### Abstract

In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical results demonstrate that the algorithm is effective. In addition, two kinds of numerical comparisons are presented which provides numerical evidence about why the smoothing Newton method is employed and also suggests a suitable smoothing function for future numerical implementations. Finally, we point out that although the main idea for proving the convergence is similar to the one used in the literature, the analysis is indeed more subtle and involves more techniques due to the feature of second-order cone.

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

Pages (from-to) | 82-96 |

Number of pages | 15 |

Journal | Applied Numerical Mathematics |

Volume | 120 |

DOIs | |

Publication status | Published - 2017 Oct 1 |

### Fingerprint

### Keywords

- Absolute value equations
- Second-order cone
- Smoothing Newton algorithm

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Numerical Mathematics*,

*120*, 82-96. https://doi.org/10.1016/j.apnum.2017.04.012

**A smoothing Newton method for absolute value equation associated with second-order cone.** / Miao, Xin He; Yang, Jian Tao; Saheya, B.; Chen, Jein-Shan.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 120, pp. 82-96. https://doi.org/10.1016/j.apnum.2017.04.012

}

TY - JOUR

T1 - A smoothing Newton method for absolute value equation associated with second-order cone

AU - Miao, Xin He

AU - Yang, Jian Tao

AU - Saheya, B.

AU - Chen, Jein-Shan

PY - 2017/10/1

Y1 - 2017/10/1

N2 - In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical results demonstrate that the algorithm is effective. In addition, two kinds of numerical comparisons are presented which provides numerical evidence about why the smoothing Newton method is employed and also suggests a suitable smoothing function for future numerical implementations. Finally, we point out that although the main idea for proving the convergence is similar to the one used in the literature, the analysis is indeed more subtle and involves more techniques due to the feature of second-order cone.

AB - In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical results demonstrate that the algorithm is effective. In addition, two kinds of numerical comparisons are presented which provides numerical evidence about why the smoothing Newton method is employed and also suggests a suitable smoothing function for future numerical implementations. Finally, we point out that although the main idea for proving the convergence is similar to the one used in the literature, the analysis is indeed more subtle and involves more techniques due to the feature of second-order cone.

KW - Absolute value equations

KW - Second-order cone

KW - Smoothing Newton algorithm

UR - http://www.scopus.com/inward/record.url?scp=85019344946&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019344946&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2017.04.012

DO - 10.1016/j.apnum.2017.04.012

M3 - Article

VL - 120

SP - 82

EP - 96

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -