I have an Error with nonlinear constraint on cplex. the code is as follows
`forall(t in time, z in kkk )
X[z][t]* R[t] == sum (i in source)
(sq[i][z]* Z[i][t]);
Where X[z][t], R[t], and Z[i][t] are continuous variables.
Is there a possible way that Cplex can deal with this non linearity directly or it should be linearized?
/*
Here, I want to help those who want to stay MIP and need to deal with
dvar float x;
dvar float y;
subject to
{
x*y<=10;
}
What you can do is remember that
4*x*y=(x+y)*(x+y)-(x-y)(x-y)
So if you do a variable change X=x+y and Y=x-y
x*y
becomes
1/4*(X*X-Y*Y)
which is separable.
And then you are able to interpolate the function x*x by piecewise linear function:
// y=x*x interpolation
*/
int sampleSize=10000;
float s=0;
float e=100;
float x[i in 0..sampleSize]=s+(e-s)*i/sampleSize;
int nbSegments=20;
float x2[i in 0..nbSegments]=(s)+(e-s)*i/nbSegments;
float y2[i in 0..nbSegments]=x2[i]*x2[i];
float firstSlope=0;
float lastSlope=0;
tuple breakpoint // y=f(x)
{
key float x;
float y;
}
sorted { breakpoint } breakpoints={<x2[i],y2[i]> | i in 0..nbSegments};
float slopesBeforeBreakpoint[b in breakpoints]=
(b.x==first(breakpoints).x)
?firstSlope
:(b.y-prev(breakpoints,b).y)/(b.x-prev(breakpoints,b).x);
pwlFunction f=piecewise(b in breakpoints)
{ slopesBeforeBreakpoint[b]->b.x; lastSlope } (first(breakpoints).x, first(breakpoints).y);
assert forall(b in breakpoints) f(b.x)==b.y;
float maxError=max (i in 0..sampleSize) abs(x[i]*x[i]-f(x[i]));
float averageError=1/(sampleSize+1)*sum (i in 0..sampleSize) abs(x[i]*x[i]-f(x[i]));
execute
{
writeln("maxError = ",maxError);
writeln("averageError = ",averageError);
}
dvar float a in 0..10;
dvar float b in 0..10;
dvar float squareaplusb;
dvar float squareaminusb;
maximize a+b;
dvar float ab;
subject to
{
ab<=10;
ab==1/4*(squareaplusb-squareaminusb);
squareaplusb==f(a+b);
squareaminusb==f(a-b);
}
from
How to multiply two float decision variables
from https://www.linkedin.com/pulse/how-opl-alex-fleischer/
and if you need pairwise multiplication:
int sampleSize=10000;
float s=0;
float e=100;
float x[i in 0..sampleSize]=s+(e-s)*i/sampleSize;
int nbSegments=500;
float x2[i in 0..nbSegments]=(s)+(e-s)*i/nbSegments;
float y2[i in 0..nbSegments]=x2[i]*x2[i];
float firstSlope=0;
float lastSlope=0;
tuple breakpoint // y=f(x)
{
key float x;
float y;
}
sorted { breakpoint } breakpoints={<x2[i],y2[i]> | i in 0..nbSegments};
float slopesBeforeBreakpoint[b in breakpoints]=
(b.x==first(breakpoints).x)
?firstSlope
:(b.y-prev(breakpoints,b).y)/(b.x-prev(breakpoints,b).x);
pwlFunction f=piecewise(b in breakpoints)
{ slopesBeforeBreakpoint[b]->b.x; lastSlope } (first(breakpoints).x, first(breakpoints).y);
assert forall(b in breakpoints) f(b.x)==b.y;
float maxError=max (i in 0..sampleSize) abs(x[i]*x[i]-f(x[i]));
float averageError=1/(sampleSize+1)*sum (i in 0..sampleSize) abs(x[i]*x[i]-f(x[i]));
execute
{
writeln("maxError = ",maxError);
writeln("averageError = ",averageError);
}
dvar float a[1..2] in 0..10;
dvar float b[1..2] in 0..10;
dvar float squareaplusb[1..2];
dvar float squareaminusb[1..2];
maximize sum(i in 1..2)(a[i]+b[i]);
dvar float ab[1..2];
subject to
{
forall(i in 1..2)ab[i]<=10;
forall(i in 1..2)ab[i]==1/4*(squareaplusb[i]-squareaminusb[i]);
forall(i in 1..2) squareaplusb[i]==f(a[i]+b[i]);
forall(i in 1..2) squareaminusb[i]==f(a[i]-b[i]);
abs(ab[1]-ab[2])<=1;;
abs(a[1]-a[2])>=1;
}