### Abstract

Original language | English |
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Place of Publication | Tilburg |

Publisher | CentER, Center for Economic Research |

Number of pages | 28 |

Volume | 2015-004 |

Publication status | Published - 21 Jan 2015 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2015-004 |

### Fingerprint

### Keywords

- Maximum volume inscribed ellipsoid
- chebyshev center
- polytopic projection
- adjustable robust optimization

### Cite this

*Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection*. (CentER Discussion Paper; Vol. 2015-004). Tilburg: CentER, Center for Economic Research.

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**Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection.** / Zhen, J.; den Hertog, D.

Research output: Working paper › Discussion paper › Other research output

TY - UNPB

T1 - Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

AU - Zhen, J.

AU - den Hertog, D.

PY - 2015/1/21

Y1 - 2015/1/21

N2 - 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.

AB - 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.

KW - Maximum volume inscribed ellipsoid

KW - chebyshev center

KW - polytopic projection

KW - adjustable robust optimization

M3 - Discussion paper

VL - 2015-004

T3 - CentER Discussion Paper

BT - Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

PB - CentER, Center for Economic Research

CY - Tilburg

ER -