### Abstract

We present a nonlinear least-square formulation for the second-order cone complementarity problem based on the Fischer-Burmeister (FB) function and the plus function. This formulation has two-fold advantages. First, the operator involved in the over-determined system of equations inherits the favourable properties of the FB function for local convergence, for example, the (strong) semi-smoothness; second, the natural merit function of the over-determined system of equations share all the nice features of the class of merit functions fYF studied in [J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005), pp. 293-327] for global convergence. We propose a semi-smooth Levenberg-Marquardt method to solve the arising over-determined system of equations, and establish the global and local convergence results. Among others, the superlinear (quadratic) rate of convergence is obtained under strict complementarity of the solution and a local error bound assumption, respectively. Numerical results verify the advantages of the least-square reformulation for difficult problems.

Original language | English |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Optimization Methods and Software |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 Feb 1 |

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### Keywords

- Fischer-Burmeister function
- Levenberg-Marquardt method
- second-order cone complementarity problem
- semi-smooth

### ASJC Scopus subject areas

- Software
- Control and Optimization
- Applied Mathematics