Optimization Online


Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

Areesh Mittal (areeshmittal***at***utexas.edu)
Grani Hanasusanto (grani.hanasusanto***at***utexas.edu)

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.

Keywords: minimum volume ellipsoid problem; copositive programming; semidefinite programming

Category 1: Linear, Cone and Semidefinite Programming


Download: [PDF]

Entry Submitted: 07/17/2018
Entry Accepted: 07/17/2018
Entry Last Modified: 06/17/2020

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society