Optimization Letters

, Volume 12, Issue 3, pp 455–473 | Cite as

Nesterov’s smoothing technique and minimizing differences of convex functions for hierarchical clustering

Original Paper


A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks. In this paper, we consider two different formulations of the bilevel hierarchical clustering problem, a discrete optimization problem which can be shown to be NP-hard. Our approach is to reformulate the problem as a continuous optimization problem by making some relaxations on the discreteness conditions. Then Nesterov’s smoothing technique and a numerical algorithm for minimizing differences of convex functions called the DCA are applied to cope with the nonsmoothness and nonconvexity of the problem. Numerical examples are provided to illustrate our method.


DC programming Nesterov’s smoothing technique Hierarchical clustering Subgradient Fenchel conjugate 



N. M. Nam: Research of this author was partly supported by the National Science Foundation under Grant #1411817.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Fariborz Maseeh Department of Mathematics and StatisticsPortland State UniversityPortlandUSA
  2. 2.School of General StudiesStockton UniversityGallowayUSA

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