### Abstract

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback- Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:1-19, 1993) holds.

Original language | English |
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Pages (from-to) | 477-499 |

Number of pages | 23 |

Journal | Computational Optimization and Applications |

Volume | 47 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Nov 1 |

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### Keywords

- Entropy-like distance
- Exponential multiplier method
- Proximal-like method
- Symmetric cone optimization

### ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Applied Mathematics

### Cite this

**An entropy-like proximal algorithm and the exponential multiplier method for convex symmetric cone programming.** / Chen, Jein-Shan; Pan, Shaohua.

Research output: Contribution to journal › Article

*Computational Optimization and Applications*, vol. 47, no. 3, pp. 477-499. https://doi.org/10.1007/s10589-008-9227-0

}

TY - JOUR

T1 - An entropy-like proximal algorithm and the exponential multiplier method for convex symmetric cone programming

AU - Chen, Jein-Shan

AU - Pan, Shaohua

PY - 2010/11/1

Y1 - 2010/11/1

N2 - We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback- Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:1-19, 1993) holds.

AB - We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback- Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:1-19, 1993) holds.

KW - Entropy-like distance

KW - Exponential multiplier method

KW - Proximal-like method

KW - Symmetric cone optimization

UR - http://www.scopus.com/inward/record.url?scp=78650067900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650067900&partnerID=8YFLogxK

U2 - 10.1007/s10589-008-9227-0

DO - 10.1007/s10589-008-9227-0

M3 - Article

AN - SCOPUS:78650067900

VL - 47

SP - 477

EP - 499

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 3

ER -