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In this paper, we use the concept of barrier-based smoothing approximations intro- duced 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 con- verge superlinearly to a solution of the variational inequality. We further prove that if X is proper and definable in the o-minimal structure R Ralg an , 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.

A Superlinearly Convergent Smoothing Newton Continuation Algorithm for Variational Inequalities over Definable Sets