We give some improved convergence results about the smoothing-regularization approach to mathematical programs with vanishing constraints (MPVC for short), which is proposed in Achtziger et al. (2013). We show that the Mangasarian-Fromovitz constraints qualification for the smoothing-regularization problem still holds under the VC-MFCQ (see Definition 5) which is weaker than the VC-LICQ (see Definition 7) and the condition of asymptotic nondegeneracy. We also analyze the convergence behavior of the smoothing-regularization method and prove that any accumulation point of a sequence of stationary points for the smoothing-regularization problem is still strongly-stationary under the VC-MFCQ and the condition of asymptotic nondegeneracy.

A limited memory BFGS (L-BFGS) algorithm is presented for solving large-scale symmetric nonlinear equations, where a line search technique without derivative information is used. The global convergence of the proposed algorithm is established under some suitable conditions. Numerical results show that the given method is competitive to those of the normal BFGS methods.