(177h) Overlapping Schwarz: A New Decomposition Paradigm for Large-Scale Optimization
AIChE Annual Meeting
2020
2020 Virtual AIChE Annual Meeting
Computing and Systems Technology Division
CAST Director's Student Presentation Award Finalists (Invited Talks)
Monday, November 16, 2020 - 9:45am to 10:00am
In this work we propose a different decomposition paradigm for optimization: Overlapping Schwarz Decomposition. Overlapping Schwarz methods have been traditionally used to solve linear algebra systems arising from partial differential equations [6], but we have recently shown that these methods can be generalized to solve structured optimization problems [7,8,9]. In the proposed scheme, we consider optimization problems whose algebraic structure can be represented by graphs. The graph domain is partitioned into overlapping subdomains, yielding a set of coupled subproblems. The algorithm computes the primal-dual solution of the subproblems in parallel and facilitates convergence by communicating the solution information in the coupled regions. The size of overlap is the key algorithmic parameter. In particular, we show that convergence can be guaranteed if the overlap is sufficiently large and that the convergence rate improves exponentially with the size of the overlap (i.e., the larger the overlap, the faster the convergence). However, increasing the size of overlap increases the subproblem complexity at the same time; thus, one needs to tune the size of overlap to achieve optimal convergence performance. Overlapping Schwarz schemes provide a bridge between fully decentralized (no overlap) and centralized algorithms (the overlap is the entire domain).
Our convergence results rely on a fundamental property of structured optimization problems that we call exponential decay of sensitivity [9,10,11]. This property states that the sensitivity of the primal-dual solution at a given node decays exponentially with respect to the distance from the node on which parametric perturbation is applied. We establish sufficient conditions for this property to hold and provide an explicit estimate of the sensitivity coefficients. Our results reveal that local convexity and generalized controllability provide sufficient conditions for exponential decay of sensitivity to hold.
Overlapping Schwarz decomposition is a flexible and versatile approach for decomposing structured optimization problems. Specifically, we show that this paradigm can be applied to decompose problems at three different levels: (i) problem level, (ii) subproblem level (within an augmented Lagrangian or sequential quadratic programming method), and (iii) linear algebra level (within an interior point method). Problem level decomposition enables fully decentralized and asynchronous implementation (no centralized coordinator is necessary), thus advantageous for decentralized control, but the convergence may not be robust for non-convex problems [7,9]. The linear algebra level decomposition requires centralized implementation but enables efficient computations and robust convergence (inherit the convergence of the outer loop) [12]. The effectiveness of the proposed algorithms is demonstrated with a wide variety of problem instances that arise in energy systems [13,14].
References:
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