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Active Set Strategies for the Computation of Minimum-Volume Enclosing Ellipsoids

Fulltext:


Authors:

Linus Källberg, Daniel Andrén

Publication Type:

Report - MRTC

Publisher:

Mälardalen Real-Time Research Centre, Mälardalen University

ISRN:

MDH-MRTC-328/2019-1-SE


Abstract

We describe and evaluate several variants of an active set algorithm for the problem of computing a (1+ε)-approximation to the minimum-volume ellipsoid enclosing a given point set. The general approach is to run an existing algorithm repeatedly on smaller subsets of the points, and thereby achieve improved solution times compared to solving the whole problem directly. As the underlying algorithm, we use that of Todd and Yıldırım, which belongs to a group of algorithms based on the first-order Frank–Wolfe method. We propose multiple strategies to choose a new active set in each iteration, including an improved version of an existing strategy by Sun and Freund. In addition, we develop a variation of the elimination heuristic by Harman and Pronzato, that eliminates input points more aggressively in each iteration and then checks correctness of the solution before returning it. When used to (1 + 10^-6)-approximate the minimum-volume ellipsoid enclosing sets of 10^6 points in 2 to 25 dimensions, the proposed techniques generate speedups up to 70× compared to our baseline.

Bibtex

@techreport{Kallberg5680,
author = {Linus K{\"a}llberg and Daniel Andr{\'e}n},
title = {Active Set Strategies for the Computation of Minimum-Volume Enclosing Ellipsoids},
month = {November},
year = {2019},
publisher = {M{\"a}lardalen Real-Time Research Centre, M{\"a}lardalen University},
url = {http://www.es.mdh.se/publications/5680-}
}