# Hyperbolic constraint as second-order cone

I have a problem which simplifies to: \begin{align} \max w &\\ w&\le xy \\ x,y&\le10 \\ x,y&\ge0 \end{align} Recognizing that $$xy$$ form a hyperbolic constraint, I can solve by introducing a second-order cone: \begin{align} \max w & \\ \left\lVert\begin{bmatrix} 2w \\ x - y \end{bmatrix}\right\rVert &\le x+y \\ x,y&\le10 \\ x,y&\ge0 \end{align} And all is well with the world, except that $$w=\sqrt{xy}$$. What I would like in reality is $$w=xy$$.

Is there an SOCP way of achieving this?

If you cannot accept a general nonlinear form (and thus a general nonlinear solver), you might use a geometric programming form (and thus solve as a convex problem) if all your other constraints satisfy the requirements for a posynomial representation. In GP language, you are minimizing the posynomial $$w^{-1}$$ under the posynomial constraint $$wx^{-1}y^{-1}\leq 1$$.
• Why are you talking about linear Mccormick outer approximations now?The question was if it is possible to represent $w\leq xy$ using convex programming, i.e. writing a hypograph representation of $xy$, and the answer is no since the function $xy$ is not concave. An epigraph $w \geq xy$ is not possible either since $xy$ is not convex either. – Johan Löfberg Mar 27 at 13:47