An accelerated Newton–Dinkelbach method and its application to two variables per inequality systems

Dadush, Daniel, Koh, Zhuan Khye, Natura, Bento and Végh, László A. ORCID: 0000-0003-1152-200X (2023) An accelerated Newton–Dinkelbach method and its application to two variables per inequality systems. Mathematics of Operations Research, 48 (4). 1934 - 1958. ISSN 0364-765X

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Abstract

We present an accelerated or “look-ahead” version of the Newton–Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains. (i) For linear fractional combinatorial optimization, we show a convergence bound of O(m log m) iterations; the previous best bound was O(m 2logm) by Wang, Yang, and Zhang from 2006. (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with n variables and m constraints, our algorithm runs in O(mn) iterations. Every iteration takes O(mn) time for general 2VPI systems and O(m + n log n) time for the special case of deterministic Markov decision processes (DMDPs). This extends and strengthens a previous result by Madani from 2002 that showed a weakly polynomial bound for a variant of the Newton–Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result from 2017 by Goemans, Gupta, and Jaillet.

Item Type:	Article
Official URL:	https://pubsonline.informs.org/journal/moor
Additional Information:	© 2022 Institute for Operations Research and the Management Sciences
Divisions:	Statistics Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	28 Oct 2022 09:39
Last Modified:	30 Nov 2023 16:12
URI:	http://eprints.lse.ac.uk/id/eprint/117202

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