It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this short paper, we first study the global convergence of a sequential injective algorithm for weakly univalent vector equation. Then, we apply the convergence analysis to the regularized smoothing Newton algorithm for mixed nonlinear second-order cone complementarity problems. We prove the global convergence property under the (Cartesian) $P_0$ assumption, which is strictly weaker than the original monotonicity assumption.

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View Global convergence of sequential injective algorithm for weakly univalent vector equation: application to regularized smoothing Newton algorithm