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

2018-11-04

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Question

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!

Answer

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