A least-square semismooth Newton method for the second-order cone complementarity problem

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.