Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

Areesh Mittal, Grani A. Hanasusanto

Research output: Contribution to journalArticlepeer-review


We study the problem of finding the Löwner–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.

Original languageEnglish (US)
Pages (from-to)2867-2882
Number of pages16
JournalOperations Research
Issue number5
StatePublished - Sep 1 2022
Externally publishedYes


  • copositive programming
  • distributionally robust optimization
  • minimum volume ellipsoids
  • semidefinite programming

ASJC Scopus subject areas

  • Computer Science Applications
  • Management Science and Operations Research


Dive into the research topics of 'Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming'. Together they form a unique fingerprint.

Cite this