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) $$ y_2=\frac{1}{2}(x_1x_2) $$ y_1^2y_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?
Approach to handle a quadratic constraint xy <= z
optimization;quadratic programming;nonconvex