Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

TL;DR

This paper introduces a novel approach using generalized copositive programming to efficiently compute minimum volume ellipsoids containing convex sets, improving solution quality and speed over existing methods.

Contribution

It reformulates the minimum volume ellipsoid problem as a generalized copositive program and develops tractable semidefinite approximations, outperforming current schemes in speed and accuracy.

Findings

Method produces high-quality ellipsoids faster than optimal solutions.

Outperforms existing approximation schemes in solution time and quality.

Applicable to dynamic distributionally robust problems and control systems.

Abstract

We study the problem of finding the Lowner-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 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.