An interior point algorithm of$O(\sqrt m \left| {\ln \varepsilon } \right|)$ iterations forC ¹-convex programming

Abstract

We present a theoretical result on a path-following algorithm for convex programs. The algorithm employs a nonsmooth Newton subroutine. It starts from a near center of a restricted constraint set, performs a partial nonsmooth Newton step in each iteration, and converges to a point whose cost is withinε accuracy of the optimal cost in$O(\sqrt m \left| {\ln \varepsilon } \right|)$ iterations, wherem is the number of constraints in the problem. Unlike other interior point methods, the analyzed algorithm only requires a first-order Lipschitzian condition and a generalized Hessian similarity condition on the objective and constraint functions. Therefore, our result indicates the theoretical feasibility of applying interior point methods to certainC ¹-optimization problems instead ofC ²-problems. Since the complexity bound is unchanged compared with similar algorithms forC ²-convex programming, the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods.