Applying a type of SOC-functions to solve a system of equalities and inequalities under the order induced by second-order cone

In this paper, we introduce a special type of SOC-functions which is a vectorvalued function associated with second-order cone. By using it, we construct a type of smoothing functions which converges to the projection function onto second-order cone. Then, we reformulate the system of equalities and inequalities under the order induced by second-order cone as a system of parameterized smooth equations. Accordingly, we propose a smoothing-type Newton algorithm to solve the reformulation, and show that the proposed algorithm is globally convergent and locally quadratically convergent under suitable assumptions. Preliminary numerical results demonstrate that the approach is effective. Numerical comparison based on various smoothing functions is reported as well.