### Abstract

We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the Fischer-Burmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function _{FB} as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan P-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104(2005), pp. 293-327), which confirm the theoretical results and the effectiveness of the algorithm.

Original language | English |
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Pages (from-to) | 1173-1197 |

Number of pages | 25 |

Journal | Optimization |

Volume | 59 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2010 Oct 29 |

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### Keywords

- Derivative-free methods
- Descent algorithms
- Fischer- burmeister function
- Linear convergence
- Second-order cone complementarity problem

### ASJC Scopus subject areas

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