Reference
Category : GAMS NOA library
Mainfile : chain.gms
$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
Tested with nh=3000, 4000, 5000; May 26, 2005
References:
Neculai Andrei, "Models, Test Problems and Applications for
Mathematical Programming". Technical Press, Bucharest, 2003.
Application A7, page 350.
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.
$offText
$if set n $set nh %n%
$if not set nh $set nh 400
set nh /i0 * i%nh%/;
alias(nh,i);
scalars 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;
variables
x(i) height of the chain
u(i) derivative of x
energy potential energy ;
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);
*Equations
equations 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;
model chain /all/;
$ifThenI x%mode%==xbook
$onEcho >minos.opt
superbasics limit = 5000
$offEcho
chain.workspace=120;
$endIf
solve chain using nlp minimizing energy;
$ifThenI x%mode%==xbook
file res /chain.dat/;
put res
loop(i, put x.l(i):10:5, put/)
$endIf
*------------ January 26, 2011
* For nh=1000 I obtained the following results:
* CONOPT: 20 iterations, 2.654 seconds, vfo=5.0685102
* KNITRO: 8 iterations, 0.380 seconds, vfo=5.06850999
* MINOS : 202 iterations, 73.145 seconds, vfo=5.068510
* End Hanging Chain