A Generalized Heron-Waist Problem: Optimality Conditions and Convergence Analysis

TL;DR

This paper formulates and analyzes the Generalized Heron-Waist Problem, combining optimal hub location with minimal-perimeter polygonal configurations, providing existence, uniqueness, optimality conditions, and a convergent algorithm with numerical validation.

Contribution

It introduces the GHWP, establishes theoretical properties, derives optimality conditions, and develops a convergent computational algorithm for this new problem.

Findings

Existence of optimal solutions under certain conditions

Uniqueness for strictly convex constraint sets

Convergence of the proposed algorithm with numerical validation

Abstract

This paper introduces and solves the Generalized Heron-Waist Problem (GHWP), that integrates the classical Heron problem of optimal hub location and the waist problem of minimal-perimeter configuration. The GHWP seeks an optimal closed polygonal chain with weights whose vertices are constrained to lie in the given nonempty, closed, and convex sets, while simultaneously minimizing weighted distances to a central hub point. This coupled formulation naturally models systems in which cyclic internal connectivity and radial access to a hub must be optimized jointly a structural feature that arises in applications such as supply-chain design, transportation planning, and communication infrastructures. Using modern convex analysis tools, we establish existence of optimal solutions under boundedness and general position assumptions of sets, we prove uniqueness when constraint sets are strictly convex with positive weights. We also derive first order necessary and sufficient optimality conditions using subdifferential calculus. For computation, we develop a Projected Subgradient Algorithm (PSA) and we prove convergence of the best-iterate sequence under classical diminishing step size rules. Numerical illustrations in $\mathbb{R}^2$ and $\mathbb{R}^3$ are provided to validate the algorithm's robustness across diverse geometries and weighting schemes.