NewVEM: A Newton Vertex Exchange Method for a Class of Constrained Self-Concordant Minimization Problems
Ling Liang, Kim-Chuan Toh, Haizhao Yang
TL;DR
NewVEM tackles generalized simplex-constrained self-concordant minimization by a robust double-loop framework: an outer projected Newton step and an inner vertex exchange method for solving strongly convex QP subproblems. A fast semismooth Newton-based projection onto the generalized simplex provides a feasible starting point with precise convergence guarantees, enabling efficient overall progress. The approach yields local linear convergence for both loops and demonstrates strong practical performance in large-scale D-optimal design and related problems, outperforming several state-of-the-art solvers in accuracy and speed. This work broadens scalable second-order methods to constrained self-concordant problems with generalized simplex sets, suggesting wide applicability in experimental design and other domains requiring efficient projection and Newton-type steps.
Abstract
We propose \textbf{NewVEM}, a Newton vertex exchange method for efficiently solving self-concordant minimization problems under generalized simplex constraints. The algorithm features a two-level structure: the outer loop employs a projected Newton method, and the inner loop uses a vertex exchange approach to solve strongly convex quadratic subproblems. Both loops converge locally at linear rates under technical conditions, resulting in a ``fast $\times$ fast'' framework that demonstrates high efficiency and scalability in practice. To get a feasible initial point to execute the algorithm, we also present and analyze a highly efficient semismooth Newton method for computing the projection onto the generalized simplex. The excellent practical performance of the proposed algorithms is demonstrated by a set of numerical experiments. Our results further motivate the potential real-world applications of the considered model and the proposed algorithms.