### Abstract

The semi definite programming (SDP) problem is one of the central problems in mathematical optimization. The primal-dual interior-point method (PDIPM) is one of the most powerful algorithms for solving SDP problems, and many research groups have employed it for developing software packages. However, two well-known major bottlenecks, i.e., the generation of the Schur complement matrix (SCM) and its Cholesky factorization, exist in the algorithmic framework of the PDIPM. We have developed a new version of the semi definite programming algorithm parallel version (SDPARA), which is a parallel implementation on multiple CPUs and GPUs for solving extremely large-scale SDP problems with over a million constraints. SDPARA can automatically extract the unique characteristics from an SDP problem and identify the bottleneck. When the generation of the SCM becomes a bottleneck, SDPARA can attain high scalability using a large quantity of CPU cores and some processor affinity and memory interleaving techniques. SDPARA can also perform parallel Cholesky factorization using thousands of GPUs and techniques for overlapping computation and communication if an SDP problem has over two million constraints and Cholesky factorization constitutes a bottleneck. We demonstrate that SDPARA is a high-performance general solver for SDPs in various application fields through numerical experiments conducted on the TSUBAME 2.5 supercomputer, and we solved the largest SDP problem (which has over 2.33 million constraints), thereby creating a new world record. Our implementation also achieved 1.713 PFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.

Original language | English |
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Title of host publication | Proceedings - IEEE 28th International Parallel and Distributed Processing Symposium, IPDPS 2014 |

Publisher | IEEE Computer Society |

Pages | 1171-1180 |

Number of pages | 10 |

ISBN (Print) | 9780769552071 |

DOIs | |

Publication status | Published - Jan 1 2014 |

Event | 28th IEEE International Parallel and Distributed Processing Symposium, IPDPS 2014 - Phoenix, AZ, United States Duration: May 19 2014 → May 23 2014 |

### Publication series

Name | Proceedings of the International Parallel and Distributed Processing Symposium, IPDPS |
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ISSN (Print) | 1530-2075 |

ISSN (Electronic) | 2332-1237 |

### Other

Other | 28th IEEE International Parallel and Distributed Processing Symposium, IPDPS 2014 |
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Country | United States |

City | Phoenix, AZ |

Period | 5/19/14 → 5/23/14 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computational Theory and Mathematics
- Computer Networks and Communications
- Hardware and Architecture
- Software

### Cite this

*Proceedings - IEEE 28th International Parallel and Distributed Processing Symposium, IPDPS 2014*(pp. 1171-1180). [6877345] (Proceedings of the International Parallel and Distributed Processing Symposium, IPDPS). IEEE Computer Society. https://doi.org/10.1109/IPDPS.2014.121

**Petascale general solver for semidefinite programming problems with over two million constraints.** / Fujisawa, Katsuki; Endo, Toshio; Yasui, Yuichiro; Sato, Hitoshi; Matsuzawa, Naoki; Matsuoka, Satoshi; Waki, Hayato.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - IEEE 28th International Parallel and Distributed Processing Symposium, IPDPS 2014.*, 6877345, Proceedings of the International Parallel and Distributed Processing Symposium, IPDPS, IEEE Computer Society, pp. 1171-1180, 28th IEEE International Parallel and Distributed Processing Symposium, IPDPS 2014, Phoenix, AZ, United States, 5/19/14. https://doi.org/10.1109/IPDPS.2014.121

}

TY - GEN

T1 - Petascale general solver for semidefinite programming problems with over two million constraints

AU - Fujisawa, Katsuki

AU - Endo, Toshio

AU - Yasui, Yuichiro

AU - Sato, Hitoshi

AU - Matsuzawa, Naoki

AU - Matsuoka, Satoshi

AU - Waki, Hayato

PY - 2014/1/1

Y1 - 2014/1/1

N2 - The semi definite programming (SDP) problem is one of the central problems in mathematical optimization. The primal-dual interior-point method (PDIPM) is one of the most powerful algorithms for solving SDP problems, and many research groups have employed it for developing software packages. However, two well-known major bottlenecks, i.e., the generation of the Schur complement matrix (SCM) and its Cholesky factorization, exist in the algorithmic framework of the PDIPM. We have developed a new version of the semi definite programming algorithm parallel version (SDPARA), which is a parallel implementation on multiple CPUs and GPUs for solving extremely large-scale SDP problems with over a million constraints. SDPARA can automatically extract the unique characteristics from an SDP problem and identify the bottleneck. When the generation of the SCM becomes a bottleneck, SDPARA can attain high scalability using a large quantity of CPU cores and some processor affinity and memory interleaving techniques. SDPARA can also perform parallel Cholesky factorization using thousands of GPUs and techniques for overlapping computation and communication if an SDP problem has over two million constraints and Cholesky factorization constitutes a bottleneck. We demonstrate that SDPARA is a high-performance general solver for SDPs in various application fields through numerical experiments conducted on the TSUBAME 2.5 supercomputer, and we solved the largest SDP problem (which has over 2.33 million constraints), thereby creating a new world record. Our implementation also achieved 1.713 PFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.

AB - The semi definite programming (SDP) problem is one of the central problems in mathematical optimization. The primal-dual interior-point method (PDIPM) is one of the most powerful algorithms for solving SDP problems, and many research groups have employed it for developing software packages. However, two well-known major bottlenecks, i.e., the generation of the Schur complement matrix (SCM) and its Cholesky factorization, exist in the algorithmic framework of the PDIPM. We have developed a new version of the semi definite programming algorithm parallel version (SDPARA), which is a parallel implementation on multiple CPUs and GPUs for solving extremely large-scale SDP problems with over a million constraints. SDPARA can automatically extract the unique characteristics from an SDP problem and identify the bottleneck. When the generation of the SCM becomes a bottleneck, SDPARA can attain high scalability using a large quantity of CPU cores and some processor affinity and memory interleaving techniques. SDPARA can also perform parallel Cholesky factorization using thousands of GPUs and techniques for overlapping computation and communication if an SDP problem has over two million constraints and Cholesky factorization constitutes a bottleneck. We demonstrate that SDPARA is a high-performance general solver for SDPs in various application fields through numerical experiments conducted on the TSUBAME 2.5 supercomputer, and we solved the largest SDP problem (which has over 2.33 million constraints), thereby creating a new world record. Our implementation also achieved 1.713 PFlops in double precision for large-scale Cholesky factorization using 2,720 CPUs and 4,080 GPUs.

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U2 - 10.1109/IPDPS.2014.121

DO - 10.1109/IPDPS.2014.121

M3 - Conference contribution

AN - SCOPUS:84906672755

SN - 9780769552071

T3 - Proceedings of the International Parallel and Distributed Processing Symposium, IPDPS

SP - 1171

EP - 1180

BT - Proceedings - IEEE 28th International Parallel and Distributed Processing Symposium, IPDPS 2014

PB - IEEE Computer Society

ER -