On the Douglas–Rachford algorithm

Bauschke, Heinz H.; Moursi, Walaa M.

doi:10.1007/s10107-016-1086-3

Cited by 60 publications

(79 citation statements)

References 44 publications

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“…Other splitting schemes can be found in [13,14,16] and the references therein.When applied to two normal cone operators, the DR algorithm can be used to solve the feasibility problem of finding a common point of two sets. In this context, the DR algorithm possesses many good properties, for example, it finds a best approximation point when the intersection of sets is empty [3,5,8], it finds an exact solution after only a finite number of iterations under verifiable conditions [1,4,7], and it converges globally in some nonconvex settings [9,19] while converges locally with linear or sublinear rate under some regularity assumptions [11,29,33]. In the absence of constraint qualifications, [6] suggests that the DR algorithm outperforms the well-known method of alternating projections.…”

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confidence: 99%

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Dao¹,

Phan²

2019

SIAM J. Optim.

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The Douglas-Rachford algorithm is a classical and powerful splitting method for minimizing the sum of two convex functions and, more generally, finding a zero of the sum of two maximally monotone operators. Although this algorithm has been well understood when the involved operators are monotone or strongly monotone, the convergence theory for weakly monotone settings is far from being complete. In this paper, we propose an adaptive Douglas-Rachford splitting algorithm for the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With appropriately chosen parameters, the algorithm converges globally to a fixed point from which we derive a solution of the problem. When one operator is Lipschitz continuous, we prove global linear convergence which sharpens recent known results. entire mathematical model. It is worth mentioning (see, e.g., [23]) that several splitting methods such as the method of partial inverses [35] and the alternating direction method of multipliers (ADMM) [25] can be written in the form of the DR algorithm, which itself can be transformed into the proximal point algorithm [34]. Other splitting schemes can be found in [13,14,16] and the references therein.When applied to two normal cone operators, the DR algorithm can be used to solve the feasibility problem of finding a common point of two sets. In this context, the DR algorithm possesses many good properties, for example, it finds a best approximation point when the intersection of sets is empty [3,5,8], it finds an exact solution after only a finite number of iterations under verifiable conditions [1,4,7], and it converges globally in some nonconvex settings [9,19] while converges locally with linear or sublinear rate under some regularity assumptions [11,29,33]. In the absence of constraint qualifications, [6] suggests that the DR algorithm outperforms the well-known method of alternating projections. In attempting to generalize the DR algorithm for feasibility problems, several parameters were added to its formulation [12,18,17,24]. In this case, one has the freedom to modify the parameters that are associated with the projections without giving up the solution. This approach is possible because the underlying normal cone operators have homogeneous values, which allows for scaling them independently. The situation changes completely when working with general problems where two involved operators may no longer have such homogeneity. In this case, a naive scaling may destroy the ability to solve the original problem. Therefore, we aim to overcome this hurdle by proposing an adaptive approach.The paper is devoted to the convergence analysis of the adaptive DR algorithm for finding a zero of the sum of α-and β-monotone operators, in which α-monotonicity is a unification of strong and weak monotonicity (see Definition 3.1). This situation arises in various important applications; see [27] for a brief discussion. The main contribution is summarized below.(R1) We incorporate parameters into the DR algorith...

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confidence: 99%

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Dao¹,

Phan²

2019

SIAM J. Optim.

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“…Among them, we are particularly interested in convex feasibility problems as in [1] and general convex inclusions (e.g. [19][20][21]). Simple examples, however, show that the circumcenter C(x) may not be defined for general convex sets at some "pathological" points x.…”

Section: Discussionmentioning

confidence: 99%

On the linear convergence of the circumcentered-reflection method

Behling

Bello-Cruz

Santos

2018

Operations Research Letters

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In order to accelerate the Douglas-Rachford method we recently developed the circumcentered-reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.

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“…Then it is known (see [27] or [7]) that (b n ) n∈N converges weakly to P A∩B (z). Note that (28) simplifies to…”

Section: Douglas-rachford Algorithmmentioning

confidence: 99%

“…In the current paper, we deal with continuous-time optimal control problems, which are infinite-dimensional optimization problems that are set in Hilbert spaces. After splitting the constraints of the problem, we apply Dykstra's algorithm [9], the Douglas-Rachford (DR) method [4,7,15,16,23,27], and the Aragón Artacho-Campoy (AAC) algorithm [2], all of which solve the underlying best approximation problem.…”

Section: Introductionmentioning

confidence: 99%

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

Bauschke

Burachik

Kaya

2019

Splitting Algorithms, Modern Operator Theory, and Applications

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We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i) Dykstra's algorithm, (ii) the Douglas-Rachford (DR) method and (iii) the Aragón Artacho-Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.

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On the Douglas–Rachford algorithm

Cited by 60 publications

References 44 publications

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

Adaptive Douglas--Rachford Splitting Algorithm for the Sum of Two Operators

On the linear convergence of the circumcentered-reflection method

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

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