A Superlinearly Convergent Smoothing Newton Continuation Algorithm for Variational Inequalities over Definable Sets
Chua, Chek Beng
Hien, L. T. K.
Date of Issue2015
School of Physical and Mathematical Sciences
In this paper, we use the concept of barrier-based smoothing approximations introduced by Chua and Li [SIAM J. Optim., 23 (2013), pp. 745--769] to extend the smoothing Newton continuation algorithm of Hayashi, Yamashita, and Fukushima [SIAM J. Optim., 15 (2005), pp. 593--615] to variational inequalities over general closed convex sets X. We prove that when the underlying barrier has a gradient map that is definable in some o-minimal structure, the iterates generated converge superlinearly to a solution of the variational inequality. We further prove that if X is proper and definable in the o-minimal structure Ran, then the gradient map of its universal barrier is definable in the o-minimal expansion Ran,exp. Finally, we consider the application of the algorithm to complementarity problems over epigraphs of matrix operator norm and nuclear norm and present preliminary numerical results.
SIAM Journal on Optimization
© 2015 Society for Industrial and Applied Mathematics (SIAM). 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 (SIAM). The published version is available at: [http://dx.doi.org/10.1137/140957615]. 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.