Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection
提出一种融合傅里叶-莫茨金消去和可调鲁棒优化的方法，无需显式描述投影即可计算多面体投影的最大体积内接椭球，并给出上界评估近似质量。
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. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0763 .