# Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP

## Abstract

The Jacobian consistency of smoothing functions plays an important role for achieving the rapid convergence of Newton methods or Newton-like methods with an appropriate parameter control. In this paper, we study the properties, derive the computable formula for the Jacobian matrix and prove the Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for second-order cone complementarity problems proposed by Tang et al. (Comput Appl Math 33:655–669, 2014). Then we apply its Jacobian consistency to a smoothing Newton method with the appropriate parameter control presented by Chen et al. (Math Comput 67:519–540, 1998), and show the global convergence and local quadratic convergence of the algorithm for solving the SOCCP under rather weak assumptions.

- Authors:

- Guilin University of Electronic Technology, School of Mathematics and Computing Science, Guangxi Key Laboratory of Automatic Detecting Technology and Instruments (China)
- Guilin University of Electronic Technology, School of Mathematics and Computing Science, Guangxi Key Laboratory of Cryptography and Information Security (China)
- Guilin University of Electronic Technology, School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation (China)
- Wuhan University, School of Mathematics and Statistics (China)

- Publication Date:

- OSTI Identifier:
- 22769379

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 1; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; CONVERGENCE; FUNCTIONS; MATRICES; NEWTON METHOD

### Citation Formats

```
Chi, Xiaoni, Wang, Yang, Zhu, Zhibin, and Wan, Zhongping.
```*Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0352-6.

```
Chi, Xiaoni, Wang, Yang, Zhu, Zhibin, & Wan, Zhongping.
```*Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP*. United States. doi:10.1007/S40314-016-0352-6.

```
Chi, Xiaoni, Wang, Yang, Zhu, Zhibin, and Wan, Zhongping. Thu .
"Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP". United States. doi:10.1007/S40314-016-0352-6.
```

```
@article{osti_22769379,
```

title = {Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP},

author = {Chi, Xiaoni and Wang, Yang and Zhu, Zhibin and Wan, Zhongping},

abstractNote = {The Jacobian consistency of smoothing functions plays an important role for achieving the rapid convergence of Newton methods or Newton-like methods with an appropriate parameter control. In this paper, we study the properties, derive the computable formula for the Jacobian matrix and prove the Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for second-order cone complementarity problems proposed by Tang et al. (Comput Appl Math 33:655–669, 2014). Then we apply its Jacobian consistency to a smoothing Newton method with the appropriate parameter control presented by Chen et al. (Math Comput 67:519–540, 1998), and show the global convergence and local quadratic convergence of the algorithm for solving the SOCCP under rather weak assumptions.},

doi = {10.1007/S40314-016-0352-6},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 1,

volume = 37,

place = {United States},

year = {2018},

month = {3}

}