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December 2014 Least quantile regression via modern optimization
Dimitris Bertsimas, Rahul Mazumder
Ann. Statist. 42(6): 2494-2525 (December 2014). DOI: 10.1214/14-AOS1223


We address the Least Quantile of Squares (LQS) (and in particular the Least Median of Squares) regression problem using modern optimization methods. We propose a Mixed Integer Optimization (MIO) formulation of the LQS problem which allows us to find a provably global optimal solution for the LQS problem. Our MIO framework has the appealing characteristic that if we terminate the algorithm early, we obtain a solution with a guarantee on its sub-optimality. We also propose continuous optimization methods based on first-order subdifferential methods, sequential linear optimization and hybrid combinations of them to obtain near optimal solutions to the LQS problem. The MIO algorithm is found to benefit significantly from high quality solutions delivered by our continuous optimization based methods. We further show that the MIO approach leads to (a) an optimal solution for any dataset, where the data-points $(y_{i},\mathbf{x}_{i})$’s are not necessarily in general position, (b) a simple proof of the breakdown point of the LQS objective value that holds for any dataset and (c) an extension to situations where there are polyhedral constraints on the regression coefficient vector. We report computational results with both synthetic and real-world datasets showing that the MIO algorithm with warm starts from the continuous optimization methods solve small ($n=100$) and medium ($n=500$) size problems to provable optimality in under two hours, and outperform all publicly available methods for large-scale ($n=10,000$) LQS problems.


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Dimitris Bertsimas. Rahul Mazumder. "Least quantile regression via modern optimization." Ann. Statist. 42 (6) 2494 - 2525, December 2014.


Published: December 2014
First available in Project Euclid: 12 November 2014

zbMATH: 1302.62154
MathSciNet: MR3277669
Digital Object Identifier: 10.1214/14-AOS1223

Primary: 62G35 , 62J05
Secondary: 90C11 , 90C26

Keywords: algorithms , continuous optimization , global optimization , least median of squares , least quantile regression , mixed integer programming , robust statistics

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 6 • December 2014
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