An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems
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(mlog m) iterations; the previous best bound was O(m²log m) by Wang et al. (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 + nlog n) time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (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 by Goemans et al. (2017).
Newton-Dinkelbach method
fractional optimization
parametric optimization
strongly polynomial algorithms
two variables per inequality systems
Markov decision processes
submodular function minimization
Theory of computation~Design and analysis of algorithms
Mathematics of computing~Mathematical optimization
36:1-36:15
Regular Paper
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement nos. 757481-ScaleOpt and 805241-QIP).
https://arxiv.org/abs/2004.08634
The fourth author would like to thank Neil Olver for several inspiring discussions on 2VPI systems, in particular, on symmetries of the problem.
Daniel
Dadush
Daniel Dadush
CWI, Amsterdam, The Netherlands
https://orcid.org/0000-0001-5577-5012
Zhuan Khye
Koh
Zhuan Khye Koh
Department of Mathematics, London School of Economics and Political Science, UK
https://orcid.org/0000-0002-4450-8506
Bento
Natura
Bento Natura
Department of Mathematics, London School of Economics and Political Science, UK
https://orcid.org/0000-0002-8068-3280
László A.
Végh
László A. Végh
Department of Mathematics, London School of Economics and Political Science, UK
https://orcid.org/0000-0003-1152-200X
10.4230/LIPIcs.ESA.2021.36
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Daniel Dadush, Zhuan Khye Koh, Bento Natura, and László A. Végh
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