In this paper, we use the concept of barrier-based smoothing approximations introduced by Chua and Li to extend various smoothing Newton continuation algorithms 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 ℜ_{an}^{ℜ_{alg}}, then the gradient map of its universal barrier is definable in the o-minimal expansion ℜ_{an,exp}.

## Citation

Research Report, Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, February 2014.