A Generalized Waist Problem: Optimality Condition and Algorithm

TL;DR

This work generalizes the classical waist problem by replacing the three lines with $m$ closed convex sets in $\mathbb{R}^n$ and seeks a cyclic chain of points $a_i\in C_i$ minimizing $D(a)=\sum_{i=1}^m \|a_i-a_{i+1}\|$. The authors establish existence in two broad cases (bounded sets or coercive growth when some set is unbounded) and prove uniqueness when at least one $C_i$ is strictly convex under a general-position assumption. They derive a complete set of necessary and sufficient optimality conditions expressed via subdifferentials and normal cones, with a clear geometric interpretation linked to angle bisectors and reflection principles. A projected subgradient descent algorithm is developed and proven convergent, and its efficiency is demonstrated through numerical experiments on discs and spheres, with acceleration via Aitken’s Delta^2 transform markedly reducing iterations. The work provides both theoretical insights and practical algorithms applicable to facility location, network design, robotics, and spatial planning, and suggests future extensions to broader spaces and non-convex constraints.

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.

Figures (9)

• Figure 5: Illustration of the generalized waist problem for three convex sets.
• Figure 6: Illustration of the closed polygonal chain in $\mathbb{R}^3$
• Figure 7: Real-world analogy of the waist problem using four islands in the Andaman region.
• Figure 8: Convex sets $A$ and $B$ are closed and disjoint, but cannot be strictly separated by a hyperplane.
• Figure 9: Three parallel lines not lying in the same plane. Multiple optimal solutions may occur.

Theorems & Definitions (45)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Theorem 2.7: Strict Separation Theorem
• Definition 2.8
• Theorem 2.9
• Definition 2.10