Let S denote the convex hull of m full-dimensional ellipsoids in Rn. Given > 0 and δ > 0, we study the problems of computing a (1 + )-approximation to the minimum volume covering ellipsoid of S and a (1 + δ)n-rounding of S. 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 ...

Given a set of points S = {x1, . . . , xm} ⊂ R and > 0, we propose and analyze an algorithm for the problem of computing a (1 + )-approximation to the minimum-volume axis-aligned ellipsoid enclosing S . We establish that our algorithm is polynomial for fixed . In addition, the algorithm returns a small core set X ⊆ S , whose size is independent of the number of points m, with the property that ...

In this paper, we present a new formulation for constructing an ellipsoid which generalizes the computation of the minimum volume covering ellipsoid, based on the CVaR minimization technique proposed by Rockafellar and Uryasev (2002). The proposed ellipsoid construction is formulated as a convex optimization and an interior point algorithm for the solution can be developed. In addition, the opt...

The Ellipsoid algorithm was developed by (formerly) Soviet mathematicians (Shor (1970), Yudin and Nemirovskii (1975)). Khachian (1979) proved that it provides a polynomial time algorithm for linear programming. The average behavior of the Ellipsoid algorithm is too slow, making it not competitive with the simplex algorithm. However, the theoretical implications of the algorithm are very importa...

The MVE estimator is an important estimator in robust statistics and characterized by an ellipsoid which contains inside 100β percent of given points in IR n and attains the minimal volume, where β ∈ [0.5, 1.0) is of usual interest. The associated minimal ellipsoid can be considered as a generalization of the minimum volume covering ellipsoid which covers all the given points since the latter e...

5 Algorithms 8 5.1 Fourier-Motzkin Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 The Simplex Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.3 Seidel’s Randomized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.4 The Ellipsoid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5.5 Using the Ellipsoid...

In high bit rate optical fiber communication systems, Polarization mode dispersion (PMD) is one of the main factors to signal distortion and needs to be compensated. PMD monitoring system is the key integral part of an adaptive PMD compensator. The degree of polarization (DOP) ellipsoid obtained by using a polarization scrambler can be used as a feedback signal for automatic PMD compensation. G...

This lecture provides an in-depth explanation of ellipsoid algorithm which was developed in 1979 by a Soviet mathematician L.G. Khachian. Ellipsoid algorithm is the first polynomial-time linear programming algorithm which also knows as Khachiyan’s algorithm to acknowledge Khachiyan’s discovery. This algorithm is fundamentally different from Simplex Algorithm in a sense that it does not exploit ...