A Smoothing Algorithm for the Smallest Intersecting Ball Problem (original) (raw)
Related papers
Optimization Letters, 2012
In this paper we study the following problems: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects all of the sets. These problems can be viewed as generalized versions of the smallest enclosing circle problem introduced in the 19th century by Sylvester [12] which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We will focus on the sufficient conditions for the existence and uniqueness of an optimal solution for each problem, while the study of optimality conditions and numerical implementation will be addressed in our next projects.
Two Algorithms for the Minimum Enclosing Ball Problem
SIAM Journal on Optimization, 2008
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 with the former one, it allows "dropping" points from the working core set at each iteration. Our computational results indicate that the latter algorithm indeed returns 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 other input sets. In particular, we establish the existence of a core set of size O(1/) for a much wider class of input sets.
Efficient Speed-Up of the Smallest Enclosing Circle Algorithm
Informatica, 2022
The smallest enclosing circle is a well-known problem. In this paper, we propose modifications to speed-up the existing Weltzl’s algorithm. We perform the preprocessing to reduce as many input points as possible. The reduction step has lower computational complexity than the Weltzl’s algorithm and thus speed-ups its computation. Next, we propose some changes to Weltzl’s algorithm. In the end are summarized results, that show the speed-up for 106{10^{6}}106 input points up to 100 times compared to the original Weltzl’s algorithm. Even more, the proposed algorithm is capable to process significantly larger data sets than the standard Weltzl’s algorithm.
Applications of Convex Analysis to the Smallest Intersecting Ball Problem
Arxiv preprint arXiv:1105.2132, 2011
Abstract: The smallest enclosing circle problem asks for the circle of smallest radius enclosing a given set of finite points on the plane. This problem was introduced in the 19th century by Sylvester [17]. After more than a century, the problem remains very active. This ...
Solution Methodologies for the Smallest Enclosing Circle Problem
2003
Tribute. We would like to dedicate this paper to Elijah Polak. Professor Polak has made substantial contributions to a truly broad spectrum of topics in nonlinear optimization, including optimization for engineering design centering, multi-criteria optimization, optimal control, feasible directions methods, quasi-Newton and Newton methods, non-differential optimization, semi-infinite optimization, conjugate directions methods, gradient projection and reduced gradient methods, and barrier methods, among many other topics. His many and varied contributions to our field are important today and will influence the research in our field well into the future.
Identification and Elimination of Interior Points for the Minimum Enclosing Ball Problem
Siam Journal on Optimization, 2008
Given A := {a 1 , . . . , a m } ⊂ R n , we consider the problem of reducing the input set for the computation of the minimum enclosing ball of A. In this note, given an approximate solution to the minimum enclosing ball problem, we propose a simple procedure to identify and eliminate points in A that are guaranteed to lie in the interior of the minimum-radius ball enclosing A. Our computational results reveal that incorporating this procedure into the two recent algorithms proposed by Yıldırım leads to significant speed-ups in running times especially for randomly generated large-scale problems. We also illustrate that the extra overhead due to the elimination procedure remains at an acceptable level for spherical or almost spherical input sets.
A dual algorithm for the minimum covering ball problem in
Operations Research Letters, 2009
A dual type algorithm constructs the minimum covering ball of a given finite set of points in R n by finding the minimum covering balls of a sequence of subsets, each with no more than n + 1 points and with strictly increasing radius, until all points are covered. (P.M. Dearing). center, without violating the covering property, until optimality is reached. Elzinga and Hearn [4] developed a dual approach in which the minimum covering circle is found for a sequence of subsets S ⊆ P, each of at most 3 points and with increasing radius, until some circle covers the entire set P. Other approaches use Voronoi diagrams . Meggido [6] developed a theoretical linear time algorithm for solving the problem.
Computing the smallest k-enclosing circle and related problems
Computational Geometry, 1994
We present an efficient algorithm for solving the "smallest k-enclosing circle" (RSC) problem: Given a set of n points in the plane and an integer k < n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nklog'n). When only O(nlogn) storage is allowed, the running time is O(nk log' n log n/k). We also extend our technique to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P which contains k points from a given planar set, and finding the smallest disk intersecting k segments from a given planar set of non-intersecting segments.
2020
Primal and dual algorithms are developed for solving the n-dimensional convex optimization problem of finding the Euclidean ball of minimum radius that covers m given Euclidean balls, each with given center and radius. Each algorithm is based on a directional search method in which a search path may be a ray or a two-dimensional conic section in IRn. At each iteration, a search path is constructed by the intersection of bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional hyperboloids. The optimal step size along each search path is determined explicitly.