Abstract
We consider the problem of optimally covering plane domains by a given number of circles. The mathematical modeling of this problem leads to a min–max–min formulation which, in addition to its intrinsic multi-level nature, has the significant characteristic of being non-differentiable. In order to overcome these difficulties, we have developed a smoothing strategy using a special class C∞ smoothing function. The final solution is obtained by solving a sequence of differentiable subproblems which gradually approach the original problem. The use of this technique, called Hyperbolic Smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essential of the method is presented. For the purpose of illustrating both the actual working and the potentialities of the method, a set of computational results is presented.
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Xavier, A.E., Oliveira, A.A.F.D. Optimal Covering of Plane Domains by Circles Via Hyperbolic Smoothing. J Glob Optim 31, 493–504 (2005). https://doi.org/10.1007/s10898-004-0737-8
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DOI: https://doi.org/10.1007/s10898-004-0737-8