A new merit function and its related properties for the second-order cone complementarity problem.

*(English)*Zbl 1178.90324Summary: Recently, J.-S. Chen and P. Tseng extended two merit functions for the nonlinear complementarity problem (NCP) and the semidefinite complementarity problem (SDCP) to the second-order cone commplementarity problem (SOCCP) and showed several favorable properties. In this paper, we extend a merit function for the NCP studied by Yamada, Yamashita, and Fukushima to the SOCCP and show that the SOCCP is equivalent to an unconstrained smooth minimization via this new merit function. Furthermore, we study conditions under which the new merit function provides a global error bound which plays an important role in analyzing the convergence rate of iterative methods for solving the SOCCP; and conditions under which the new merit function has bounded level sets which ensures that the sequence generated by a descent method has at least one accumulation point.

##### MSC:

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

26B05 | Continuity and differentiation questions |

26B35 | Special properties of functions of several variables, HĂ¶lder conditions, etc. |

65K05 | Numerical mathematical programming methods |