An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds
Mathematical Programming: Series A and B
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
Convex Optimization
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Core Vector Machines: Fast SVM Training on Very Large Data Sets
The Journal of Machine Learning Research
Excessive Gap Technique in Nonsmooth Convex Minimization
SIAM Journal on Optimization
New algorithms for singly linearly constrained quadratic programs subject to lower and upper bounds
Mathematical Programming: Series A and B
Simpler core vector machines with enclosing balls
Proceedings of the 24th international conference on Machine learning
Computational Geometry: Theory and Applications
Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Efficient projections onto the l1-ball for learning in high dimensions
Proceedings of the 25th international conference on Machine learning
Coresets for polytope distance
Proceedings of the twenty-fifth annual symposium on Computational geometry
Efficient Euclidean projections in linear time
ICML '09 Proceedings of the 26th Annual International Conference on Machine Learning
Maximum margin coresets for active and noise tolerant learning
IJCAI'07 Proceedings of the 20th international joint conference on Artifical intelligence
Two Algorithms for the Minimum Enclosing Ball Problem
SIAM Journal on Optimization
Mirror descent and nonlinear projected subgradient methods for convex optimization
Operations Research Letters
A fast iterative nearest point algorithm for support vector machine classifier design
IEEE Transactions on Neural Networks
No dimension independent core-sets for containment under homothetics
Proceedings of the twenty-seventh annual symposium on Computational geometry
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Given n points in a d dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all n points. We give two approximation algorithms for producing an enclosing ball whose radius is at most ε away from the optimum. The first requires O(nd£/√ε) effort, where £ is a constant that depends on the scaling of the data. The second is a O*(ndQ/√ε) approximation algorithm, where Q is an upper bound on the norm of the points. This is in contrast with coresets based algorithms which yield a O(nd/ε) greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present O(mnd£/ε) and O* (mndQ/ε) approximation algorithms, where m is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of Nesterov [19] to obtain our results.