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

Manohar Choudhary, Triloki Nath, Ram K. Pandey

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.

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

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 and are provided to validate the algorithm's robustness across diverse geometries and weighting schemes.
Paper Structure (13 sections, 12 theorems, 106 equations, 6 figures, 4 tables, 1 algorithm)

This paper contains 13 sections, 12 theorems, 106 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Modern Generalizations via Convex Analysis
  3. Problem Formulation
  4. Preliminaries
  5. Characterization of Solutions
  6. Existence of Optimal Solutions
  7. Uniqueness of Optimal Solutions
  8. Optimality Conditions
  9. Computational Algorithms
  10. Convergence Analysis
  11. Numerical Illustrations
  12. Experimental Setup:
  13. Conclusion

Key Result

Theorem 2.8

Let $C \subset \mathbb{R}^n$ be a nonempty closed convex set and let $y \notin C$. Then there exist a vector $a \in \mathbb{R}^n$, $a \neq 0$, and a scalar $\alpha \in \mathbb{R}$ such that

Figures (6)

  • Figure 1: Illustration of the Generalized Heron Problem (GHP).
  • Figure 2: Illustration of the generalized waist problem in $\mathbb{R}^2$.
  • Figure 3: Illustration of the Generalized Heron-Waist Problem (GHWP).
  • Figure 4: Illustration of the nondegeneracy condition in the Generalized Heron--Waist Problem.
  • Figure 6: Final optimal configuration for Example \ref{['GHWPEX1']} in $\mathbb{R}^2$.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Remark 1.1: Necessity of the Nondegeneracy Condition
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 25 more