We study the problem of computing a (1 + )-approximation to the minimum volume enclosing ellipsoid of a given point set S = {p 1 , p 2 , . . . , p n } ⊆ R d .Based on a simple, initial volume approximation method, we propose a modification of Khachiyan's first-order algorithm. Our analysis leads to a slightly improved complexity bound of O(nd 3 / ) operations for ∈ (0, 1). As a byproduct, our algorithm returns a core set X ⊆ S with the property that the minimum volume enclosing ellipsoid of X provides a good approximation to that of S. Furthermore, the size of X depends only on the dimension d and , but not on the number of points n. In particular, our results imply that |X | = O(d 2 / ) for ∈ (0, 1).

We study the minimum enclosing ball (MEB) problem for sets of points or balls in high dimensions. Using techniques of second-order cone programming and "core-sets", we have developed (1 + )-approximation algorithms that perform well in practice, especially for very high dimensions, in addition to having provable guarantees. We prove the existence of core-sets of size O(1/ ), improving the previous bound of O(1/ 2 ), and we study empirically how the core-set size grows with dimension. We show that our algorithm, which is simple to implement, results in fast computation of nearly optimal solutions for point sets in much higher dimension than previously computable using exact techniques.

Cataloged from PDF version of article.Given A := {a(1),..., a(m)} subset of R(d) whose affine hull is R(d), 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 V. 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-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of,91, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A.. Our algorithm is a modification of the algorithm of Kumar and Yildirim, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yildirim or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique. (C) 2007 Elsevier B.V. All rights reserved

Abstract. We study the situation in which, having solved a linear program with an interiorpoint method, we are presented with a new problem instance whose data is slightly perturbed from the original. We describe strategies for recovering a warm-start" point for the perturbed problem instance from the iterates of the original problem instance. We obtain worst-case estimates of the number of iterations required to converge to a solution of the perturbed instance from the warm-start points, showing that these estimates depend on the size of the perturbation and on the conditioning and other properties of the problem instances.

This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical properties of eigenvalue functions can be put into use to derive piece-wise quadratic functions that underestimate the eigenvalue functions. These piece-wise quadratic under-estimators lead us to a global minimization algorithm, originally due to Breiman and Cutler. We prove the global convergence of the algorithm, and show that it can be effectively used for the minimization of extreme eigenvalues, e.g., the largest eigenvalue or the sum of the largest specified number of eigenvalues. This is particularly facilitated by the analytical formulas for the first derivatives of eigenvalues, as well as analytical lower bounds on the second derivatives that can be deduced for extreme eigenvalue functions. The applications that we have in mind also include the ${\rm H}_\infty$-norm of a linear dynamical system, numerical radius, distance to uncontrollability and various other non-convex eigenvalue optimization problems, for which, generically, the eigenvalue function involved is simple at all points.Comment: 25 pages, 3 figure

Abstract. Given A := {a 1 , . . . , a m } ⊂ R n and > 0, we propose and analyze two algorithms for the problem of computing a (1 + )-approximation to the radius of the minimum enclosing ball of A. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of the minimum enclosing ball problem. We establish that this algorithm converges in O(1/ ) iterations with an overall complexity bound of O(mn/ ) arithmetic operations. In addition, the algorithm returns a "core set" of size O(1/ ), which is independent of both m and n. The latter algorithm is obtained by incorporating "away" steps into the former one at each iteration and achieves the same asymptotic complexity bound as the first one. While the asymptotic bound on the size of the core set returned by the second algorithm also remains the same as the first one, the latter algorithm has the potential to compute even smaller core sets in practice, since, in contrast to the former one, it allows "dropping" points from the working core set at each iteration. Our analysis reveals that the leading terms in the asymptotic complexity analysis are reasonably small. In contrast to the first algorithm, we also establish that the second algorithm asymptotically exhibits linear convergence, which provides further insight into our computational results, indicating that the latter algorithm indeed terminates faster with smaller core sets in comparison with the first one. We also discuss how our algorithms can be extended to compute an approximation to the minimum enclosing ball of more general input sets without sacrificing the iteration complexity and the bound on the core set size. In particular, we establish the existence of a core set of size O(1/ ) for a much wider class of input sets. We adopt the real number model of computation in our analysis.

We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify the state-of-the-art interior-point solver PCx in our implementation. We evaluate the effectiveness of each warm-start strategy based on the number of iterations and the computation time in comparison with "cold start" on the NETLIB test suite. Our experiments reveal that each of the warm-start strategies leads to a reduction in the number of interior-point iterations especially for smaller perturbations and for perturbations of fewer data components in comparison with cold start. On the other hand, only one of the warm-start strategies exhibits better performance than cold start in terms of computation time. Based on the insight gained from the computational results, we discuss several potential improvements to enhance the performance of warm-start strategies.

In this study, we investigate the brain networks during positive and negative emotions for different types of stimulus (audio only, video only and audio + video) in [Formula: see text], and [Formula: see text] bands in terms of phase locking value, a nonlinear method to study functional connectivity. Results show notable hemispheric lateralization as phase synchronization values between channels are significant and high in right hemisphere for all emotions. Left frontal electrodes are also found to have control over emotion in terms of functional connectivity. Besides significant inter-hemisphere phase locking values are observed between left and right frontal regions, specifically between left anterior frontal and right mid-frontal, inferior-frontal and anterior frontal regions; and also between left and right mid frontal regions. ANOVA analysis for stimulus types show that stimulus types are not separable for emotions having high valence. PLV values are significantly different only for negative emotions or neutral emotions between audio only/video only and audio only/audio + video stimuli. Finding no significant difference between video only and audio + video stimuli is interesting and might be interpreted as that video content is the most effective part of a stimulus.