(705d) Overlapping Domain Decomposition Schemes for Graph-Structured Optimization Problems
Decomposition schemes have been increasingly used for solving large-scale optimization problems as they can help exploit parallel computing architectures and practical limitations of centralized schemes. In a typical decomposition scheme, the problem graph domain is decomposed into multiple subdomains and the subproblems associated with the subdomains are solved iteratively by exchanging primal/dual information at their boundaries. This type of paradigm is used by methods such as the alternating direction method of multipliers (ADMM) , Lagrangian decomposition , Benders decomposition , and Gauss-Seidel schemes . ADMM, in particular, has been widely used in applications due to its convenience of implementation . However, the convergence of ADMM is often slow and the algorithm parameters are difficult to tune [1, Section 3.2.2].
In this work, we present a new parallel decomposition paradigm for the solution of graph-structured optimization problems. Specifically, we propose to use the overlapping domain decomposition method which is widely used for the solution of PDEs - to address the slow convergence issue of existing decomposition schemes. A key and novel design concept of the proposed scheme is that it uses overlapping subdomains to promote and accelerate convergence. We show that the algorithm converges if the size of the overlap is sufficiently large and that the convergence rate improves exponentially with the size of the overlap . In particular, we demonstrate that the proposed method achieves superior computational performance over ADMM. The high efficiency is achieved thanks to a more efficient exploitation of the graph structure.
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