A DC programming to two-level hierarchical clustering with ℓ1 norm

The main challenge in solving clustering problems using mathematical optimization techniques is the non-smoothness of the distance measure used. To overcome this challenge, we used Nesterov's smoothing technique to find a smooth approximation of the ℓ 1 norm. In this study, we consider a bi-level hierarchical clustering problem where the similarity distance measure is induced from the ℓ 1 norm. As a result, we are able to design algorithms that provide optimal cluster centers and headquarter (HQ) locations that minimize the total cost, as evidenced by the obtained numerical results.