Can $x_1 x_2 \ge x_3 x_4$ be represented as a second-order conic constraint?

Question

I'll like to know if the constraint $x_1 x_2 \ge x_3 x_4$ can be represented as a SOCP constraint.

Let

$$A = \begin{bmatrix} x_1 & x_3 \\ x_4 & x_2 \end{bmatrix}$$

The constraint is equivalent to $\det(A) \geq 0$, so it is an SDP constraint.

Answer 1

Unfortunately, the set described by your constraint is not convex, which prevents both SDP and SOCP formulations. To see this, you can for example intersect it with hyperplanes $x_1=1$ and $x_2=2$: the resulting set $x_3 x_4 \le 1$ is clearly not convex, which implies that the original set was not either.

The $\det(A) \geq 0$ argument invoked above does not work because matrix $A$ is not symmetric.