# Nesterov's Smoothing Technique and Minimizing Differences of Convex Functions for Hierarchical Clustering

## TL;DR

This paper reformulates a complex bilevel hierarchical clustering problem as a continuous optimization task, applying Nesterov's smoothing and DCA algorithms to effectively handle nonsmoothness and nonconvexity, demonstrated through numerical examples.

## Contribution

It introduces a novel continuous reformulation of NP-hard hierarchical clustering models and applies advanced smoothing and optimization techniques to solve them.

## Key findings

- Successful application of Nesterov's smoothing to hierarchical clustering
- Effective use of DCA for nonsmooth nonconvex optimization
- Numerical results demonstrate method's viability

## Abstract

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.