A superlinearly convergent smoothing newton continuation algorithm for variational inequalities over definable sets - Chua, Chek Beng (2015)
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 e RRalg an, then the gradient map of its universal barrier is definable in the o-minimal expansion n 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.