteration of the proposed…

## 39 Citations

### Convergence of an Inertial Shadow Douglas-Rachford Splitting Algorithm for Monotone Inclusions

- MathematicsNumerical Functional Analysis and Optimization
- 2021

Abstract An inertial shadow Douglas-Rachford splitting algorithm for finding zeros of the sum of monotone operators is proposed in Hilbert spaces. Moreover, a three-operator splitting algorithm for…

### Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions

- Mathematics, Computer ScienceAppl. Math. Comput.
- 2020

### A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity

- MathematicsSIAM J. Optim.
- 2020

This work proposes a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators that does not require cocoercivity of the single-valued operator.

### Strengthened splitting methods for computing resolvents

- Computer Science, MathematicsComput. Optim. Appl.
- 2021

A systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually, which can be viewed as a type of regularisation that preserves computational tractability.

### A continuous dynamical splitting method for solving ‘strongly+weakly’ convex programming problems

- Mathematics
- 2019

ABSTRACT The Douglas–Rachford splitting method is a classical and powerful method for minimizing the sum of two convex functions. In this paper, we introduce two dynamical systems based on this…

### Forward-partial inverse-half-forward splitting algorithm for solving monotone inclusions

- MathematicsSet-Valued and Variational Analysis
- 2022

Abstract. In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone,…

### An adaptive Douglas-Rachford dynamic system for finding the zero of sum of two operators

- Mathematics
- 2020

The Douglas-Rachford splitting method is a popular method for finding a zero of the sum of two operators, and has been well studied as a numerical method. Nevertheless, the convergence theory for its…

### An outer reflected forward-backward splitting algorithm for solving monotone inclusions

- Mathematics
- 2020

Monotone inclusions have wide applications in solving various convex optimization problems arising in signal and image processing, machine learning, and medical image reconstruction. In this paper,…

### A second-order adaptive Douglas–Rachford dynamic method for maximal $$\alpha $$ α -monotone operators

- Mathematics
- 2021

The Douglas–Rachford splitting method is a classical and powerful method that is widely used in engineering fields for finding a zero of the sum of two operators. In this paper, we begin by proposing…

### Convergence of Adaptive Operator Extrapolation Method for Operator Inclusions In Banach Spaces

- MathematicsCMIS
- 2022

A novel iterative splitting algorithm for solving operator inclusions problem is considered in a real Banach space setting. The operator is a sum of the multivalued maximal monotone operator and the…

## References

SHOWING 1-10 OF 27 REFERENCES

### A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity

- MathematicsSIAM J. Optim.
- 2020

This work proposes a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators that does not require cocoercivity of the single-valued operator.

### Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators

- Mathematics, Computer Science
- 2011

This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods and the application to convex minimization problems is given special attention.

### A Modified Forward-Backward Splitting Method for Maximal Monotone Mappings

- MathematicsSIAM J. Control. Optim.
- 2000

A modification to the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings is proposed, under which the method converges assuming only the forward mapping is (Lipschitz) continuous on some closed convex subset of its domain.

### Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces

- MathematicsJ. Optim. Theory Appl.
- 2014

Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions, which prove a descent minimizing property and weak convergence to equilibria of the trajectories.

### A forward-backward-forward differential equation and its asymptotic properties

- Mathematics
- 2015

In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit…

### Projected Reflected Gradient Methods for Monotone Variational Inequalities

- Mathematics, Computer ScienceSIAM J. Optim.
- 2015

The projected reflected gradient algorithm with a constant stepsize is proposed, which requires only one projection onto the feasible set and only one value of the mapping per iteration and has R-linear rate of convergence under the strong monotonicity assumption.

### A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy

- Mathematics
- 2015

In this paper, we propose variants of Forward-Backward splitting method for finding a zero of the sum of two operators. A classical modification of Forward-Backward method was proposed by Tseng,…

### Continuous Gradient Projection Method in Hilbert Spaces

- Mathematics
- 2003

This paper is concerned with the asymptotic analysis of the trajectories of some dynamical systems built upon the gradient projection method in Hilbert spaces. For a convex function with locally…

### Convex Analysis and Monotone Operator Theory in Hilbert Spaces

- MathematicsCMS Books in Mathematics
- 2011

This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is…

### Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time

- Mathematics
- 2009

This survey is devoted to the asymptotic behavior of solutions of evolution equations generated by maximal monotone operators in Hilbert spaces. The emphasis is in the comparison of the continuous…