# How to deal with an $xy\le 1$ constraint?

I have to solve the following optimization problem: \begin{align*} \min_{x,y} &\{-x-y\} \\ \text{such that} \\ y &\ge 3 \\ y &\le 30 \\ x &\ge 0 \\ xy &\le 1 \\ \end{align*} I want to use a second order cone programming (SOCP) solver because the rest of my problem (not shown here) can be formulated as a second order cone program. However, my problem is the $xy\le 1$ hyperbolic constraint that has the inequality "the wrong way" to be written as a second order cone. It would be great if you could show me for this toy problem how one might deal with the $xy\le 1$ constraint. Thanks a lot!

• $0\le x\le\frac{1}{y}$ and $y\ge 3$ imply $0\le x\le\frac{1}{3}$. So we clearly minimise by taking $y=30,x=\frac{1}{30}$. – almagest Apr 26 '16 at 9:06
• Yeah, but my question is how to reformulate it for a computer convex solver to solve, not by hand. This is an example problem of a much more complicated problem that I'm dealing with. – space_voyager Apr 26 '16 at 9:09
• I would say that it is not possible to formulate your problem as a convex optimization problem. – gerw Apr 26 '16 at 10:35
• @gerw is correct. It's not convex. – Michael Grant Apr 27 '16 at 16:00
• @MichaelGrant I've not gone over the theory of convex optimization. Is it generally true that if a constraint is non-convex, then it is impossible to make it convex? Perhaps there are techniques to convexify this constraint? – space_voyager Apr 28 '16 at 21:33

The constraint $$xy\le1$$ is non-convex, so the problem cannot be solved as stated using SOCP techniques.