Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

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.

Centered solutions for uncertain linear equations

“…3.1, we extend the method of Zhen and den Hertog (2017) to compute the maximum size inscribed convex body (MCB) of a polytopic projection. Since the obtained MCB is an under approximation of the optimal MCB, in Sect.…”

“…Due to the existence of y in the description of H, finding the MCB in a polytopic projection is generally a non-convex optimization problem (see Zhen and den Hertog 2017). One can use elimination methods, e.g., Fourier-Motzkin elimination (see Fourier 1824; Motzkin 1936), to eliminate all y in H. This is equivalent to deriving a description of H that does not contain y.…”

“…We extend the method of Zhen and den Hertog (2017) to compute the MCB (and its center) of a polytopic projection by solving the following adjustable robust optimization problem:…”

“…The obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We extend the method developed in Zhen and den Hertog (2017) to compute the MCB of the solution set X . This method is independently developed in Zhang et al (2017) and applied to optimal control problems.…”

Robust Solutions for Systems of Uncertain Linear Equations

“…Since for a general convex set, finding the MVE can be computationally intractable, for this method, we focus on polyhedral U j for all j. We apply the method developed in Zhen and den Hertog (2015) to compute the MVE center of the solution set X .…”