On the global minimum in a balanced circular packing problem
- 167 Downloads
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.
KeywordsBalanced packing Circles Nonsmooth optimization Shor’s r-algorithm Dual bound
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.
- 1.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
- 2.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)Google Scholar
- 5.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)Google Scholar
- 6.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)Google Scholar
- 7.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)Google Scholar
- 11.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)Google Scholar
- 13.Stetsyuk, P.I.: Ellipsoid Methods and \(r\)-Algorithms. Evrika, Chisinau (2014). (in Russian)Google Scholar
- 15.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
- 16.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)Google Scholar