Lotka Volterra fishing problem (AMPL)

AMPL

The model in AMPL code for a fixed control discretization grid with a collocation method. We need a model file lotka_ampl.mod,

# ----------------------------------------------------------------
# Lotka Volterra fishing problem with collocation (explicit Euler)
# (c) Sebastian Sager
# ----------------------------------------------------------------

param T    > 0;
param nt   > 0;
param nu   > 0;
param ntperu > 0;
param c1   > 0;
param c2   > 0;
param ref1 > 0;
param ref2 > 0;
param dt := T / (nt-1);
set I:= 0..nt;
set U:= 0..nu-1;

param uidx {I};

var x {I, 1..2} >= 0;
var w {U} >= 0, <= 1 binary;

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

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

a data file lotka_ampl.dat,

# ------------------------------------
# Data: Lotka Volterra fishing problem
# ------------------------------------

# Algorithmic parameters
param ntperu := 1;
param nu := 100;
param nt := 100;

# 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;

param mysum;

# Set indices of controls corresponding to time points
for {i in 0..nu-1} {
	for {j in 0..ntperu-1} {
		let uidx[i*ntperu+j] := i; 
	}
}
let uidx[nt] := nu-1;

and a running script lotka_ampl.run,

# ----------------------------------------------------
# Solve Lotka Volterra fishing problem via collocation
# ----------------------------------------------------

model ampl_lotka.mod;
data ampl_lotka.dat;

option solver bonmin;

solve;

Preliminary Results

Default values for all the options are used in all the solvers under NEOS Server environment using AMPL.

The preliminary results are as follows:

• MINLP : Infeasible problem
• FilMINT : Error in MINTO
• Bonmin : (options = B-BB, B-OA, B-QG and B-Hyb) Infeasible problem
• KNITRO : problem solved with objective value 1.5847 when using Branch and Bound; the solution can be obtained around 15 seconds (CPU time) by some branching strategies without parallel features

The preliminary results are tested by Henry Kar Ming Chan