A globally and quadratically convergent smoothing Newton method for solving second-order cone optimization

Second-order cone optimization (denoted by SOCO) is a class of convex optimization problems and it contains the linear optimization problem, convex quadratic optimization problem and quadratically constrained convex quadratic optimization problem as special cases. In this paper, we propose a new smoothing Newton method for solving the SOCO based on a non-symmetrically perturbed smoothing Fischer–Burmeister function. At each iteration, a system of linear equations is solved only approximately by using the inexact Newton method. It is shown that any accumulation point of the iteration sequence generated by the proposed algorithm is a solution of the SOCO. Furthermore, we prove that the generated sequence is bounded and hence it has at least one accumulation point. Under the assumption of nonsingularity, we establish the local quadratic convergence of the proposed algorithm without strict complementarity condition. Numerical experiments indicate that our method is effective.