An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization

Handle URI:
http://hdl.handle.net/10754/597514
Title:
An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization
Authors:
Petra, Cosmin G.; Schenk, Olaf; Lubin, Miles; Gäertner, Klaus
Abstract:
We present a scalable approach and implementation for solving stochastic optimization problems on high-performance computers. In this work we revisit the sparse linear algebra computations of the parallel solver PIPS with the goal of improving the shared-memory performance and decreasing the time to solution. These computations consist of solving sparse linear systems with multiple sparse right-hand sides and are needed in our Schur-complement decomposition approach to compute the contribution of each scenario to the Schur matrix. Our novel approach uses an incomplete augmented factorization implemented within the PARDISO linear solver and an outer BiCGStab iteration to efficiently absorb pivot perturbations occurring during factorization. This approach is capable of both efficiently using the cores inside a computational node and exploiting sparsity of the right-hand sides. We report on the performance of the approach on highperformance computers when solving stochastic unit commitment problems of unprecedented size (billions of variables and constraints) that arise in the optimization and control of electrical power grids. Our numerical experiments suggest that supercomputers can be efficiently used to solve power grid stochastic optimization problems with thousands of scenarios under the strict "real-time" requirements of power grid operators. To our knowledge, this has not been possible prior to the present work. © 2014 Society for Industrial and Applied Mathematics.
Citation:
Petra CG, Schenk O, Lubin M, Gäertner K (2014) An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization. SIAM Journal on Scientific Computing 36: C139–C162. Available: http://dx.doi.org/10.1137/130908737.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
Issue Date:
Jan-2014
DOI:
10.1137/130908737
Type:
Article
ISSN:
1064-8275; 1095-7197
Sponsors:
We acknowledge Jeff Hammond for his guidance in BG/Pimplementation and informative discussions. We also acknowledge Edward Rothbergof Gurobi for an early discussion on the use of the augmented approach. K. G¨artneralso acknowledges King Abdullah University of Science and Technology (KAUST) forproviding access to a BG/P platform.
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Full metadata record

DC FieldValue Language
dc.contributor.authorPetra, Cosmin G.en
dc.contributor.authorSchenk, Olafen
dc.contributor.authorLubin, Milesen
dc.contributor.authorGäertner, Klausen
dc.date.accessioned2016-02-25T12:41:14Zen
dc.date.available2016-02-25T12:41:14Zen
dc.date.issued2014-01en
dc.identifier.citationPetra CG, Schenk O, Lubin M, Gäertner K (2014) An Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimization. SIAM Journal on Scientific Computing 36: C139–C162. Available: http://dx.doi.org/10.1137/130908737.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/130908737en
dc.identifier.urihttp://hdl.handle.net/10754/597514en
dc.description.abstractWe present a scalable approach and implementation for solving stochastic optimization problems on high-performance computers. In this work we revisit the sparse linear algebra computations of the parallel solver PIPS with the goal of improving the shared-memory performance and decreasing the time to solution. These computations consist of solving sparse linear systems with multiple sparse right-hand sides and are needed in our Schur-complement decomposition approach to compute the contribution of each scenario to the Schur matrix. Our novel approach uses an incomplete augmented factorization implemented within the PARDISO linear solver and an outer BiCGStab iteration to efficiently absorb pivot perturbations occurring during factorization. This approach is capable of both efficiently using the cores inside a computational node and exploiting sparsity of the right-hand sides. We report on the performance of the approach on highperformance computers when solving stochastic unit commitment problems of unprecedented size (billions of variables and constraints) that arise in the optimization and control of electrical power grids. Our numerical experiments suggest that supercomputers can be efficiently used to solve power grid stochastic optimization problems with thousands of scenarios under the strict "real-time" requirements of power grid operators. To our knowledge, this has not been possible prior to the present work. © 2014 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipWe acknowledge Jeff Hammond for his guidance in BG/Pimplementation and informative discussions. We also acknowledge Edward Rothbergof Gurobi for an early discussion on the use of the augmented approach. K. G¨artneralso acknowledges King Abdullah University of Science and Technology (KAUST) forproviding access to a BG/P platform.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectEconomic dispatchen
dc.subjectParallel linear algebraen
dc.subjectParallelinterior pointen
dc.subjectStochastic optimizationen
dc.subjectStochastic programmingen
dc.subjectUnit commitmenten
dc.titleAn Augmented Incomplete Factorization Approach for Computing the Schur Complement in Stochastic Optimizationen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.contributor.institutionArgonne National Laboratory, Argonne, United Statesen
dc.contributor.institutionUniversita della Svizzera italiana, Lugano, Switzerlanden
dc.contributor.institutionMassachusetts Institute of Technology, Cambridge, United Statesen
dc.contributor.institutionWeierstrass Institute for Applied Analysis and Stochastics, Berlin, Germanyen
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