cplexnonlinear-functionsopl

Cross product multiplication non-linearity error


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?


Solution

  • /*
     
     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;
    }