A Feasible Direction Algorithm for Nonlinear Second-Order Cone Optimization Problems

In this work we present a new feasible direction algorithm for solving smooth nonlinear second-order cone programs. These problems consist of minimizing a nonlinear differentiable objective function subject to some nonlinear second-order cone constraints. Given a point interior to the feasible set defined by the nonlinear constraints, the proposed approach computes a feasible and descent direction for the objective function. The search direction is computed by using a formulation that is similar to the algorithm FDIPA for nonlinear programming. A line search along the search direction finds a new feasible point that has a lower value of the objective function. Repeating this process, the algorithm generates a feasible sequence with a monotone decrease of the objective function. Under mild assumptions we prove that the present algorithm converge globally to stationary points of the nonlinear second-order cone program. We test our algorithm with several instances of robust classification of support vector machines.