Circumcentering the Douglas–Rachford method

We introduce and study a geometric modification of the Douglas–Rachford method called the Circumcentered–Douglas–Rachford method. This method iterates by taking the intersection of bisectors of reflection steps for solving certain classes of feasibility problems. The convergence analysis is established for best approximation problems involving two (affine) subspaces and both our theoretical and numerical results compare favorably to the original Douglas–Rachford method. Under suitable conditions, it is shown that the linear rate of convergence of the Circumcentered–Douglas–Rachford method is at least the cosine of the Friedrichs angle between the (affine) subspaces, which is known to be the sharp rate for the Douglas–Rachford method. We also present a preliminary discussion on the Circumcentered–Douglas–Rachford method applied to the many set case and to examples featuring non-affine convex sets.

[1]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[2]  D. Russell Luke,et al.  Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility , 2013, IEEE Transactions on Signal Processing.

[3]  P. L. Combettes,et al.  Foundation of set theoretic estimation , 1993 .

[4]  Raniere Silva,et al.  Perprof-py: A Python Package for Performance Profile of Mathematical Optimization Software , 2016 .

[5]  Benar Fux Svaiter,et al.  On Weak Convergence of the Douglas-Rachford Method , 2010, SIAM J. Control. Optim..

[6]  Heinz H. Bauschke,et al.  On the local convergence of the Douglas–Rachford algorithm , 2014, 1401.6188.

[7]  Alan Edelman,et al.  Julia: A Fresh Approach to Numerical Computing , 2014, SIAM Rev..

[8]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[9]  Joël Benoist,et al.  The Douglas–Rachford algorithm for the case of the sphere and the line , 2015, J. Glob. Optim..

[10]  Jonathan M. Borwein,et al.  A Cyclic Douglas–Rachford Iteration Scheme , 2013, J. Optim. Theory Appl..

[11]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[12]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[13]  Heinz H. Bauschke,et al.  Optimal Rates of Linear Convergence of Relaxed Alternating Projections and Generalized Douglas-Rachford Methods for Two Subspaces , 2015, Numerical Algorithms.

[14]  Jonathan M. Borwein,et al.  Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem , 2015, J. Glob. Optim..

[15]  Jonathan M. Borwein,et al.  Global convergence of a non-convex Douglas–Rachford iteration , 2012, J. Glob. Optim..

[16]  Jonathan M. Borwein,et al.  Recent Results on Douglas–Rachford Methods for Combinatorial Optimization Problems , 2013, J. Optim. Theory Appl..

[17]  Hung M. Phan,et al.  Linear convergence of the Douglas–Rachford method for two closed sets , 2014, 1401.6509.

[18]  Heinz H. Bauschke,et al.  The rate of linear convergence of the Douglas-Rachford algorithm for subspaces is the cosine of the Friedrichs angle , 2013, J. Approx. Theory.

[19]  Jonathan M. Borwein,et al.  The Douglas-Rachford Algorithm in the Absence of Convexity , 2011, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[20]  P. L. Combettes.  The foundations of set theoretic estimation , 1993 .

[21]  Andrzej Cegielski,et al.  Projection methods: an annotated bibliography of books and reviews , 2014, 1406.6143.

[22]  D. Russell Luke,et al.  Nonconvex Notions of Regularity and Convergence of Fundamental Algorithms for Feasibility Problems , 2012, SIAM J. Optim..

[23]  Heinz H. Bauschke,et al.  On the Douglas–Rachford algorithm , 2016, Mathematical Programming.

[24]  Howard L. Weinert,et al.  Error bounds for the method of alternating projections , 1988, Math. Control. Signals Syst..

[25]  Heinz H. Bauschke,et al.  Linear and strong convergence of algorithms involving averaged nonexpansive operators , 2014, Journal of Mathematical Analysis and Applications.

[26]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[27]  Karin Schwab,et al.  Best Approximation In Inner Product Spaces , 2016 .

[28]  C. Badea,et al.  The rate of convergence in the method of alternating projections , 2010, 1006.2047.

[29]  F. Deutsch.  The Angle Between Subspaces of a Hilbert Space , 1995 .