A barrier-based smoothing proximal point algorithm for NCPs over closed convex cones
Chua, Chek Beng.
Date of Issue2013
School of Physical and Mathematical Sciences
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 cone 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 ill-conditioning 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 show that our algorithm is comparable with the Newton-CG augmented Lagrangian algorithm proposed in [X. Y. Zhao, D. Sun, and K.-C. Toh, SIAM J. Optim., 20 (2010), pp. 1737--1765].
SIAM journal on optimization
© 2013 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Optimization and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/12087565X]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.