We study the problem of computing a (1+ )-approximation to the minimum volume covering ellipsoid of a given set S of the convex hull of m full-dimensional ellipsoids in Rn. We extend the first-order algorithm of Kumar and Yıldırım that computes an approximation to the minimum volume covering ellipsoid of a finite set of points in Rn, which, in turn, is a modification of Khachiyan’s algorithm. F...

In this chapter a randomized ellipsoid algorithm is described that can be used for finding solutions to robust Linear Matrix Inequality (LMI) problems. The iterative algorithm enjoys the property that the convergence speed is independent on the number of uncertain parameters. Other advantages, as compared to the deterministic algorithms, are that the uncertain parameters can enter the LMIs in a...

Methods for simulation from multivariate Gaussian distributions restricted to be from outside an arbitrary ellipsoidal region are often needed in applications. A standard rejection algorithm that draws a sample from a multivariate Gaussian distribution and accepts it if it is outside the ellipsoid is often employed: however, this is computationally inefficient if the probability of that ellipso...

Given A := {a1, . . . , am} ⊂ Rd whose affine hull is Rd, we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum volume enclosing ellipsoid of A. In the case of centrally symmetric sets, we first establish that Khachiyan’s barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two...

The problem of fitting ellipsoids occurs in many areas of science. It is useful in pattern recognition, particle physics, computer graphics, medical imaging of organs, and statistical error analysis. We describe an algorithm for finding an equation for a multi-dimensional ellipsoid fit to a distribution of discrete data points. In general, we find that the algorithm works well for fitting ellip...

We show that with recently developed derandomization techniques, one can convert Clarkson's randomized algorithm for linear programming in xed dimension into a lineartime deterministic one. The constant of proportionality is d, which is better than for previously known such algorithms. We show that the algorithm works in a fairly general abstract setting, which allows us to solve various other ...

We consider the problem of approximating xed points of non smooth con tractive functions with using of the absolute error criterion In we proved that the upper bound on the number of function evaluations to compute approximations is O n ln ln q ln n in the worst case where q is the contraction factor and n is the dimension of the problem This upper bound is achieved by the circumscribed ellipso...

We present a new implementation of the almost optimal Circumscribed Ellipsoid (CE) Algorithm for approximating fixed points of nonexpanding functions, as well as of functions that may be globally expanding, however, are nonexpanding/contracting in the direction of fixed points. Our algorithm is based only on function values, i.e., it does not require computing derivatives of any order. We utili...

We present an affine-invariant random walk for drawing uniform random samples from a convex body K Ă Rn for which the maximum volume inscribed ellipsoid, known as John’s ellipsoid, may be computed. We consider a polytope P “ x P Rn ˇ̌ Ax ď 1 ( where A P R as a special case. Our algorithm makes steps using uniform sampling from the John’s ellipsoid of the symmetrization of K at the current point....