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douglas-rachford algorithm for control-constrained minimum-energy control problems

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2024年第1期30卷 18-18页

作者： Burachik, Regina S. Caldwell, Bethany I. Yalcin Kaya, C. Univ South Australia Math UniSA STEM Mawson Lakes SA 5095 Australia

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.

关键词： Optimal control harmonic oscillator douglas-rachford algorithm control constraints numerical methods

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Solving Graph Coloring Problems with the douglas-rachford algorithm

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SET-VALUED AND VARIATIONAL ANALYSIS 2018年第2期26卷 277-304页

作者： Aragon Artacho, Francisco J. Campoy, Ruben Univ Alicante Alicante Spain

We present the douglas-rachford algorithm as a successful heuristic for solving graph coloring problems. Given a set of colors, these types of problems consist in assigning a color to each node of a graph, in such a way that every pair of adjacent nodes are assigned with different colors. We formulate the graph coloring problem as an appropriate feasibility problem that can be effectively solved by the douglas-rachford algorithm, despite the nonconvexity arising from the combinatorial nature of the problem. Different modifications of the graph coloring problem and applications are also presented. The good performance of the method is shown in various computational experiments.

关键词： douglas-rachford algorithm Graph coloring Feasibility problem Non-convex

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On Slater's condition and finite convergence of the douglas-rachford algorithm for solving convex feasibility problems in Euclidean spaces

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JOURNAL OF GLOBAL OPTIMIZATION 2016年第2期65卷 329-349页

作者： Bauschke, Heinz H. Dao, Minh N. Noll, Dominikus Phan, Hung M. Univ British Columbia Dept Math Kelowna BC V1V 1V7 Canada Hanoi Natl Univ Educ Dept Math & Informat 136 Xuan Thuy Hanoi Vietnam Univ Toulouse Inst Math 118 Route Narbonne F-31062 Toulouse France Univ Massachusetts Dept Math Sci 265 Riverside StOlney Hall 428 Lowell MA 01854 USA

The douglas-rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater's condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the douglas-rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection-projection.

关键词： Alternating projections Convex feasibility problem Convex set douglas-rachford algorithm Epigraph Finite convergence Method of reflection-projection Monotone operator Partial inverse Polyhedral set Projector Slater's condition

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ON THE FINITE CONVERGENCE OF THE douglas-rachford algorithm FOR SOLVING (NOT NECESSARILY CONVEX) FEASIBILITY PROBLEMS IN EUCLIDEAN SPACES

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SIAM JOURNAL ON OPTIMIZATION 2017年第1期27卷 507-537页

作者： Bauschke, Heinz H. Dao, Minh N. Univ British Columbia Math Kelowna BC V1V 1V7 Canada Univ Newcastle CARMA Callaghan NSW 2308 Australia Hanoi Natl Univ Educ Dept Math & Informat 136 Xuan Thuy Hanoi Vietnam

Solving feasibility problems is a central task in mathematics and the applied sciences. One particularly successful method is the douglas-rachford algorithm. In this paper, we provide many new conditions sufficient fo... 详细信息

关键词： averaged alternating reflections douglas-rachford algorithm epigraph feasibility problem finite convergence global convergence halfspace polyhedron projector reflector

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On the douglas-rachford algorithm

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MATHEMATICAL PROGRAMMING 2017年第1-2期164卷 263-284页

作者： Bauschke, Heinz H. Moursi, Walaa M. Univ British Columbia Math Kelowna BC V1V 1V7 Canada Mansoura Univ Math Dept Fac Sci Mansoura 35516 Egypt

The douglas-rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. The behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. However, more than a decade ago, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and its weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Our proof relies on a new convergence principle for Fej,r monotone sequences. Numerous examples illustrate our results.

关键词： Attouch-Thera duality douglas-rachford algorithm Inconsistent case Maximally monotone operator Nonexpansive mapping Paramonotone operator Sum problem Weak convergence

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An enhanced formulation for solving graph coloring problems with the douglas-rachford algorithm

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JOURNAL OF GLOBAL OPTIMIZATION 2020年第2期77卷 383-403页

作者： Aragon Artacho, Francisco J. Campoy, Ruben Elser, Veit Univ Alicante Dept Math Alicante Spain Cornell Univ Dept Phys Ithaca NY 14853 USA

We study the behavior of the douglas-rachford algorithm on the graph vertex-coloring problem. Given a graph and a number of colors, the goal is to find a coloring of the vertices so that all adjacent vertex pairs have different colors. In spite of the combinatorial nature of this problem, the douglas-rachford algorithm was recently shown to be a successful heuristic for solving a wide variety of graph coloring instances, when the problem was cast as a feasibility problem on binary indicator variables. In this work we consider a different formulation, based on semidefinite programming. The much improved performance of the douglas-rachford algorithm, with this new approach, is demonstrated through various numerical experiments.

关键词： douglas-rachford algorithm Graph coloring Feasibility problem Nonconvex constraints

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Affine Nonexpansive Operators, Attouch-Thera Duality and the douglas-rachford algorithm

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SET-VALUED AND VARIATIONAL ANALYSIS 2017年第3期25卷 481-505页

作者： Bauschke, Heinz H. Lukens, Brett Moursi, Walaa M. Univ British Columbia Math Kelowna BC V1V 1V7 Canada 3990 Lansdowne Rd Armstrong BC V0E 1B3 Canada Mansoura Univ Math Dept Fac Sci Mansoura 35516 Egypt

The douglas-rachford splitting algorithm was originally proposed in 1956 to solve a system of linear equations arising from the discretization of a partial differential equation. In 1979, Lions and Mercier brought forward a very powerful extension of this method suitable to solve optimization problems. In this paper, we revisit the original affine setting. We provide a powerful convergence result for finding a zero of the sum of two maximally monotone affine relations. As a by product of our analysis, we obtain results concerning the convergence of iterates of affine nonexpansive mappings as well as Attouch-Th,ra duality. Numerous examples are presented.

关键词： Affine mapping Attouch-Thera duality douglas-rachford algorithm Linear convergence Maximally monotone operator Nonexpansive mapping Paramonotone operator Strong convergence Toeplitz matrix Tridiagonal matrix

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A Lyapunov-type approach to convergence of the douglas-rachford algorithm for a nonconvex setting

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第1期73卷 83-112页

作者： Dao, Minh N. Tam, Matthew K. Univ Newcastle CARMA Callaghan NSW 2308 Australia Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany

The douglas-rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been experimentally observed to solve various nonconvex feasibility problems;an observation which current theory cannot sufficiently explain. In this paper, we prove convergence of the douglas-rachford algorithm in a potentially nonconvex setting. Our analysis relies on the existence of a Lyapunov-type functional whose convexity properties are not tantamount to convexity of the original constraint sets. Moreover, we provide various nonconvex examples in which our framework proves global convergence of the algorithm.

关键词： douglas-rachford algorithm Feasibility problem Global convergence Graph of a function Linear convergence Lyapunov function Method of alternating projections Newton's method Nonconvex set Projection Stability Zero of a function

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A Parallel douglas-rachford algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

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SIAM JOURNAL ON IMAGING SCIENCES 2016年第3期9卷 901-937页

作者： Bergmann, Ronny Persch, Johannes Steidl, Gabriele Tech Univ Kaiserslautern Dept Math D-67663 Kaiserslautern Germany

We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation like regularizing term. To solve the convex minimization problem, we extend the douglas-rachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the douglas-rachford algorithm is reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for Krasnoselski-Mann iterations imply the convergence of the douglas-rachford algorithm. Unfortunately, these general results do not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions-namely, several distance-like functions and an indicator function of a special convex set. We prove that the reflections of certain distance-like functions on Hadamard manifolds are nonexpansive, which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is non-expansive on Hadamard manifolds with constant curvature so that the douglas-rachford algorithm converges here. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods such as the cyclic proximal point algorithm and half-quadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric, which does not have a constant curvature.

关键词： manifold-valued data variational restoration methods douglas-rachford algorithm symmetric Hadamard manifold DT-MRI

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The douglas-rachford algorithm for a hyperplane and a doubleton

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第1期74卷 79-93页

作者： Bauschke, Heinz H. Dao, Minh N. Lindstrom, Scott B. Univ British Columbia Dept Math Kelowna BC V1V 1V7 Canada Univ Newcastle CARMA Callaghan NSW 2308 Australia

The douglas-rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case whichsomewhat surprisinglydepends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.

关键词： Closed-form expressions Cycling douglas-rachford algorithm Feasibility problem Finite set Hyperplane Method of alternating projections Projector Reflector Primary 47H10 49M27 Secondary 65K05 65K10 90C26

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