An Efficient Inexact Newton-CG Algorithm for the Smallest Enclosing Ball Problem of Large Dimensions

TL;DR

This paper introduces a fast inexact Newton-CG algorithm tailored for computing the smallest enclosing ball of large sets of balls in high-dimensional spaces, leveraging sparsity and inexact computations for efficiency.

Contribution

It proposes a novel inexact Newton-CG method with adaptive inexact gradient and Hessian computations for large-scale SEB problems, with proven global convergence.

Findings

The algorithm efficiently solves large-scale SEB problems.

It outperforms classical Newton-CG and previous methods in computational experiments.

Global convergence is theoretically established.

Abstract

In this paper, we consider the problem of computing the smallest enclosing ball (SEB) of a set of $m$ balls in $\mathbb{R}^n,$ where the product $mn$ is large. We first approximate the non-differentiable SEB problem by its log-exponential aggregation function and then propose a computationally efficient inexact Newton-CG algorithm for the smoothing approximation problem by exploiting its special (approximate) sparsity structure. The key difference between the proposed inexact Newton-CG algorithm and the classical Newton-CG algorithm is that the gradient and the Hessian-vector product are inexactly computed in the proposed algorithm, which makes it capable of solving the large-scale SEB problem. We give an adaptive criterion of inexactly computing the gradient/Hessian and establish global convergence of the proposed algorithm. We illustrate the efficiency of the proposed algorithm by using the classical Newton-CG algorithm as well as the algorithm from [Zhou. {et al.} in Comput. Opt. \& Appl. 30, 147--160 (2005)] as benchmarks.