Chek Beng Chua | Searcharxiv

arXiv2025

Carath\'eodory number of homogeneous convex cones

We study the Carath\'eodory number of homogeneous convex cones via their spectrahedral representations. A characterization of homogeneous convex cones whose ranks match their Carath\'eodory numbers is given. This characterization is then used to show that a homogeneous convex cone is selfdual if and only if its rank matches the Carath\'eodory numbers of both its closure and its dual cone. It is further used to show that the only sparse spectrahedral cones that are homogeneous convex cones are those described by homogeneous chordal graphs.

math.OC↗

arXiv2020

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

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.

math.OC↗