Vertex Exchange Method for a Class of Quadratic Programming Problems

2024-07-03

arXiv.org

AI Summary

This paper addresses strongly convex quadratic programming under generalized simplex constraints. To overcome the computational difficulty of the associated projection subproblem and the low efficiency of existing algorithms, we propose— for the first time—a vertex-exchange algorithm, coupled with an efficient semismooth Newton method for solving the generalized simplex projection. Theoretically, we establish global convergence and linear convergence rate guarantees for the proposed algorithm. Methodologically, we uncover the fundamental and broadly applicable role of the vertex-exchange strategy in various classes of constrained convex optimization problems. Extensive numerical experiments demonstrate that our approach significantly outperforms state-of-the-art solvers in solution accuracy, robustness, and scalability, offering a novel paradigm for structured constrained optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A vertex exchange method is proposed for solving the strongly convex quadratic program subject to the generalized simplex constraint. We conduct rigorous convergence analysis for the proposed algorithm and demonstrate its essential roles in solving some important classes of constrained convex optimization. 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 extensive numerical experiments. Our theoretical and numerical results further motivate the potential applications of the considered model and the proposed algorithms.

Problem

• Solving self-concordant minimization with generalized simplex constraints
• Developing a fast Newton-vertex exchange algorithm structure
• Providing efficient projection methods for feasible initialization

Innovation

• Newton vertex exchange method for self-concordant minimization
• Two-level structure with projected Newton outer loop
• Vertex exchange approach for quadratic subproblems