How to linearize the product of two float variables - c++

Question

I would like linearizate the product of two float variables. Suppose that a model has the product x * y, where x and y are float, with 0 <= x <= 1 and 0 <= y <= 1. How linearizate this product?

Answer 1

I gave an example in OPL/CPLEX here
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);
}