A SUPERLINEARLY CONVERGENT SMOOTHING NEWTON CONTINUATION ALGORITHM FOR VARIATIONAL INEQUALITIES OVER DEFINABLE SETS

作者：Chua C B^*; Hien L T K

来源：SIAM Journal on Optimization, 2015, 25(2): 1034-1063.

DOI：10.1137/140957615

• variational inequalities
• smoothing Newton continuation
• superlinear convergence
• barrier-based smoothing approximation

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. 593615] 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(Ralg), then the gradient map of its universal barrier is definable in the o-minimal expansion R-an,R-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.

• 出版日期2015
• 单位南阳理工学院