A global linear and local superlinear/quadratic inexact non-interior continuation method for variational inequalities

TL;DR

This paper introduces an inexact non-interior continuation method for variational inequalities over convex sets, combining barrier-based smoothing with Newton's method, achieving global linear and local superlinear/quadratic convergence.

Contribution

It extends previous non-interior continuation methods to inexact solutions for high-dimensional problems over general convex sets using barrier-based smoothing.

Findings

Method converges globally linearly.

Achieves local superlinear/quadratic convergence.

Effective for various convex sets including cones and epigraphs.

Abstract

We use the concept of barrier-based smoothing approximations introduced in [ C. B. Chua and Z. Li, A barrier-based smoothing proximal point algorithm for NCPs over closed convex cones, SIOPT 23(2), 2010] to extend the non-interior continuation method proposed in [B. Chen and N. Xiu, A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, SIOPT 9(3), 1999] to an inexact non-interior continuation method for variational inequalities over general closed convex sets. Newton equations involved in the method are solved inexactly to deal with high dimension problems. The method is proved to have global linear and local superlinear/quadratic convergence under suitable assumptions. We apply the method to non-negative orthants, positive semidefinite cones, polyhedral sets, epigraphs of matrix operator norm cone and epigraphs of matrix nuclear norm cone.