Smoothing Newton algorithm for the second-order cone programming with a nonmonotone line search

The smoothing‐type algorithms, which are in general designed based on some monotone line search, have been successfully applied to solve the second‐order cone programming (denoted by SOCP). In this paper, we propose a nonmonotone smoothing Newton algorithm for solving the SOCP. Under suitable assumptions, we show that the proposed algorithm is globally and locally quadratically convergent. To compare with the existing smoothing‐type algorithms for the SOCP, our algorithm has the following special properties: (i) it is based on a new smoothing function of the vector‐valued natural residual function; (ii) it uses a nonmonotone line search scheme which contains the usual monotone line search as a special case. Preliminary numerical results demonstrate that the smoothing‐type algorithm using the nonmonotone line search is promising for solving the SOCP.