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

Fujishige, Satoru, Kitahara, Tomonari and Végh, László A. ORCID: 0000-0003-1152-200X (2024) An update-and-stabilize framework for the minimum-norm-point problem. Mathematical Programming. ISSN 0025-5610

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Abstract

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.

Item Type:	Article
Official URL:	https://link.springer.com/journal/10107
Additional Information:	© 2024 The Author(s)
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	28 Feb 2024 15:12
Last Modified:	12 Dec 2024 04:04
URI:	http://eprints.lse.ac.uk/id/eprint/122122

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