I have non-linear constraints like
$ x_1x_2\leq x_3 $
where $ x_1,x_2,x_3\geq 0 $. The objective is linear, and all other constraints are linear, too. I know that I can transform the product as
$ y_1=\frac{1}{2}(x_1+x_2) $
$ y_2=\frac{1}{2}(x_1−x_2) $
$ y_1^2−y_2^2\leq x_3 $
But when it comes to the last constraint, it is not convex (matrix is not PSD), and thus not suitable for commercial solvers like CPLEX and Gurobi (as far as I know). Moreover, they are not conic quadratic representable. At least I don't know how to reformulate them, or to find a suitable approximation ("good for practical purposes"). Now, my question.
Is there an efficient (to some degree) approach to deal with these kind of constraints?
I am asking this because they look quite simple and the constraint expressions are the difference of convex functions $ y_1^2−(y_2^2+x_3) $ (the sum of two convex functions on different domains is convex).
Maybe some relaxation technique has been proven to be useful? Convexification? In other words, how to circumvent this?