### Abstract

In the conic optimization problems, it is well-known that a positive duality gap may occur, and that solving such a problem is numerically difficult or unstable. For such a case, we propose a facial reduction algorithm to find a primal-dual pair of conic optimization problems having the zero duality gap and the optimal value equal to one of the original primal or dual problems. The conic expansion approach is also known as a method to find such a primal-dual pair, and in this paper we clarify the relationship between our facial reduction algorithm and the conic expansion approach. Our analysis shows that, although they can be regarded as dual to each other, our facial reduction algorithm has ability to produce a finer sequence of faces of the cone including the feasible region. A simple proof of the convergence of our facial reduction algorithm for the conic optimization is presented. We also observe that our facial reduction algorithm has a practical impact by showing numerical experiments for graph partition problems; our facial reduction algorithm in fact enhances the numerical stability in those problems.

Original language | English |
---|---|

Pages (from-to) | 188-215 |

Number of pages | 28 |

Journal | Journal of Optimization Theory and Applications |

Volume | 158 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics

### Cite this

*Journal of Optimization Theory and Applications*,

*158*(1), 188-215. https://doi.org/10.1007/s10957-012-0219-y

**Facial Reduction Algorithms for Conic Optimization Problems.** / Waki, Hayato; Muramatsu, Masakazu.

Research output: Contribution to journal › Article

*Journal of Optimization Theory and Applications*, vol. 158, no. 1, pp. 188-215. https://doi.org/10.1007/s10957-012-0219-y

}

TY - JOUR

T1 - Facial Reduction Algorithms for Conic Optimization Problems

AU - Waki, Hayato

AU - Muramatsu, Masakazu

PY - 2013/7/1

Y1 - 2013/7/1

N2 - In the conic optimization problems, it is well-known that a positive duality gap may occur, and that solving such a problem is numerically difficult or unstable. For such a case, we propose a facial reduction algorithm to find a primal-dual pair of conic optimization problems having the zero duality gap and the optimal value equal to one of the original primal or dual problems. The conic expansion approach is also known as a method to find such a primal-dual pair, and in this paper we clarify the relationship between our facial reduction algorithm and the conic expansion approach. Our analysis shows that, although they can be regarded as dual to each other, our facial reduction algorithm has ability to produce a finer sequence of faces of the cone including the feasible region. A simple proof of the convergence of our facial reduction algorithm for the conic optimization is presented. We also observe that our facial reduction algorithm has a practical impact by showing numerical experiments for graph partition problems; our facial reduction algorithm in fact enhances the numerical stability in those problems.

AB - In the conic optimization problems, it is well-known that a positive duality gap may occur, and that solving such a problem is numerically difficult or unstable. For such a case, we propose a facial reduction algorithm to find a primal-dual pair of conic optimization problems having the zero duality gap and the optimal value equal to one of the original primal or dual problems. The conic expansion approach is also known as a method to find such a primal-dual pair, and in this paper we clarify the relationship between our facial reduction algorithm and the conic expansion approach. Our analysis shows that, although they can be regarded as dual to each other, our facial reduction algorithm has ability to produce a finer sequence of faces of the cone including the feasible region. A simple proof of the convergence of our facial reduction algorithm for the conic optimization is presented. We also observe that our facial reduction algorithm has a practical impact by showing numerical experiments for graph partition problems; our facial reduction algorithm in fact enhances the numerical stability in those problems.

UR - http://www.scopus.com/inward/record.url?scp=84878771087&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878771087&partnerID=8YFLogxK

U2 - 10.1007/s10957-012-0219-y

DO - 10.1007/s10957-012-0219-y

M3 - Article

AN - SCOPUS:84878771087

VL - 158

SP - 188

EP - 215

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

IS - 1

ER -