A Convex Optimization Approach to the Discrete Hanging Chain Problem
Russell Gabrys, Stefan Sremac
TL;DR
The paper generalizes the discrete hanging chain to nonuniform link masses $m_i$ and lengths $\ell_i$ and shows that the equilibrium shape minimizes potential energy under the given constraints. It recasts the problem as a convex program $\min_y \sum_{i=1}^n c_i y_i$ subject to $\sum y_i=0$ and $d - \sum_{i=1}^n \sqrt{\ell_i^2 - y_i^2} \le 0$, with $c_i$ defined as $c_i = \frac{1}{2} m_i + \sum_{j=i+1}^n m_j$ (and $c_n = \frac{1}{2} m_n)$. In the symmetric case, the optimal solution reduces to solving a single univariate equation in $\mu$, via a KKT-based representation $y_i = -\frac{\ell_i (c_i - \bar c)}{\sqrt{(c_i - \bar c)^2 + \mu^2}}$, where $\bar c = \frac{1}{2}\sum m_i$, and a monotone scalar function $\phi(\mu)$; existence and uniqueness of the root follow from IVT and monotonicity, enabling Newton-type methods. The results provide a rigorous and computationally efficient framework for nonuniform discrete hanging chains and connect energy minimization, convex analysis, and univariate root finding.
Abstract
In this paper we investigate the discrete version of the classical hanging chain problem. We generalize the problem, by allowing for arbitrary mass and length of each link. We show that the shape of the chain can be obtained by solving a convex optimization problem. Then we use optimality conditions to show that the problem can be further reduced to solving a single non-linear equation, when the links of the chain have symmetric mass and length.