\begin{equation}
K^1 := \{ x \mid 2 x_1 x_2 \geq ||x_{3:n}||^2, x_1, x_2 \geq 0 \}
\end{equation}
for the rotated quadratic cone, Occasionally users of MOSEK ask why there is the 2 in front of the product $x_1 x_2$. Why not use the definition
\begin{equation}
K^2 := \{ x \mid x_1 x_2 \geq ||x_{3:n}||^2, x_1, x_2 \geq 0 \} ?
\end{equation}
The reason is that the dual cone plays an important role and the dual cone of $K^1$ is $K^1$ i.e. it is self-dual. That is pretty! Now the dual cone of $K^2$ is
\begin{equation}
\{ x \mid 4 s_1 s_2 \geq ||s_{3:n}||^2, s_1, s_2 \geq 0 \}.
\end{equation}
Hence, $K^2$ is not self-dual! That is somewhat ugly and inconvenient.
To summarize the definition $K^1$ for the rotated quadratic cone is preferred because the alternative definition $K^2$ is not self-dual
A couple of historical notes are:
A couple of historical notes are:
- In the classical paper by Lobo et. al. the cone $K^2$ is called a hyperbolic constraint in Section 2.3.
- MOSEK is highly inspired by the important work of the late Jos Sturm on the code SeDuMi . Now SeDuMi is short for self-dual minimization and for that reason Sturm employs the definition $K^1$ too.
