Is a Finite Intersection of Balls Covered by a Finite Union of Balls in Euclidean Spaces?
Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming problems. For each problem, we study the intersection betwee...
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| Published in | Journal of optimization theory and applications Vol. 187; no. 2; pp. 431 - 447 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.11.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-3239 1573-2878 |
| DOI | 10.1007/s10957-020-01762-2 |
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| Abstract | Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming problems. For each problem, we study the intersection between a sphere and a Voronoi-like polyhedron. That way, we get information about a possible overlap between the frontier of the union and the intersection of balls. If the polyhedra are non-degenerate, the initial nonconvex geometric problem, which is NP-hard in general, is tractable in polynomial time by convex optimization tools and vertex enumeration. Under some mild conditions, the vertex enumeration can be skipped. Simulations highlight the accuracy and efficiency of our approach compared with competing algorithms in Python for nonconvex quadratically constrained quadratic programming. This work is motivated by an application in statistics to the problem of multidimensional changepoint detection using pruned dynamic programming algorithms. |
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| AbstractList | Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming problems. For each problem, we study the intersection between a sphere and a Voronoi-like polyhedron. That way, we get information about a possible overlap between the frontier of the union and the intersection of balls. If the polyhedra are non-degenerate, the initial nonconvex geometric problem, which is NP-hard in general, is tractable in polynomial time by convex optimization tools and vertex enumeration. Under some mild conditions, the vertex enumeration can be skipped. Simulations highlight the accuracy and efficiency of our approach compared with competing algorithms in Python for nonconvex quadratically constrained quadratic programming. This work is motivated by an application in statistics to the problem of multidimensional changepoint detection using pruned dynamic programming algorithms. |
| Author | Runge, Vincent |
| Author_xml | – sequence: 1 givenname: Vincent orcidid: 0000-0002-4857-1799 surname: Runge fullname: Runge, Vincent email: vincent.runge@univ-evry.fr organization: Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry |
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| Cites_doi | 10.1109/TSP.2016.2637317 10.1007/s10107-012-0602-3 10.1017/CBO9780511546587 10.1007/BF02293050 10.1007/s00454-018-0010-4 10.1007/BF01587086 10.1080/01621459.2017.1385466 10.1007/3-540-61576-8_77 10.1007/s11222-016-9636-3 10.1109/MSP.2010.936019 10.1007/s00454-004-2916-2 10.1109/TSP.2013.2297683 10.2140/pjm.1967.23.1 10.1561/2200000016 10.1109/MSP.2010.936015 10.1007/BF02187730 10.1007/BF02458835 10.1007/BF01582241 10.1007/BF01901190 10.1109/LSP.2014.2370033 10.1145/2049662.2049665 10.1023/A:1010295711303 |
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| DOI | 10.1007/s10957-020-01762-2 |
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| Keywords | Computational geometry Nonconvex quadratically constrained quadratic programming Voronoi-like polyhedron Vertex enumeration 90C26 Polynomial time complexity 68U05 52C17 Ball covering problem 62L10 |
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| SubjectTerms | Algorithms Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Computational geometry Convexity Dynamic programming Engineering Enumeration Euclidean geometry Euclidean space Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Polyhedra Polynomials Programming languages Quadratic programming Theory of Computation |
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| Title | Is a Finite Intersection of Balls Covered by a Finite Union of Balls in Euclidean Spaces? |
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