A Barrier-Based Smoothing Proximal Point Algorithm for NCPs over Closed Convex Cones

We present a new barrier-based method of constructing smoothing approximations for the Euclidean projector onto closed convex cones. These smoothing approximations are used in a smoothing proximal point algorithm to solve monotone nonlinear complementarity problems (NCPs) over a convex cones via the normal map equation. The smoothing approximations allow for the solution of the smoothed normal map equations with Newton’s method, and do not require additional analytical properties of the Euclidean projector. The use of proximal terms in the algorithm adds stability to the solution of the smoothed normal map equation, and avoids numerical issues due to illconditioning at iterates near the boundary of the cones. We prove a sufficient condition on the barrier used that guarantees the convergence of the algorithm to a solution of the NCP. The sufficient condition is satisfied by all logarithmically homogeneous barriers. Preliminary numerical tests on semidefinite programming problems (SDPs) shows that our algorithm is comparable with the Newton-CG augmented Lagrangian algorithm (SDPNAL) proposed in [X. Y. Zhao, D. Sun, and K.-C.Toh, SIAM J. Optim. 20 (2010), 1737–1765].