CPLEX vs Constraint Programming Solution for a linearly constrained MIP with a nonlinear objective(division)

Question

I am trying to solve an MIP with a non-linear objective such as sum(a(i)*x(i))/sum(b(i)*x(i)) where a(i) and b(i) are parameters. Since CPLEX cannot extract this expression, I tried using CP.

However, I have constraint stating that the decision variable x(i) should be a multiplier of 2.5, therefore a float. Since CP cannot handle floats, I set x to be an integer, and implied to be multiplier of 25. In every other constraint and expression I divided x by 10, so that the calculations remained the same.

First, I solved the model with the multiplier 5, I got a solution with a very small gap, which is fine. When I changed the multiplier to 25, the model cannot terminated within 2 hours (stucked with a gap of 90%). I believe this happened due to scaling, but still cannot figure it out, since I am not familiar with the CP algorithm.

I am still working to linearize the oblective but, any suggestion on both CPLEX and CP engine solutions will be highly appreciated. Thanks in advance.

Answer 1

have you tried to turn equalities into inequalities to allow some flexibility ?

For instance

using CP;

int scale=1000;
dvar int scalex in 0..2000;
dexpr float x=scalex/scale;

subject to
 {
   x*x==2;
 }

gives no solution whereas

using CP;

float epsilon=0.001;

int scale=1000;
dvar int scalex in 0..2000;
dexpr float x=scalex/scale;

subject to
 {
   abs(x*x-2)<=epsilon;
 }

gives

x=1.414

and

using CP;

{int} hm=asSet(1..4);
float epsilon=0.001;

dvar int scalex[1..card(hm)] in 0..20000;
int scale=10000;

dexpr float A[ i in 1..card(hm)]=scalex[i]/scale;
subject to
{
  forall(i in 1..card(hm))abs(A[i]*A[i]-2)<=epsilon;
}

works fine too