View source for Lotka Volterra fishing problem (AMPL)

This page contains a discretized version of the MIOCP [[Lotka Volterra fishing problem]] in [http://www.ampl.org AMPL] format. You should be aware of the comments regarding discretization made on the [[:Category:AMPL|AMPL overview]] page.

=== AMPL ===

The model in AMPL code for a fixed control discretization grid with a collocation method. We need a model file lotka_ampl.mod,
<source lang="AMPL">
# ----------------------------------------------------------------
# Lotka Volterra fishing problem with collocation (implicit Euler)
# (c) Sebastian Sager
# ----------------------------------------------------------------
var x {I, 1..nx} >= 0;
param c1 > 0; 
param c2 > 0; 
param ref1 > 0; 
param ref2 > 0;

minimize Deviation:
 0.5 * (dt[0]/ntperu) * ( (x[0,1]-ref1)^2 + (x[0,2]-ref2)^2 )
 + 0.5 * (dt[nu-1]/ntperu) * ((x[nt,1]-ref1)^2 + (x[nt,2]-ref2)^2)
 + sum {i in I diff {0,nt} } ( (dt[uidx[i]]/ntperu) * 
   ( (x[i,1] - ref1)^2 + (x[i,2] - ref2)^2 ) ) ;

subj to ODE_DISC_1 {i in I diff {0}}:
 x[i,1] = x[i-1,1] + (dt[uidx[i]]/ntperu) * 
  ( x[i,1] - x[i,1]*x[i,2] - x[i,1]*c1*w[uidx[i]] );
	 
subj to ODE_DISC_2 {i in I diff {0}}:
 x[i,2] = x[i-1,2] + (dt[uidx[i]]/ntperu) * 
  ( - x[i,2] + x[i,1]*x[i,2] - x[i,2]*c2*w[uidx[i]] );

subj to overall_stage_length:
 sum {i in U} dt[i] = T;
</source>
a data file lotka_ampl.dat,
<source lang="AMPL">
# ------------------------------------
# Data: Lotka Volterra fishing problem
# ------------------------------------

# Algorithmic parameters
param ntperu := 100;
param nu := 100;
param nt := 10000;
param nx := 2;
param fix_w := 0;
param fix_dt := 1;

# Problem parameters
param T := 12.0;
param c1 := 0.4;   
param c2 := 0.2;
param ref1 := 1.0; 
param ref2 := 1.0;

# Initial values differential states
let x[0,1] := 0.5; let x[0,2] := 0.7;
fix x[0,1]; fix x[0,2];

# Initial values control
let {i in U} w[i] := 0.0;
for {i in 0..(nu-1) / 2} { let w[i*2] := 1.0; }
let {i in U} dt[i] := T / nu;

</source>
and a running script lotka_ampl.run,
<source lang="AMPL">
# ------------------------------------
# Solve Lotka Volterra fishing problem
# ------------------------------------

model ../general/ampl_general.mod;
model ampl_lotka_paper.mod;
data ampl_lotka_paper.dat;
data ../general/ampl_general.dat;

option presolve_eps 1e-10;
option solver ...;

solve;

param myt;
let myt := 0;
for {i in I} { 
	if ( dt[uidx[i]] > 1e-6 ) then {
		print myt, w[uidx[i]], x[i,1], x[i,2] > resultInt.txt;
	}
	let myt := myt + (dt[uidx[i]]/ntperu);
}

display w;
</source>

== Results ==

The solution calculated by Bonmin (subversion revision number 1453, default settings, 3 GHz, Linux 2.6.28-13-generic, with ASL(20081205)) has an objective function value of <MATH>\Phi= 1.34434</MATH>, while the optimum of the relaxation is <MATH>\Phi=1.3423368</MATH>. Bonmin needs 35301 iterations and 2741 nodes (4899.97 seconds). Strong branching is done 81 times (1859 iterations), with 0 fathomed nodes and 0 fixed variables. The intervals on the equidistant grid on which <MATH>w(t) = 1</MATH> holds, counting from 0 to 99, are 20-32, 34, 36, 38, 40, 44, 53.

[[Category:AMPL]]