Condensed-space methods for nonlinear programming on GPUs
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909736104886272 |
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| author | Pacaud, François Shin, Sungho Montoison, Alexis Schanen, Michel Anitescu, Mihai |
| author_facet | Pacaud, François Shin, Sungho Montoison, Alexis Schanen, Michel Anitescu, Mihai |
| contents | This paper explores two condensed-space interior-point methods to efficiently solve large-scale nonlinear programs on graphics processing units (GPUs). The interior-point method solves a sequence of symmetric indefinite linear systems, or Karush-Kuhn-Tucker (KKT) systems, which become increasingly ill-conditioned as we approach the solution. Solving a KKT system with traditional sparse factorization methods involve numerical pivoting, making parallelization difficult. A solution is to condense the KKT system into a symmetric positive-definite matrix and solve it with a Cholesky factorization, stable without pivoting. Although condensed KKT systems are more prone to ill-conditioning than the original ones, they exhibit structured ill-conditioning that mitigates the loss of accuracy. This paper compares the benefits of two recent condensed-space interior-point methods, HyKKT and LiftedKKT. We implement the two methods on GPUs using MadNLP.jl, an optimization solver interfaced with the NVIDIA sparse linear solver cuDSS and with the GPU-accelerated modeler ExaModels.jl. Our experiments on the PGLIB and the COPS benchmarks reveal that GPUs can attain up to a tenfold speed increase compared to CPUs when solving large-scale instances. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14236 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Condensed-space methods for nonlinear programming on GPUs Pacaud, François Shin, Sungho Montoison, Alexis Schanen, Michel Anitescu, Mihai Optimization and Control This paper explores two condensed-space interior-point methods to efficiently solve large-scale nonlinear programs on graphics processing units (GPUs). The interior-point method solves a sequence of symmetric indefinite linear systems, or Karush-Kuhn-Tucker (KKT) systems, which become increasingly ill-conditioned as we approach the solution. Solving a KKT system with traditional sparse factorization methods involve numerical pivoting, making parallelization difficult. A solution is to condense the KKT system into a symmetric positive-definite matrix and solve it with a Cholesky factorization, stable without pivoting. Although condensed KKT systems are more prone to ill-conditioning than the original ones, they exhibit structured ill-conditioning that mitigates the loss of accuracy. This paper compares the benefits of two recent condensed-space interior-point methods, HyKKT and LiftedKKT. We implement the two methods on GPUs using MadNLP.jl, an optimization solver interfaced with the NVIDIA sparse linear solver cuDSS and with the GPU-accelerated modeler ExaModels.jl. Our experiments on the PGLIB and the COPS benchmarks reveal that GPUs can attain up to a tenfold speed increase compared to CPUs when solving large-scale instances. |
| title | Condensed-space methods for nonlinear programming on GPUs |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2405.14236 |