Sl1QP Based Algorithm with Trust Region Technique for Solving Nonlinear Second-Order Cone Programming Problems

In this paper, we propose an algorithm based on Fletcher’s Sl1QP method and the trust region technique for solving Nonlinear Second-Order Cone Programming (NSOCP) problems. The Sl1QP method was originally developed for nonlinear optimization problems with inequality constraints. It converts a constrained optimization problem into an unconstrained problem by using the l1 exact penalty function, and then finds an optimum by solving approximate quadratic programming subproblems successively. In order to apply the Sl1QP method to the NSOCP problem, we introduce an exact penalty function with respect to second-order cone constraints and reformulate the NSOCP problem as an unconstrained optimization problem. However, since each subproblem generated by the Sl1QP method is not differentiable, we reformulate it as a second-order cone programming problem whose objective function is quadratic and constraint functions are affine. We analyze the convergence property of the proposed algorithm, and show that the generated sequence converge to a stationary point of the NSOCP problem under mild assumptions. We also confirm the efficiency of the algorithm by means of numerical experiments.