Some comments of Wolfe's `away step'
Mathematical Programming: Series A and B
A subexponential bound for linear programming
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Mathematical Programming: Series A and B
On the complexity of approximating the maximal inscribed ellipsoid for a polytope
Mathematical Programming: Series A and B
Rounding of polytopes in the real number model of computation
Mathematics of Operations Research
Exact primitives for smallest enclosing ellipses
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Determinant Maximization with Linear Matrix Inequality Constraints
SIAM Journal on Matrix Analysis and Applications
On linear-time deterministic algorithms for optimization problems in fixed dimension
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Computational Optimization and Applications
Bounding ellipsoids for ray-fractal intersection
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Improved Complexity for Maximum Volume Inscribed Ellipsoids
SIAM Journal on Optimization
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
On Numerical Solution of the Maximum Volume Ellipsoid Problem
SIAM Journal on Optimization
Approximate minimum enclosing balls in high dimensions using core-sets
Journal of Experimental Algorithmics (JEA)
Faster core-set constructions and data stream algorithms in fixed dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Practical methods for shape fitting and kinetic data structures using core sets
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Computation of Minimum-Volume Covering Ellipsoids
Operations Research
Optimization Methods & Software
Uniqueness results for minimal enclosing ellipsoids
Computer Aided Geometric Design
Coresets for polytope distance
Proceedings of the twenty-fifth annual symposium on Computational geometry
Modified algorithms for the minimum volume enclosing axis-aligned ellipsoid problem
Discrete Applied Mathematics
Elliptic indexing of multidimensional databases
ADC '09 Proceedings of the Twentieth Australasian Conference on Australasian Database - Volume 92
Perspective Reconstruction of a Spheroid from an Image Plane Ellipse
International Journal of Computer Vision
An algebraic approach to continuous collision detection for ellipsoids
Computer Aided Geometric Design
A portfolio selection model using fuzzy returns
Fuzzy Optimization and Decision Making
INFORMS Journal on Computing
Rank-two update algorithms for the minimum volume enclosing ellipsoid problem
Computational Optimization and Applications
Performance comparison of accelerometer calibration algorithms based on 3D-ellipsoid fitting methods
Computer Methods and Programs in Biomedicine
Overlap of convex polytopes under rigid motion
Computational Geometry: Theory and Applications
On the maximal singularity-free ellipse of planar 3-RP R parallel mechanisms via convex optimization
Robotics and Computer-Integrated Manufacturing
Hi-index | 0.04 |
Given A@?{a^1,...,a^m}@?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 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-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 A, 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.