Using Nemirovski's Mirror-Prox method as basic procedure in Chubanov's method for solving homogeneous feasibility problems

We introduce a new variant of Chubanov's method for solving linear homogeneous systems with positive variables. In the Basic Procedure we use a recently introduced cut in combination with Nemirovski's Mirror-Prox method. We show that the cut requires at most time, just as Chubanov's c...

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Bibliographic Details
Published in:Optimization methods & software 2022-07, Vol.37 (4), p.1447-1470
Main Authors: Wei, Zhang, Roos, Kees
Format: Article
Language:English
Subjects:
Algorithms
Chubanov's method
homogeneous
Linear optimization
Mirror-Prox method
Optimization
Rescaling
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Summary:We introduce a new variant of Chubanov's method for solving linear homogeneous systems with positive variables. In the Basic Procedure we use a recently introduced cut in combination with Nemirovski's Mirror-Prox method. We show that the cut requires at most time, just as Chubanov's cut. In an earlier paper it was shown that the new cuts are at least as sharp as those of Chubanov. Our Modified Main Algorithm is in essence the same as Chubanov's Main Algorithm, except that it uses the new Basic Procedure as a subroutine. The new method has time complexity, where is a suitably defined condition number. As we show, a simplified version of the new Basic Procedure competes well with the Smooth Perceptron Scheme of Peña and Soheili and, when combined with Rescaling, also with two commercial codes for linear optimization.
ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2021.2023523