Open Access
2013 On the convergence of simulation-based iterative methods for solving singular linear systems
Mengdi Wang, Dimitri P. Bertsekas
Stoch. Syst. 3(1): 38-95 (2013). DOI: 10.1214/12-SSY074


We consider the simulation-based solution of linear systems of equations, $Ax=b$, of various types frequently arising in large-scale applications, where $A$ is singular. We show that the convergence properties of iterative solution methods are frequently lost when they are implemented with simulation (e.g., using sample average approximation), as is often done in important classes of large-scale problems. We focus on special cases of algorithms for singular systems, including some arising in least squares problems and approximate dynamic programming, where convergence of the residual sequence $\{Ax_{k}-b\}$ may be obtained, while the sequence of iterates $\{x_{k}\}$ may diverge. For some of these special cases, under additional assumptions, we show that the iterate sequence is guaranteed to converge. For situations where the iterates diverge but the residuals converge to zero, we propose schemes for extracting from the divergent sequence another sequence that converges to a solution of $Ax=b$.


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Mengdi Wang. Dimitri P. Bertsekas. "On the convergence of simulation-based iterative methods for solving singular linear systems." Stoch. Syst. 3 (1) 38 - 95, 2013.


Published: 2013
First available in Project Euclid: 24 February 2014

zbMATH: 1295.65037
MathSciNet: MR3353468
Digital Object Identifier: 10.1214/12-SSY074

Primary: 15A06
Secondary: 60H99 , 65C05

Keywords: approximate dynamic programming , Monte-Carlo estimation , proximal method , regularization , simulation , singular system , Stochastic algorithm

Rights: Copyright © 2013 INFORMS Applied Probability Society

Vol.3 • No. 1 • 2013
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