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Douglas-Rachford method for the feasibility problem involving a circle and a disc

Suvendu Pattanaik(suvendu.pattanaik***at***gmail.com)
Sweta Shrivastav(swetasrivastava68***at***gmail.com)

Abstract: The Douglas-Rachford algorithm is a classical and a successful method for solving the feasibility problems. Here, we provide a region for global convergence of the algorithm for the feasibility problem involving a disc and a circle in the Euclidean space of dimension two.

Keywords: Douglas-Rachford algorithm, global convergence, feasibility problem, projector, reflector

Category 1: Global Optimization (Applications )

Citation: 1. Borwein, J.M., Sims, B.: The Douglas-Rachford algorithm in the absence of convexity. Fixed-point Algorithms for Inverse Problems in Science and Engineering. 49, 93-109 (2011). 2. Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Transactions of the AMS. 82, 421-439 (1956). 3. Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM Journal on Numerical Analysis. 16, 964-979 (1979). 4. Elser, V., Rankenburg I., Thibault, P.: Searching with iterated maps. Proceedings of the National Academy of Sciences. 104(2), 418-423 (2007). 5. Aragon Artacho, F.J., Borwein J.M., Tam, M.K.: Recent results on Douglas-Rachford methods for the combinatorial optimization problem. J. Optim. Theory Appl. 163(1), 1-30, (2014). 6. Aragon Artacho, F.J., Borwein J.M., Tam, M.K.: Douglas-Rachford feasibility methods for matrix completion problems. ANZIAM J. 55(4), 299-326 (2014). 7. Borwein, J.M., Tam, M.K.: Reflection methods for inverse problems with applications to protein conformation determination. In: Springer Volume on the CIMPA school Generalized Nash Equilibrium Problems, Bilevel Programming and MPEC, New Delhi (2012). 8. Benoist, J.: The DouglasRachford algorithm for the case of the sphere and the line. J. Global Optimization. 63, 363-380 (2015). 9. Gravel, S., Elser, V.: Divide and concur: A general approach constraint satisfaction. Phys. Rev. E. 78, 036706, 15 (2008). 10. Bauschke, H.H., Combettes, P.L., Luke, D.R.: Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization. J. Opt. Soc. Amer. A. 19, 1334-1345 (2002). 11. Aragon Artacho, F.J., Borwein, J.M., Tam, M.K.: Global behaviour of the Douglas-Rachford method for a nonconvex feasibility problem. J. Global Optimization. 65(2), 309-327 (2016). 12. Svaiter, B.F.: On the weak convergence of the DouglasRachford method. SIAM J. Control Optimization. 49(1), 280-287 (2011). 13. Bauschke, H.H., Dao, M.N.: On the nite convergence of the Douglas-Rachford algorithm for solving (not necessarily convex) feasibility problems in Euclidean spaces. SIAM Journal on Optimization. 27 (1), 507-537 (2017).

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Entry Submitted: 09/07/2018
Entry Accepted: 09/07/2018
Entry Last Modified: 09/07/2018

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