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

• URL: http://arxiv.org/abs/2407.03294v2
• Date: Tue, 14 Oct 2025 19:23:26 GMT
• Title: NewVEM: A Newton Vertex Exchange Method for a Class of Constrained Self-Concordant Minimization Problems
• Authors: Ling Liang, Kim-Chuan Toh, Haizhao Yang,
• Abstract summary: We present and analyze a highly efficient semismooth Newton method for computing the projection onto the generalized simplex.<n>The excellent practical performance of the proposed algorithms is demonstrated by a set of numerical experiments.
• Score: 13.365308428809849
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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