NewVEM: A Newton Vertex Exchange Method for a Class of Constrained Self-Concordant Minimization Problems

TL;DR

NewVEM introduces a two-level Newton vertex exchange algorithm for efficiently solving self-concordant minimization problems with simplex constraints, demonstrating high efficiency and scalability through theoretical convergence and practical experiments.

Contribution

It presents a novel two-level Newton vertex exchange method with convergence analysis and an efficient projection technique for generalized simplex constraints.

Findings

Algorithm converges locally at linear rates.

Demonstrates high efficiency and scalability in numerical experiments.

Potential for real-world applications in constrained optimization.

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.