Abstract
The nonlinear programming problem of finding the minimum covering ball of a finite set of points in \(\mathbb {R}^n\), with a positive weight corresponding to each point, is solved by a directional search method. At each iteration, the search path is either a ray or the arc of a circle and is determined by bisectors of points. Each step size along the search path is determined explicitly. The primal algorithm is shown to search along the farthest point Voronoi diagram of the given points. We provide computational results that show the efficiency of the algorithm when compared to general convex nonlinear optimization solvers.
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References
Aurenhammer F, Klein R (2000) Voronoi diagrams. In: Sack J-R, Urrutia J (ed) Handbook of computational geometry. Elsevier, North Holland, pp 201–290
Bland RG (1977) New finite pivoting rules for the simplex method. Math Oper Res 2(2):103–107
Botkin N, Turova-Botkina V (1994) An algorithm for finding the Chebyshev center of a convex polyhedron. Appl Math Optim 29(2):211–222
Chrystal G (1885) On the problem to construct the minimum circle enclosing \(n\) given points in the plane. Proc Edinb Math Soc 3:30–33
Dearing PM, Zeck CR (2009) A dual algorithm for the minimum covering ball problem in \({\mathbb{R}}^n\). Oper Res Lett 37:171–175
Dearing PM, Smith A (2013) A dual algorithm for the weighted minimum covering ball problem in \({\mathbb{R}}^n\). J Glob Optim 55(2):261–278
Dyer ME (1992) A class of convex programs with applications to computational geometry. In: Proceedings 8th annual ACM symposium on computational geometry, pp 9–15
Elzinga J, Hearn DW (1972a) Geometrical solutions for some minimax location problems. Transp Sci 6:379–394
Elzinga J, Hearn DW (1972b) The minimum covering sphere problem. Manag Sci 19:96–104
Fischer K, Gärtner B, Kutz M (2003) Fast smallest-enclosing-ball computation in high dimensions. In: Proceedings of the 11th annual European symposium on algorithms (ESA), lecture notes in computer science, vol 2832. Springer, pp 630–641
Fourer R, Gay DM, Kernighan BW (1993) AMPL: a modeling language for mathematical programming. Boyd and Fraser Publishing Company, San Francisco
Gärtner B (1999) Fast and robust smallest enclosing balls. In: Proceedings of the 7th annual European symposium on algorithms (ESA), lecture notes in computer science, vol 1643. Springer, pp 325–338
Gärtner B, Schönherr S (2000) An efficient, exact, and generic quadratic programming solver for geometric optimization. In: Proceedings of the 16th annual ACM symposium on computational geometry, pp 110–118
Hearn DW, Vijay J (1982) Efficient algorithms for the weighted minimum circle problem. Oper Res 30:777–795
Hopp TH, Reeve CP (1996) An algorithm for computing the minimum covering sphere in any dimension. Technical report NISTIR 5831, National Institute of Standards and Technology
Megiddo N (1983a) Linear time algorithm for linear programming in \({\mathbb{R}}^3\) and related problems. SIAM J Comput 12:759–776
Megiddo N (1983b) The weighted Euclidean 1-center problem. Math Oper Res 8:498–504
Megiddo N (1984) Linear programming in linear time when the dimension is fixed. J ACM 31:114–127
Plastria F (2002) Continuous covering location problems. In: Drezner Z, Hamacher HW (eds) Facility location: application and theory. Springer, New York, pp 37–79
Sylvester JJ (1857) A question in the geometry of situation. Q J Pure Appl Math 1:79
Sylvester JJ (1860) On Poncelet’s approximation linear valuation of surd forms. Philos Mag (Fourth Ser) XX:203–222
The MathWorks, Inc. (1997) Using matlab. The MathWorks, Inc., Natick
Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Program 106(1):25–57
Welzl E (1991) Smallest enclosing disks (balls and ellipsoids). In: Maurer H (ed) New results and new trends in computer science, lecture notes in computer science, vol 555. Springer, Berlin, pp 359–370
Zhou G, Toh K, Sun J (2005) Efficient algorithms for the smallest enclosing ball problem. Comput Optim Appl 30:147–160
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Dearing, P.M., Belotti, P. & Smith, A.M. A primal algorithm for the weighted minimum covering ball problem in \(\mathbb {R}^n\) . TOP 24, 466–492 (2016). https://doi.org/10.1007/s11750-015-0405-9
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DOI: https://doi.org/10.1007/s11750-015-0405-9