A Generalized Waist Problem: Optimality Condition and Algorithm

TL;DR

This paper generalizes the waist problem by replacing lines with convex sets and seeks to find minimal connecting polygons, establishing existence, uniqueness, optimality conditions, and developing a convergent algorithm with practical applications.

Contribution

It introduces a generalized geometric optimization problem, derives comprehensive optimality conditions, and proposes a convergent projected subgradient method with numerical validation.

Findings

Solutions exist and are unique under certain convexity and general position conditions.

Derived necessary and sufficient optimality conditions with geometric interpretations.

Developed and proved convergence of a projected subgradient descent algorithm.

Abstract

Many years ago John Tyrell a lecturer at King's college London challenged his Ph.D. students with the following puzzle: show that there is a unique triangle of minimal perimeter with exactly one vertex to lie on one of three given lines, pairwise disjoint and not all parallel in the space. The problem in literature is known as the waist problem, and only convexity rescued in this case. Motivated by this we generalize it by replacing lines with a number of convex sets in the Euclidean space and ask to minimize the sum of distances connecting the sets by means of closed polygonal curve. This generalized problem significantly broadens its geometric and practical scope in view of modern convex analysis. We establish the existence of solutions and prove its uniqueness under the condition that at least one of the convex sets is strictly convex and all are in general position: each set can be separated by convex hull of others. A complete set of necessary and sufficient optimality conditions is derived, and their geometric interpretations are explored to link these conditions with classical principles such as the reflection law of light. To address this problem computationally, we develop a projected subgradient descent method and prove its convergence. Our algorithm is supported by detailed numerical experiments, particularly in cases involving discs and spheres. Additionally, we present a real-world analogy of the problem in the form of inter-island connectivity, illustrating its practical relevance. This work not only advances the theory of geometric optimization but also contributes effective methods and insights applicable to facility location, network design, robotics., computational geometry, and spatial planning.