We define the \((m,n)\)-generalized power cone with parameter \(\alpha\) as
\begin{eqnarray} \label{eq:GPC-1}
K^{(m,n)}_\alpha := \left\{(s,u) \in \mathbb R_+^m \oplus \mathbb R^n: \prod_{i=1}^m s_i^{\alpha_i} \geq \|u\|_2 \right\},
\end{eqnarray}
where \(\alpha\) belongs to the simplex \(\{\alpha \in \mathbb R^m_+: \sum_{i=1}^m \alpha_i = 1\}\). Note that the rotated second
order cone is a special case where \(m=2\) and \(\alpha_1 = \alpha_2 = \frac 12\). Different s.c. barriers for this cone
or special cases of it have been considered Chares' thesis. The s.c. barrier DDS uses is:
\begin{eqnarray} \label{eq:GPC-2}
\Phi(s,u) = -\ln\left(\prod_{i=1}^m s_i^{2\alpha_i} - u^\top u \right) - \sum_{i=1}^{m} (1-\alpha_i) \ln(s_i).
\end{eqnarray}
To add generalized power cone constraints to DDS, we use the abbreviation 'GPC'.
To see the details of how to add a GPC constraint, please see the users' guide.