Cross product multiplication non-linearity error

Question

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?

Answer 1

/*
 
 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;
}