Douglas Rachford Research Articles

An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier–Stokes equations

We consider an optimization-based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier–Stokes system is split into the compressible Euler equations, which are solved by the positivity-preserving Runge–Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, which is solved by Crank–Nicolson in time with interior penalty DG method. Such a scheme is at most second order accurate in time, high order accurate in space, conservative, and preserves positivity of density. To further enforce the positivity of internal energy, we impose an optimization-based limiter for the total energy variable to post-process DG polynomial cell averages. The optimization-based limiter can be efficiently implemented by the popular first order convex optimization algorithms such as the Douglas–Rachford splitting method by using nearly optimal algorithm parameters. Numerical tests suggest that the DG method with Qk basis and the optimization-based limiter is robust for demanding low-pressure problems such as high-speed flows.

Accelerated iterative splitting methods on Hadamard manifolds

This paper aims to solve the monotone inclusion problem, minimization problem of multiple summands and the generalized Heron problem. We present an innovative approach, the modified normal S-iteration method, designed to approximate common fixed points of nearly nonexpansive sequences and families of operators via the property ( A ) . Some deductions of our results improve some existing results in the literature. To show the applicability of our result, we give application to the inclusion problem via forward–backward splitting method version of our algorithm and minimization problem via Douglas–Rachford splitting method version of our algorithm. To demonstrate the practical utility of the algorithm, we apply it to the generalized Heron problem.

Non-separable multidimensional multiresolution wavelets: A Douglas-Rachford approach

After re-casting the wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas–Rachford algorithm is employed in the search for multi-dimensional, non-separable, compactly supported, smooth, orthogonal, multiresolution wavelets in the case of translations along the integer lattice and isotropic dyadic dilations. An algorithm for the numerical construction of such wavelets is described. By applying the algorithm, new one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets.

A randomized block Douglas–Rachford method for solving linear matrix equation

A randomized block Douglas–Rachford method for solving linear matrix equation

Graph and Distributed Extensions of the Douglas–Rachford Method

Graph and Distributed Extensions of the Douglas–Rachford Method

Splitting algorithms for solving state-dependent maximal monotone inclusion problems

We consider the state-dependent maximal monotone inclusion problem and propose forward-backward splitting algorithms for solving it. Strong convergence of the proposed algorithms is established under suitable conditions. For the special separable case, we present an improved Douglas–Rachford variant that can be easily implemented. Moreover, some accelerated forward-backward splitting algorithms are also presented. Preliminary numerical experiments with comparisons to existing results are presented, illustrating the advantages of our methods.

Forward-backward-half forward splitting algorithm with deviations

ABSTRACT In this paper, we present a forward-backward-half forward splitting algorithm with deviations for solving the structured monotone inclusion problem composed of a maximally monotone operator, a maximally monotone and Lipschitz continuous operator and a cocoercive operator. The weak and linear convergence of the proposed algorithms is established. The aim of introducing deviations is to improve the performance of the forward-backward-half forward splitting algorithm through suitable choices of the deviations. A numerical example is given to show that the two-step inertial variant of the forward-backward-half forward splitting algorithms with deviations outperforms the existing methods in Briceño-Arias and Davis [Forward-backward-half forward algorithm for solving monotone inclusions, SIAM J Optimiz, 2017;28(4):2839–2871. doi: 10.1137/17M1120099], Zong et al. [An accelerated forward-backward-half forward splitting algorithm for monotone inclusion with applications to image restoration, Optimization, 2022;73(2):401–428. doi: 10.1080/02331934.2022.2107926] and Fan et al. [Convergence of an inertial shadow Douglas–Rachford splitting algorithm for monotone inclusions, Numer Func Anal Optim, 2021;42(14):1627–1644. doi: 10.1080/01630563.2021.2001749].

Randomized Douglas–Rachford Methods for Linear Systems: Improved Accuracy and Efficiency

Randomized Douglas–Rachford Methods for Linear Systems: Improved Accuracy and Efficiency

An Inertial Parametric Douglas–Rachford Splitting Method for Nonconvex Problems

In this paper, we propose an inertial parametric Douglas–Rachford splitting method for minimizing the sum of two nonconvex functions, which has a wide range of applications. The proposed algorithm combines the inertial technique, the parametric technique, and the Douglas–Rachford method. Subsequently, in theoretical analysis, we construct a new merit function and establish the convergence of the sequence generated by the inertial parametric Douglas–Rachford splitting method. Finally, we present some numerical results on nonconvex feasibility problems to illustrate the efficiency of the proposed method.

Optimal Control Duality and the Douglas–Rachford Algorithm

Optimal Control Duality and the Douglas–Rachford Algorithm

Douglas–Rachford algorithm for control- and state-constrained optimal control problems

<abstract><p>We consider the application of the Douglas–Rachford (DR) algorithm to solve linear-quadratic (LQ) control problems with box constraints on the state and control variables. We have split the constraints of the optimal control problem into two sets: one involving the ordinary differential equation with boundary conditions, which is affine, and the other, a box. We have rewritten the LQ control problems as the minimization of the sum of two convex functions. We have found the proximal mappings of these functions, which we then employ for the projections in the DR iterations. We propose a numerical algorithm for computing the projection onto the affine set. We present a conjecture for finding the costates and the state constraint multipliers of the optimal control problem, which can, in turn, be used to verify the optimality conditions. We conducted numerical experiments with two constrained optimal control problems to illustrate the performance and the efficiency of the DR algorithm in comparison with the traditional approach of direct discretization.</p></abstract>

Douglas–Rachford algorithm for control-constrained minimum-energy control problems

Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize the Douglas–Rachford (DR) algorithm to solve control-constrained minimum-energy optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the DR algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.

A Partially Inertial Customized Douglas–Rachford Splitting Method for a Class of Structured Optimization Problems

A Partially Inertial Customized Douglas–Rachford Splitting Method for a Class of Structured Optimization Problems

The variable metric three-operator algorithms for solving monotone inclusions

In this paper, we propose an inexact variable metric three-operator algorithm and a viscous version of it for finding a zero of the sum of three monotone operators in a real Hilbert space. Under some appropriate conditions, we establish the weak or strong convergence of the proposed algorithms. As a special case of the proposed problem, we present a variable metric Douglas–Rachford splitting algorithm and obtain a strong convergence result. The algorithms and the results in this paper are further extensions of the algorithms and results for solving the zero inclusion problems of monotone operators. To illustrate the efficiency of the proposed algorithms, we construct the numerical experiments and give some choices for the variable metric matrix.

Linearized Douglas–Rachford method for variational inequalities with Lipschitz mappings

Linearized Douglas–Rachford method for variational inequalities with Lipschitz mappings

Strict pseudocontractions and demicontractions, their properties, and applications

We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary and sufficient conditions under which a convex combination and composition of strict pseudocontractions as well as demicontractions that share a common fixed point is again a strict pseudocontraction or a demicontraction, respectively. Moreover, we introduce a generalized relaxation of composition of demicontraction and give its properties. We“ apply these properties to prove the weak convergence of a class of algorithms that is wider than the Douglas–Rachford algorithm and projected Landweber algorithms. We have also presented two numerical examples, where we compare the behavior of the presented methods with the Douglas–Rachford method.

Tikhonov regularized iterative methods for nonlinear problems

ABSTRACT We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward–backward-type algorithm and a Douglas–Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward–backward-type primal–dual algorithm and a Douglas–Rachford-type primal–dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.

Strong Convergence for the Alternating Halpern–Mann Iteration in CAT(0) Spaces

In this paper we consider, in the general context of CAT(0) spaces, an iterative schema which alternates between Halpern and Krasnoselskii–Mann style iterations. We prove, under suitable conditions, the strong convergence of this algorithm, benefiting from ideas from the proof mining program. We give quantitative information in the form of effective rates of asymptotic regularity and of metastability (in the sense of Tao). Motivated by these results, we are also able to obtain strongly convergent versions of the forward-backward and the Douglas–Rachford algorithms. Our results generalize recent work by Boţ, Csetnek, and Meier and Cheval and Leuştean.

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

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. To illustrate our results, we also perform numerical experiments.

On the Douglas–Rachford Algorithm for Solving Possibly Inconsistent Optimization Problems

More than 40 years ago, Lions and Mercier introduced in a seminal paper the Douglas–Rachford algorithm. Today, this method is well-recognized as a classic and highly successful splitting method to find minimizers of the sum of two (not necessarily smooth) convex functions. Whereas the underlying theory has matured, one case remains a mystery: the behavior of the shadow sequence when the given functions have disjoint domains. Building on previous work, we establish for the first time weak and value convergence of the shadow sequence generated by the Douglas–Rachford algorithm in a setting of unprecedented generality. The weak limit point is shown to solve the associated normal problem, which is a minimal perturbation of the original optimization problem. We also present new results on the geometry of the minimal displacement vector. Funding: The research of H. H. Bauschke and W. M. Moursi was partially supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-03703 and RGPIN-2019-04803], respectively.