chain.gms : Hanging Chain COPS 2.0 #3

**Description**

Find the chain (of uniform density) of length L suspended between two points with minimal potential energy. This model is from the COPS benchmarking suite. See http://www-unix.mcs.anl.gov/~more/cops/. The number of intervals for the discretization can be specified using the command line parameter --nh. COPS performance tests have been reported for nh = 50, 100, 200, 400

**References**

- Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS. Tech. rep., Mathematics and Computer Science Division, 2000.
- Cesari, L, Optimization - Theory and Applications. Springer Verlag, Applications of Mathematics, 1983.

**Large Model of Type :** NLP

**Category :** GAMS Model library

**Main file :** chain.gms

```
$title Hanging Chain COPS 2.0 #3 (CHAIN,SEQ=231)
$onText
Find the chain (of uniform density) of length L suspended between two
points with minimal potential energy.
This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.
The number of intervals for the discretization can be specified using
the command line parameter --nh. COPS performance tests have been
reported for nh = 50, 100, 200, 400
Dolan, E D, and More, J J, Benchmarking Optimization
Software with COPS. Tech. rep., Mathematics and Computer
Science Division, 2000.
Cesari, L, Optimization - Theory and Applications. Springer
Verlag, 1983.
Keywords: nonlinear programming, engineering, hanging chain problem, catenary
$offText
$if not set nh $set nh 50
Set nh / i0*i%nh% /;
Alias (nh,i);
Scalar
L 'length of the suspended chain' / 4 /
a 'height of the chain at t=0 (left)' / 1 /
b 'height of the chain at t=1 (left)' / 3 /
tf 'ODEs defined in [0 tf]' / 1 /
h 'uniform interval length'
n 'number of subintervals'
tmin;
if(b > a, tmin = 0.25; else tmin = 0.75;);
n = card(nh) - 1;
h = tf/n;
Variable
x(i) 'height of the chain'
u(i) 'derivative of x'
energy 'potential energy';
Equation
obj
x_eqn(i)
length_eqn;
obj..
energy =e= 0.5*h*sum(nh(i+1), x(i)*sqrt(1 + sqr(u(i))) + x(i+1)*sqrt(1 + sqr(u(i+1))));
x_eqn(i+1)..
x(i+1) =e= x(i) + 0.5*h*(u(i) + u(i+1));
length_eqn..
0.5*h*sum(nh(i+1), sqrt(1+sqr(u(i))) + sqrt(1+sqr(u(i+1)))) =e= L;
x.fx('i0') = a;
x.fx('i%nh%') = b;
x.l(i) = 4*abs(b-a)*((ord(i) - 1)/n)*(0.5*((ord(i) - 1)/n) - tmin) + a;
u.l(i) = 4*abs(b-a)*(((ord(i) - 1)/n) - tmin);
Model chain / all /;
$if set workSpace chain.workSpace = %workSpace%
solve chain using nlp minimizing energy;
```