I have been asked to add to an existing linear programming model several constraints dealing with ratios among continuous decision variables. An example ratio constraint would be like:


$x\_1\*x\_2 - x\_3\*x\_4 = 0$


They told me to preserve the lineal nature of the problem, so I am trying to find an equivalent linear formulation to the previous type of constraints. 


I have tried with Lagrange multipliers, but the system obtained from setting the derivative to zero is not linear. I also tried to do the Relaxation-Linearization Technique on it, but it did not seem to work.


I am stuck at a point where I do not really know if I am misunderstanding these techniques, or facing a problem without an evident solution. 


A simplification (real one is bigger) of the problem would be like this:


$min = a1 + a2 + b1 + b2$  

$s.t.$  

$a1 + b1 - 200 < 0$   

$a2 + b2 - 200 < 0$   

$b1 - 80 < 0$   

$b1 - 60 > 0$   

$b2 - 30 < 0$  

$b2 - 10 > 0$  

$a1 + a2 - 100 = 0$  

$b1 + b2 - 80 = 0$  

$a1 - 80 = 0$  

$a2 - 20 = 0$  

$a1\*b2 - a2\*b1 = 0$  




Nonlinear constraint and product of variables
$x\_1 x\_2 - x\_3 x\_4 = 0$ is inherently nonlinear, and maybe more importantly non-convex: e.g. the midpoint of two feasible solutions may not be (in fact, hardly ever is) a feasible solution. So in general there is no such thing as an equivalent formulation that is linear. However, in special cases something might be possible. In your example (ignoring $a\_1 - 80 = 0$ and $a\_2 - 20 = 0$ which make the problem trivial), solve the constraints $a\_1 + a\_2 - 100 = 0$ and 
$b\_1 + b\_2 - 80 = 0$ for $a\_2$ and $b\_2$ and you get $a\_1 b\_2 - a\_2 b\_1 = 80 a\_1 - 100 b\_1$ so this becomes a linear constraint. 


