Unrestricted Douglas-Rachford algorithms for solving convex feasibility problems in Hilbert space

Abstract

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted and extended the algorithm with the aim of solving CFPs and even more general problems, such as finding zeros of the sum of two maximally monotone operators. Many developments which implement various concepts concerning this algorithm have occurred during the last decade. We introduce an unrestricted DR algorithm, which provides a general framework for such concepts. Using unrestricted products of a finite number of strongly nonexpansive operators, we apply this framework to provide new iterative methods, where, inter alia, such operators may be interlaced between the operators used in the scheme of our unrestricted DR algorithm.

Keywords:

• Convex feasibility problem
• common fixed point problem
• Douglas-Rachford algorithm
• iterative method
• strongly nonexpansive operator
• unrestricted product

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their comments on the manuscript which helped them improve an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Simeon Reich was partially supported by the Israel Science Foundation (Grant 820/17), the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.

Notes on contributors

Kay Barshad

Kay Barshad obtained his PhD in Mathematics from the Technion – Israel Institute of Technology, where he also received his MSc degree. Previously, he obtained his BSc degree in Pure Mathematics from the University of Haifa. He published 7 research papers so far. His research interests lie in the fields of Optimization Theory, Projection Methods, Nonlinear Analysis and Convex Analysis.

Aviv Gibali

Aviv Gibali is an Associate Professor, head of the Mathematics Department at Braude College of Engineering since 2020. Aviv received his BSc in Mathematics from the University of Haifa (2005) and PhD in Mathematics from the Technion – Israel Institute of Technology (2012). Afterwards hid did his postdoctoral studies (2012–2014) at the Fraunhofer Institute for Industrial Mathematics (ITWM) in Kaiserslautern, Germany. His research fields focused on modelling, algorithms developments and analysis, in particular for systems of linear and nonlinear equations or inequalities and optimization techniques in radiation therapy treatment planning and image and signal processing.

Simeon Reich

Simeon Reich is a Professor Emeritus of Mathematics at the Technion – Israel Institute of Technology, where he received His BSc (1970) and DSc (1973) degrees. He taught previously at Tel Aviv University, the University of Chicago, the University of Southern California, the University of California at Berkeley, the University of California at Santa Barbara, the Tokyo Institute of Technology, and at the Maria Curie–Sklodowska University in Lublin, Poland. Professor Reich has had seventeen PhD students, and his list of publications contains four monographs and more than 500 research papers on Nonlinear Analysis and Optimization. His areas of research include nonlinear operator theory, nonlinear evolution and integral equations, infinite-dimensional holomorphy, the identification and estimation of nonlinear distributed parameter systems, and sequential and parallel algorithms in feasibility and optimization.