A coordinate gradient descent algorithm for computing the minimum volume enclosing ellipsoid

Abstract

The minimum volume enclosing ellipsoid (MVEE) is a basic convex optimization problem. This paper describes some new properties such as “algorithmic coordinate-wise smoothness” of this model and proposes a steepest descent type algorithm, the coordinate gradient descent (CGD) algorithm, to address the MVEE problem. We prove that the CGD algorithm is sublinearly convergent. Moreover, it is shown to converge in a linear rate locally, and slightly faster than the FW type algorithms, especially in the cases of large dimensions. At last, we compare our algorithm with the random coordinate descent (RCD) method and show that the RCD algorithm is less efficient than the CGD algorithm in computing the MVEE. The numerical tests support our theoretical results.