Subsections

• Implementation
• Performance

3 Hanging Chain

Find the chain (of uniform density) of length $L$ suspended between two points with minimal potential energy.

Implementation

This classical problem (see Cesari [11, pages126-127]) was suggested by Hans Mittelmann. In this problem we need to determine a function $x (t)$, the shape of the chain, that minimizes the potential energy

$$\ \int_0^1 x \sqrt{1 + x'^2} \,dt$$

subject to the constraint on the length of the chain,

$$\int_{0}^{1} \sqrt{1 + x'^2} \,dt = L,$$

and the end conditions $x(0) = a$ and $x(1) = b$. We reformulate this problem as a control problem in terms of the function $u = x'$. The optimal control version of the problem is to minimize

$$\ \int_0^1 x \sqrt{1 + u^2} \,dt$$

subject to a differential equation and a constraint on the length of the chain,

$$x' = u, \qquad \int_{0}^{1} \sqrt{1 + u^2} \,dt = L .$$

We discretize the integrals and the differential equation with the trapezoidal rule on a uniform mesh with $n_h$ intervals. Data for this problem appears in Table 3.1.

Table 3.1: Hanging chain problem data

Variables	$2 n_h$
Constraints	$n_h + 1$
Bounds	0
Linear equality constraints	$n_h$
Linear inequality constraints	0
Nonlinear equality constraints	1
Nonlinear inequality constraints	0
Nonzeros in $\nabla ^2 f(x)$	$3n_h$
Nonzeros in $c'(x)$	$5n_h$

Performance

Results for the AMPL implementation are summarized in Table 3.2 with $a = 1$, $b=3$, and $L=4$. The starting point is the quadratic

$$x(t) = \left ( 2 \vert b-a \vert \right ) t ( t - 2 t_m ) + a ,$$

where $t_m = 0.25$ if $b > a$ and $t_m = 0.75$ otherwise, evaluated at the mesh points. This choice is convex and satisfies the boundary data. The control function $u$ is set to $x'$. The optimal chain is shown in Figure 3.1.

Table 3.2: Performance on hanging chain problem

Solver	$n_h=50$	$n_h=100$	$n_h=200$	$n_h=400$
LANCELOT	13.91 s	42.79 s	268.7 s	577.24 s
$f$	5.07230e+00	5.07005e+00	5.06903e+00	5.06788e+00
$c$ violation	3.31310e-06	9.61060e-06	2.01200e-06	6.25860e-06
iterations	772	1169	3042	3203
LOQO	17.24 s	6.69 s	174.58 s	1028.35 s
$f$	5.07226e+00	5.06978e+00	5.06891e+00	5.06862e+00
$c$ violation	3.3e-08	7.3e-10	5.7e-10	2.4e-09
iterations	773	206	758	777
MINOS	1.22 s	5.52 s	14.75 s	73.9 s
$f$	5.07226e+00	5.06978e+00	5.06891e+00	5.06862e+00
$c$ violation	6.1e-08	4.0e-07	2.5e-06	3.3e-06
iterations	17	22	30	60
SNOPT	5.72 s	32.76 s	52.8 s	$\ddagger$
$f$	5.07226e+00	5.06978e+00	5.06891e+00	$\ddagger$
$c$ violation	5.9e-09	5.2e-09	1.6e-06	$\ddagger$
iterations	165	246	85	$\ddagger$
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

Figure 3.1: Hanging chain of length $L = 4$