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Moreau Envelope-Based Clustering for Generalized Multi-Source Weber Problem

Nguyen Thi Thu Van

Abstract

In this paper, we propose an efficient algorithm for data clustering based on the Moreau envelope, which approximates nonsmooth and nonconvex components of the generalized multi-source Weber problem. The number of clusters is not fixed in advance and is determined automatically by progressively removing empty or redundant clusters. The smoothing induced by the Moreau envelope transforms the original problem into a structured optimization task that can be efficiently solved using first-order methods and simple matrix vector operations. Numerical experiments on synthetic and real datasets show that the proposed approach is fast, scalable, and competitive with existing methods in both clustering quality and computational efficiency.

Moreau Envelope-Based Clustering for Generalized Multi-Source Weber Problem

Abstract

In this paper, we propose an efficient algorithm for data clustering based on the Moreau envelope, which approximates nonsmooth and nonconvex components of the generalized multi-source Weber problem. The number of clusters is not fixed in advance and is determined automatically by progressively removing empty or redundant clusters. The smoothing induced by the Moreau envelope transforms the original problem into a structured optimization task that can be efficiently solved using first-order methods and simple matrix vector operations. Numerical experiments on synthetic and real datasets show that the proposed approach is fast, scalable, and competitive with existing methods in both clustering quality and computational efficiency.

Paper Structure

This paper contains 7 sections, 6 theorems, 54 equations, 4 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Technical Tools
  3. Proposed Algorithm and Convergence Analysis
  4. Numerical Results
  5. Fixed Number of Centers
  6. Adaptive number of centers
  7. Conclusion

Key Result

Lemma 2.1

Let $\varphi:\mathbb{R}^n \to (-\infty,+\infty]$ be proper and lower semicontinuous, and let $\mu>0$. Suppose that the proximal mapping $\mathrm{prox}_{\mu\varphi}$ is single-valued and continuous in a neighborhood $U$ of $\bar{x} \in \mathbb{R}^n$. Then the Moreau envelope $\varphi^\mu$ is continuo $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Visualization of the best solution over 50 runs for the synthetic dataset under the fixed-center setting with $k_{\mathrm{init}} = 10$.
  • Figure 2: Visualization of the best solution over 50 runs for the synthetic dataset under the fixed-center setting with $k_{\mathrm{init}} = 20$.
  • Figure 3: Visualization of the best solution for the synthetic dataset under the adaptived-center setting with $k_{\mathrm{init}} = 10$.
  • Figure 4: Visualization of the best solution for the synthetic dataset under the adaptive-center setting with $k_{\mathrm{init}} = 20$.

Theorems & Definitions (14)

  • Definition 1: RockafellarWets98
  • Definition 2: RockafellarWets98
  • Lemma 2.1
  • proof
  • Lemma 3.1: Lemma 2.5, LongNamTranVan24
  • Definition 3: LongNamTranVan24, Definition 3.5
  • Lemma 3.2: LongNamTranVan24, Lemma 3.8
  • Lemma 3.3
  • proof
  • Theorem 3.1
  • ...and 4 more