Optimization Letters

, Volume 10, Issue 6, pp 1347–1360 | Cite as

On the global minimum in a balanced circular packing problem

  • Petro I. StetsyukEmail author
  • Tatiana E. Romanova
  • Guntram Scheithauer
Original Paper


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.


Balanced 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Petro I. Stetsyuk
    • 1
    Email author
  • Tatiana E. Romanova
    • 2
  • Guntram Scheithauer
    • 3
  1. 1.V.M.Glushkov Institute of Cybernetics of the NASUKievUkraine
  2. 2.Institute for Mechanical Engineering Problems of the NASUKharkovUkraine
  3. 3.Institute of Numerical MathematicsDresden University of TechnologyDresdenGermany

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