HyKKT: a hybrid direct-iterative method for solving KKT linear systems

We propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the...

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Vydané v:	Optimization methods & software Ročník 38; číslo 2; s. 332 - 355
Hlavní autori:	Regev, Shaked, Chiang, Nai-Yuan, Darve, Eric, Petra, Cosmin G., Saunders, Michael A., Świrydowicz, Kasia, Peleš, Slaven
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Abingdon Taylor & Francis 04.03.2023 Taylor & Francis Ltd
Predmet:	Cholesky factorization Factorization GPU Graphics processing units interior methods KKT systems Linear systems MATHEMATICS AND COMPUTING Optimization Performance degradation Power flow sparse matrix factorization
ISSN:	1055-6788, 1029-4937
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Abstract	We propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBLT factorization where is a lower triangular matrix and is or block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBL T factorization of the full system.
AbstractList	We propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBLT factorization where is a lower triangular matrix and is or block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBLT factorization of the full system. We propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBLT factorization where is a lower triangular matrix and is or block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBL T factorization of the full system. Here, we propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBLT factorization where L is a lower triangular matrix and B is 1×1 or 2×2 block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBLT factorization of the full system.
Author	Saunders, Michael A. Chiang, Nai-Yuan Petra, Cosmin G. Darve, Eric Peleš, Slaven Regev, Shaked Świrydowicz, Kasia
Author_xml	– sequence: 1 givenname: Shaked orcidid: 0000-0003-2896-1658 surname: Regev fullname: Regev, Shaked email: sregev@stanford.edu organization: Stanford University – sequence: 2 givenname: Nai-Yuan surname: Chiang fullname: Chiang, Nai-Yuan organization: Lawrence Livermore National Laboratory – sequence: 3 givenname: Eric surname: Darve fullname: Darve, Eric organization: Stanford University – sequence: 4 givenname: Cosmin G. surname: Petra fullname: Petra, Cosmin G. organization: Lawrence Livermore National Laboratory – sequence: 5 givenname: Michael A. surname: Saunders fullname: Saunders, Michael A. organization: Stanford University – sequence: 6 givenname: Kasia surname: Świrydowicz fullname: Świrydowicz, Kasia organization: Pacific Northwest National Laboratory – sequence: 7 givenname: Slaven surname: Peleš fullname: Peleš, Slaven organization: Oak Ridge National Laboratory
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SubjectTerms	Cholesky factorization Factorization GPU Graphics processing units interior methods KKT systems Linear systems MATHEMATICS AND COMPUTING Optimization Performance degradation Power flow sparse matrix factorization
Title	HyKKT: a hybrid direct-iterative method for solving KKT linear systems
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