Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

CentER Discussion Paper Series No. 2015-004

28 Pages Posted: 22 Jan 2015 Last revised: 14 Dec 2016

See all articles by Jianzhe Zhen

Jianzhe Zhen

Tilburg University - Center for Economic Research (CentER)

Dick den Hertog

Tilburg University - Department of Econometrics & Operations Research

Date Written: January 21, 2015

Abstract

We introduce a novel scheme based on a blending of Fourier-Motzkin elimination (FME) and adjustable robust optimization techniques to compute the maximum volume inscribed ellipsoid (MVE) in a polytopic projection. It is well-known that deriving an explicit description of a projected polytope is NP-hard. Our approach does not require an explicit description of the projection, and can easily be generalized to find a maximally sized convex body of a polytopic projection. Our obtained MVE is an inner approximation of the projected polytope, and its center is a centralized relative interior point of the projection. Since FME may produce many redundant constraints, we apply an LP-based procedure to keep the description of the projected polytopes at its minimal size. Furthermore, we propose an upper bounding scheme to evaluate the quality of the inner approximations. We test our approach on a simple polytope and a color tube design problem, and observe that as more auxiliary variables are eliminated, our inner approximations and upper bounds converge to optimal solutions.

Keywords: Fourier-Motzkin elimination; maximum volume inscribed ellipsoid; polytopic projection;

JEL Classification: C44, C61, C63

Suggested Citation

Zhen, Jianzhe and den Hertog, Dick, Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection (January 21, 2015). CentER Discussion Paper Series No. 2015-004. Available at SSRN: https://ssrn.com/abstract=2553293 or http://dx.doi.org/10.2139/ssrn.2553293

Jianzhe Zhen (Contact Author)

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

Dick Den Hertog

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands

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