An update-and-stabilize framework for the minimum-norm-point problem

• 成果类型: Article
• 署名作者: Fujishige, Satoru; Kitahara, Tomonari; Vegh, Laszlo A.
• 署名单位: Kyoto University; Kyushu University; University of London; London School Economics & Political Science
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-024-02077-0
• 发表日期: 2025
• 页码: 281-311
• 关键词: nearest point; algorithm

摘要

We consider the minimum-norm-point (MNP) problem over polyhedra, a well-studied problem that encompasses linear programming. We present a general algorithmic framework that combines two fundamental approaches for this problem: active set methods and first order methods. Our algorithm performs first order update steps, followed by iterations that aim to 'stabilize' the current iterate with additional projections, i.e., find a locally optimal solution whilst keeping the current tight inequalities. Such steps have been previously used in active set methods for the nonnegative least squares (NNLS) problem. We bound on the number of iterations polynomially in the dimension and in the associated circuit imbalance measure. In particular, the algorithm is strongly polynomial for network flow instances. Classical NNLS algorithms such as the Lawson-Hanson algorithm are special instantiations of our framework; as a consequence, we obtain convergence bounds for these algorithms. Our preliminary computational experiments show promising practical performance.