### Abstract

In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227-251] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Yamashita, M. Fukushima, New NCP functions and their properties, J. Optim. Theory Appl. 97 (1997) 115-135] to the SCCP. By exploiting the Cartesian P-properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R_{02}-property. All of these results generalize some recent important works in [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293-327; C.-K. Sim, J. Sun, D. Ralph, A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function, Math. Program. 107 (2006) 547-553; P. Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159-185] under a unified framework.

Original language | English |
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Pages (from-to) | 195-215 |

Number of pages | 21 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 355 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Jul 1 |

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### Keywords

- Cartesian P-properties
- Jordan algebra
- Lipschitz continuity
- Merit function
- Smoothness
- Symmetric cone complementarity problem

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics