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theses

Given a set of m points in the Euclidean space Rn, the problem of the minimum enclosing ball of points, "MB problem of points" for short, seeks the n-dimensional Euclidean hypersphere (ball) of smallest radius that encloses all points in the set. A generalization of this problem is the minimum enclosing ball of balls, also referred as the "MB problem of balls", where, instead of enclosing points, one wishes to enclose a set of given balls. Both are important problems in computational geometry and optimization, with applications in facility location, computer graphics, machine learning, etc.

The MB problem of points/balls shares some combinatorial properties with linear programs (LP), which makes it part of the class of LP-type problems. Methods that resemble the simplex method for LP (simplex-like methods) have been proposed to solve the MB problem of points. However, no simplex-like method has been developed for the case of balls.

We start by considering one of such simplex-like methods for the MB problem of points, the dual algorithm by Dearing and Zeck. We modify its main step, the directional search procedure, improving its computational complexity from O(n3) to O(n2). We show that this modication yields in practice faster running times as we increase the dimension.

Next, we consider the problem of partial enclosure: instead of enclosing all m given points, we want to enclose at least k with the smallest radius ball. This problem is called the minimum k-enclosing ball problem, or "MBk problem" for short, and it is NP-hard. We present a branch-and-bound (B&B) algorithm on the tree of the subsets with size k to solve this problem. The nodes on the tree are ordered in a suitable way, which, complemented with a last-in-rst-out search strategy, allows for only a small fraction of nodes to be explored. We use the dual simplex-like algorithm by Dearing and Zeck to solve the subproblem at each node of the search tree. We show that our B&B is able to solve the MBk problem exactly in a short amount of time for small and medium sized datasets.

Finally, we address the MB problem of balls, a second-order cone program, and propose a dual simplex-like algorithm to solve it. At each iteration of the algorithm, the next iterate is found by performing an exact search on a well-dened curve. The algorithm can be seen as an extension to the case of enclosing balls of the algorithm by Dearing and Zeck. The algorithm's implementation is based on the Cholesky factorization. Our computational results show that the algorithm is very effcient in solving even large instances.

ETD_10446

Ph.D.

Includes bibliographical references

doi:10.7282/t3-73qj-v961

ETD doctoral

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