Solving k-center problems involving sets based on optimization techniques
The continuous k -center problem aims at finding k balls with the smallest radius to cover a finite number of given points in R n . In this paper, we propose and study the following generalized version of the k -center problem: Given a finite number of nonempty closed convex sets in R n , find k bal...
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| Published in | Journal of global optimization Vol. 76; no. 1; pp. 189 - 209 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.01.2020
Springer Springer Nature B.V |
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| Abstract | The continuous
k
-center problem aims at finding
k
balls with the smallest radius to cover a finite number of given points in
R
n
. In this paper, we propose and study the following generalized version of the
k
-center problem: Given a finite number of nonempty closed convex sets in
R
n
, find
k
balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms. |
|---|---|
| AbstractList | The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in [Formula omitted]. In this paper, we propose and study the following generalized version of the k-center problem: Given a finite number of nonempty closed convex sets in [Formula omitted], find k balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms. The continuous k -center problem aims at finding k balls with the smallest radius to cover a finite number of given points in R n . In this paper, we propose and study the following generalized version of the k -center problem: Given a finite number of nonempty closed convex sets in R n , find k balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms. The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in \[\mathbb {R}^n\]. In this paper, we propose and study the following generalized version of the k-center problem: Given a finite number of nonempty closed convex sets in \[\mathbb {R}^n\], find k balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms. |
| Audience | Academic |
| Author | An, Nguyen Thai Nam, Nguyen Mau Qin, Xiaolong |
| Author_xml | – sequence: 1 givenname: Nguyen Thai surname: An fullname: An, Nguyen Thai organization: Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Institute of Research and Development, Duy Tan University – sequence: 2 givenname: Nguyen Mau surname: Nam fullname: Nam, Nguyen Mau organization: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University – sequence: 3 givenname: Xiaolong orcidid: 0000-0002-3381-8525 surname: Qin fullname: Qin, Xiaolong email: qxlxajh@163.com organization: Department of Mathematics, Hangzhou Normal University, Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Department of Mathematics, Zhejiang Normal University |
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| Keywords | Majorization-minimization principle 49J53 center problem Primary: 49J52 Multifacility location problem Difference of convex functions Secondary: 90C30 |
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| Snippet | The continuous
k
-center problem aims at finding
k
balls with the smallest radius to cover a finite number of given points in
R
n
. In this paper, we propose... The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in [Formula omitted]. In this paper,... The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in \[\mathbb {R}^n\]. In this paper,... |
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| SubjectTerms | Algorithms Computational geometry Computer Science Convexity Economic models Mathematical optimization Mathematics Mathematics and Statistics Methods Operations Research/Decision Theory Optimization Optimization techniques Real Functions |
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| Title | Solving k-center problems involving sets based on optimization techniques |
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