De-singularity Subgradient for the $q$-th-Powered $\ell_p$-Norm Weber Location Problem

TL;DR

This work tackles gradient singularities in the Weber location problem under the $q$-th powered $\ell_p$-norm for $1\le p<2$ and $1\le q\le p$ by introducing a de-singularity subgradient on the continuum singular set ${\mathcal S}_p$ and developing a Weiszfeld-type algorithm without singularity, denoted $q$P$p$NWAWS. It provides a complete update scheme for nonsingular iterates, a precise de-singularity subgradient framework for singular iterates, and a convergence theorem guaranteeing descent, boundedness, and convergence of the cost function (with a linear practical convergence rate). The approach is validated on six real-world datasets, showing effective resolution of singularities, fast convergence (typically under 0.03 seconds per run), and competitive investing performance in online portfolio selection tasks, with certain $(q,p)$ settings outperforming the baseline. These results establish a robust, scalable method for a broad class of $q$-th powered Weber problems and point toward extensions to multi-facility location problems.

Abstract

The Weber location problem is widely used in several artificial intelligence scenarios. However, the gradient of the objective does not exist at a considerable set of singular points. Recently, a de-singularity subgradient method has been proposed to fix this problem, but it can only handle the $q$-th-powered $\ell_2$-norm case ($1\leqslant q<2$), which has only finite singular points. In this paper, we further establish the de-singularity subgradient for the $q$-th-powered $\ell_p$-norm case with $1\leqslant q\leqslant p$ and $1\leqslant p<2$, which includes all the rest unsolved situations in this problem. This is a challenging task because the singular set is a continuum. The geometry of the objective function is also complicated so that the characterizations of the subgradients, minimum and descent direction are very difficult. We develop a $q$-th-powered $\ell_p$-norm Weiszfeld Algorithm without Singularity ($q$P$p$NWAWS) for this problem, which ensures convergence and the descent property of the objective function. Extensive experiments on six real-world data sets demonstrate that $q$P$p$NWAWS successfully solves the singularity problem and achieves a linear computational convergence rate in practical scenarios.

Figures (2)

• Figure 1: (a) When $p=2$, the singular set $\mathcal{S}_2$ (the red dots) is finite and an effective gradient-type algorithm visits each singular point for only once. Hence the iterate $\bm y_{(k)}$ (the circle) can finally escape from all the singular points. (b) When $1\leqslant p<2$, the singular set $\mathcal{S}_p$ is a continuum (the red dashed lines), hence the iterate $\bm y_{(k)}$ may revisit $\mathcal{S}_p$ for infinite times and may not escape from $\mathcal{S}_p$.
• Figure 2: An intuitive example for $U_i(\bm y_{(k)})$ and $V_t(\bm y_{(k)})$: $U_1(\bm y_{(k)})=\{1,4\}$, $U_2(\bm y_{(k)})=\{1,6,8\}$, $U_3(\bm y_{(k)})=\{3,4,6\}$, and $V_1(\bm y_{(k)})=\{1,2\}$, $V_2(\bm y_{(k)})=\emptyset$, $V_3(\bm y_{(k)})=\{3\}$, $V_4(\bm y_{(k)})=\{1,3\}$, $V_6(\bm y_{(k)})=\{2,3\}$, $V_8(\bm y_{(k)})=\{2\}$.

Theorems & Definitions (20)

• Theorem 1: Descent Property at Nonsingular Iterates
• Corollary 2
• Definition 3: Subgradient, rockafellar2009variational
• Definition 4: Singular Component(s)
• Definition 5: $\mathbf{q}$-th-Powered $\ell_p$-Norm De-singularity Subgradient
• Theorem 6: Characterization of Subgradients and Minimum
• Theorem 7: Descent Property at Singular Iterates
• Lemma 8
• Lemma 9
• Lemma 10