De-singularity Subgradient for the $q$-th-Powered $ell_p$-Norm Weber Location Problem
🤖 AI Summary
为解决韦伯定位问题中目标函数梯度缺失难题，开发新算法，优化目标函数，确保算法收敛，实验证明有效且计算高效。
Application Category
📝 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 ($1leqslant q<2$), 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 $1leqslant qleqslant p$ and $1leqslant p<2$, 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.
Problem
Research questions and friction points this paper is trying to address.
q-th Powered l_p Norm
Weber Location Problem
Gradient Non-existence
Innovation
Methods, ideas, or system contributions that make the work stand out.
De-singularity Subgradient
q-th-Powered ℓ_p-Norm Weber Location Problem
Fast Computation
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