Abstract
The paper considers balanced packing problem of a given family of circles into a larger circle of the minimal radius as a multiextremal nonlinear programming problem. We reduce the problem to unconstrained minimization problem of a nonsmooth function by means of nonsmooth penalty functions. We propose an efficient algorithm to search for local extrema and an algorithm for improvement of the lower bound of the global minimum value of the objective function. The algorithms employ nonsmooth optimization methods based on Shor’s r-algorithm. Computational results are given.
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Che, C., Wang, Y., Teng, H.: Test problems for quasi-satellite packing: cylinders packing with behavior constraints and all the optimal solutions known. Optimization Online (2008). http://www.optimization-online.org/DB_HTML/2008/09/2093.html
Fasano, G., Pinte’r, J.D. (eds.): Modeling and Optimization in Space Engineering. Series: Springer Optimization and Its Applications, Vol. 73, XII, p. 404 (2013)
Jingfa, L., Gang, L.: Basin filling algorithm for the circular packing problem with equilibrium behavioral constraints. Sci. China Inf. Sci. 53(5), 885–895 (2010)
Kappel, F., Kuntsevich, A.V.: An implementation of Shor’s \(r\)-algorithm. Comput. Optim. Appl. 15(2), 193–205 (2000)
Kovalenko, A.A., Pankratov, A.V., Romanova, T.E., Stetsyuk, P.I.: Packing circular cylinders into a cylindrical container taking into account the system behavior constraints. J. Comput. Appl. Math. Ukraine 1(111), 126–134 (2013). (in Russian)
Nenakhov, E.I., Romanova, T.E., Stetsyuk, P.I.: Balanced packing problem of circles in a circle of minimum radius. Theory of optimal solutions, Ukraine, pp. 143–153 (2013) (in Russian)
Oliveira W.A., Moretti A.C., Salles Neto L.L.: A heuristic for the nonidentical circle packing problem. Anais do CNMAC 3:626–632 (2010)
Pshenichnyi, B.N., Sobolenko, L.A.: Linearization method for inverse convex programming. Cybern. Syst. Anal. 31(6), 852–862 (1995)
Shor, N.Z.: Nondifferentiable Optimization and Polynomial Problems. Kluwer Academic Publishers, Dordrecht (1998)
Shor, N.Z., Stetsyuk, P.I.: Modified \(r\)-algorithm to find the global minimum of polynomial functions. Cybern. Syst. Anal. 33(4), 482–497 (1997)
Shor, N.Z., Stetsyuk, P.I.: Dual solution of quadratic-type problems by \(r\)-algorithm (subroutine DSQTPr), Abstracts of Second International Workshop “Recent Advances in Non-Differentiable Optimization”, Kyiv, p. 36 (2001)
Shor, N.Z., Zhurbenko, N.G., Likhovid, A.P., Stetsyuk, P.I.: Algorithms of nondifferentiable optimization: development and application. Cybern. Syst. Anal. 39(4), 537–548 (2003)
Stetsyuk, P.I.: Ellipsoid Methods and \(r\)-Algorithms. Evrika, Chisinau (2014). (in Russian)
Stetsyuk, P.I., Romanova, T.E., Schiethauer, G.: On the global minimum of the objective function in a balanced circular packing problem. Dopovidi NAN Ukraine 6, 53–57 (2014). (in Russian)
Xu, Y.-C., Dong, F.-M., Liu, Y., Xiao, R.-B., Amos, M.: Ant Colony Algorithm for the Weighted Item Layout Optimization Problem (2010). arXiv:1001.4099
Xu, Y.-C., Xiao, R.-B., Amos, M.: A novel algorithm for the layout optimization problem. In: Proceedings of the 2007 IEEE Congress on Evolutionary Computation (CEC07). IEEE Press, pp. 3938–3942 (2007)
Acknowledgments
We are extremely grateful to the associate editor and the anonymous referee for deep and useful comments that helped us considerably improve our paper. The authors acknowledge the support of the Science and Technology Centre in Ukraine and the National Academy of Sciences of Ukraine, grant 5710.
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Stetsyuk, P.I., Romanova, T.E. & Scheithauer, G. On the global minimum in a balanced circular packing problem. Optim Lett 10, 1347–1360 (2016). https://doi.org/10.1007/s11590-015-0937-9
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DOI: https://doi.org/10.1007/s11590-015-0937-9