Improved Complexity for Maximum Volume Inscribed Ellipsoids

Let P = fx j Ax bg, where A is an m n matrix. We assume that P contains a ball of radius one centered at the origin, and is contained in a ball of radius R centered at the origin. We consider the problem of approximating the maximum volume ellipsoid inscribed in P. Such ellipsoids have a number of interesting applications, including the inscribed ellipsoid method for convex optimization. We reduce the complexity of nding an ellipsoid whose volume is at least a factor e ? of the maximum possible to O(m 3:5 ln(mR==)) operations, improving on previous results of Nesterov and Nemirovskii, and Khachiyan and Todd. A further reduction in complexity is obtained by rst computing an approximation of the analytic center of P.