Solving Regularized Multifacility Location Problems with Unknown Number of Centers via Difference-of-Convex Optimization

TL;DR

This work introduces a gauge-based multifacility location model using a Minkowski gauge $\rho$ with a Laplace-type fusion penalty to jointly cluster data and determine the effective number of centers. A smoothing-based DC approach is developed, where $f$ is decomposed as $f_\mu=g_\mu-h_\mu$ and minimized via DC algorithms (DCA) with accelerated variants and an automatic cluster-deletion mechanism (LDCA-K). Theoretical results establish existence and stability of solutions, Lipschitz continuity of the data-fit and optimal value, and convergence of the proposed DC schemes. Numerical experiments on real and synthetic datasets demonstrate automatic model order selection, robustness under varying geometry, and practical scalability, illustrating that the fusion and smoothing path yields a stable, interpretable clustering structure without manual tuning.

Abstract

In this paper, we develop optimization methods for a new model of multifacility location problems defined by a Minkowski gauge with Laplace-type regularization terms. The model is analyzed from both theoretical and numerical perspectives. In particular, we establish the existence of optimal solutions and study qualitative properties of global minimizers. By combining Nesterov's smoothing technique with recent advances in difference-of-convex optimization, following the pioneering work of P. D. Tao and L. T. H. An and others, we propose efficient numerical algorithms for minimizing the objective function of this model. As an application, our approach provides an effective method for determining the number of centers in gauge-based multifacility location and clustering problems. Our results extend and complement recent developments.

Figures (8)

• Figure 1: Prototype count along the warm--started regularization path for four datasets. Each panel shows the number of surviving prototypes as a function of the fusion parameter $\lambda$, while the smoothing parameter $\mu$ simultaneously decreases along a paired geometric schedule. All trajectories begin with an intentionally over--specified initialization ($k_0=10$ prototypes). As $\lambda$ increases, redundant prototypes rapidly merge, and the model stabilizes at a data--driven number of clusters. On the three synthetic datasets (top row and bottom--left), the path consistently converges to the known ground--truth structure ($k=3$ or $k=4$), and this value remains stable across the entire range of $(\lambda,\mu)$. On the Iris dataset (bottom--right), the path similarly stabilizes at $k=3$, matching the three species. The figure illustrates the qualitative robustness of the warm--started regularization path: once a coherent cluster structure emerges, it persists over a broad region of the fusion parameter, even as smoothing weakens and cluster boundaries shift.
• Figure 2: Prototype trajectories during inner DCA optimization prior to deletion. Multiple prototypes collapse onto shared stationary points.
• Figure 3: Representative LDCA-K run illustrating convergence behavior and correct model selection despite prototype fusion.
• Figure 4: Phase diagram of the effective number of clusters over the $(\lambda,\mu)$ grid.
• Figure 5: Representative LDCA-K solutions illustrating the complementary roles of fusion and smoothing. Left:$\lambda=0.90$, $\mu=0.10$, where strong fusion with limited smoothing leads to aggressive prototype merging and a reduced effective model. Right:$\lambda=0.10$, $\mu=0.90$, where increased smoothing stabilizes the optimization and yields four well-separated prototypes aligned with the true cluster centers.

Theorems & Definitions (36)

• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Example 2.4
• Example 2.5
• Definition 2.6
• Proposition 2.7
• proof
• Example 2.8