A Generalized $(k,m)$ Heron Problem:Optimality Conditions and Algorithm

TL;DR

The paper addresses a generalized multi-set Heron problem in $\mathbb{R}^n$, seeking points $x_i \in S_i$ and $y_j \in C_j$ that minimize the total pairwise distance $\sum_{i=1}^k\sum_{j=1}^m \|x_i - y_j\|$. It formulates a convex-analytic framework, establishes existence under mild boundedness, and provides both nonuniqueness examples and a sufficiency condition for uniqueness, together with precise first-order optimality conditions via subdifferentials and normal cones. A Projected Subgradient Algorithm (PSA) with a diminishing step-size is proposed and proven convergent, and two numerical experiments in $\mathbb{R}^2$ and $\mathbb{R}^3$ confirm robustness, accuracy, and efficiency. The framework unifies classical Heron-type problems and paves the way for applications in location science, robotics, and computational geometry.

Abstract

This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The problem is formulated as finding a point in each set that minimizes the pairwise distances from the points in the $k$-feasible sets to the points in the $m$-target sets. This formulation leads to a convex optimization framework that generalizes several well-known geometric distance problems. Using tools from convex analysis, we establish fundamental results on existence, uniqueness, and first-order optimality conditions through subdifferential calculus and normal cone theory. Building on these insights, a Projected Subgradient Algorithm (PSA) is proposed for numerical solution, and its convergence is rigorously proved under a diminishing step-size rule. Numerical experiments in $\mathbb{R}^2$ and $\mathbb{R}^3$ illustrate the algorithm's stability, geometric accuracy, and computational efficiency. Overall, this work provides a comprehensive analytical and algorithmic framework for multi-set geometric optimization with promising implications for location science, robotics, and computational geometry.

Figures (4)

• Figure 1: Illustration of generalized $(k,m)$ Heron problem.
• Figure 2: Illustration of multiple optimal solutions to the generalized $(k,m)$-Heron problem.
• Figure 3: Illustration of Example \ref{['ex:2D_Heron']}
• Figure 4: Illustration of Example \ref{['ex:psa_3d']}

Theorems & Definitions (24)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Proposition 2.8
• Definition 2.9
• Proposition 2.10: Subdifferential of the distance function