Chain : Hanging Chain

Reference

  • Neculai Andrei, Nonlinear Optimization Applications Using the GAMS Technology,Springer Optimization and Its Applications, Model Chain (5.28) in chapter Applications of Mechanical Engineering , 2013

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

$onecho >bench.opt
 solvers conopt knitro minos.1
$offecho

chain.optfile=1;
chain.workspace=120;

option nlp=bench
$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