AN AUGMENTED INCOMPLETE FACTORIZATION APPROACH FOR COMPUTING THE SCHUR COMPLEMENT IN STOCHASTIC OPTIMIZATION

作     者：Petra, Cosmin G. Schenk, Olaf Lubin, Miles Gaeertner, Klaus 

作者机构：Argonne Natl Lab Math & Comp Sci Div Argonne IL 60439 USA Univ Svizzera Italiana Inst Computat Sci CH-6900 Lugano Switzerland MIT Ctr Operat Res Cambridge MA 02139 USA Weierstrass Inst Appl Anal & Stochast D-10117 Berlin Germany 

出 版 物：《SIAM JOURNAL ON SCIENTIFIC COMPUTING》 (工业与应用数学会科学计算杂志)

年 卷 期：2014年第36卷第2期

页      面：C139-C162页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：U.S. Department of Energy [DE-AC02-06CH11357] Office of Science of the U.S. Department of Energy [DE-AC02-06CH11357] Department of Energy INCITE award "Optimization of Complex Energy Systems under Uncertainty," 

主　　题：parallel linear algebra stochastic programming stochastic optimization parallel-interior point economic dispatch unit commitment 

摘      要：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 high-performance 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.