Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

• 成果类型: Article; Early Access
• 署名作者: Mittal, Areesh; Hanasusanto, Grani A.
• 署名单位: University of Texas System; University of Texas Austin
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2021.2156
• 发表日期: 2021
• 页码: 1-16
• 关键词: minimum volume ellipsoids; copositive programming; Semidefinite programming; distributionally robust optimization

摘要

We study the problem of finding the Liiwner-John ellipsoid (i.e., an ellipsoid with minimum volume that contains a given convex set). We reformulate the problem as a generalized copositive program and use that reformulation to derive tractable semidefinite programming approximations for instances where the set is defined by affine and quadratic inequalities. We prove that, when the underlying set is a polytope, our method never provides an ellipsoid of higher volume than the one obtained by scaling the maximum volume-inscribed ellipsoid. We empirically demonstrate that our proposed method generates high-quality solutions and can be solved much faster than solving the problem to optimality. Furthermore, we outperform the existing approximation schemes in terms of solution time and quality. We present applications of our method to obtain piecewise linear decision rule approximations for dynamic distributionally robust problems with random recourse and to generate ellipsoidal approximations for the set of reachable states in a linear dynamical system when the set of allowed controls is a polytope.