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

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.

• The paper proposes a novel de-singularity subgradient method that effectively handles continuum singularity issues in the q-th powered ℓp-norm Weber problem.
• The introduced qPpNWAWS algorithm leverages subgradient techniques to ensure accurate descent and linear convergence even at singular points.
• Experimental validations across financial datasets demonstrate improved convergence rates, cumulative wealth, and Sharpe Ratios in portfolio selection tasks.

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

The paper "De-singularity Subgradient for the $q$-th-Powered $\ell_p$-Norm Weber Location Problem" addresses the challenge of singularity issues in solving the Weber location problem, particularly for the $q$-th-powered $\ell_p$-norm with constraints $1 \leqslant q \leqslant p$ and $1 \leqslant p < 2$. This extension fills the gap where previous methods could only handle finite singular points for the $\ell_2$-norm case.

Context and Relevance

The Weber location problem is fundamental in areas such as AI, machine learning, and optimization, centering around minimizing the weighted sum of distances (modeled by various norms) from a set of given points to an optimal point. While the problem is well-defined for many norm configurations, singularities occur when the gradient is undefined, hindering the applicability of gradient-based solutions.

Methodology and Contribution

The authors propose a novel de-singularity subgradient approach, extending the capabilities of existing methods to scenarios where the singularity set is a continuum, as is the case with their norm constraints. Key contributions include:

1. De-singularity Subgradient Development: This methodological advancement enables handling of singularities effectively without escalating computational complexity.
2. Algorithm Design - $q$-P$p$NWAWS: The paper introduces the $q$-th-Powered $\ell_p$-Norm Weiszfeld Algorithm without Singularity (abbreviated as $q$P$p$NWAWS). The algorithm is designed to ensure convergence past singular points, leveraging subgradients for accurate descent direction even at singular iterates.
3. Convergence Proofs: The paper delivers comprehensive proofs of the algorithm's convergence, addressing not just theoretical soundness but practical implementability by showing a linear computational convergence rate in experiments.

Experimental Validation

Experiments conducted on six datasets from varied financial markets demonstrate the effectiveness of the proposed algorithm. Notably, $q$P$p$NWAWS manages to resolve singularities efficiently—requiring minimal iteration steps—and achieves favorable convergence in computational time and rate across different parameter values ($p$ and $q$). The algorithm's success in improving cumulative wealth and Sharpe Ratios in online portfolio selection tasks underscores its practical value.

Implications and Future Work

The research not only advances methodological tools for tackling singularity in optimization but also opens up new avenues for applying robust optimization techniques across complex, real-world scenarios. Importantly, it demonstrates that accommodating different distances through parameter variation can yield substantive improvements in financial contexts.

Future research could contemplate extending the framework to multi-facility location problems, where similar singularity issues could impede optimization. Additionally, further exploration of non-Euclidean geometries in optimization using these techniques could be beneficial, especially in areas like computer vision or autonomous systems.

Overall, the paper's rigorous approach and substantial improvements in both theory and practice make it a significant contribution to the field of optimization and computational algorithms.