```markdown
### Duality: SOCP

The Lagrange dual function is defined as:

\[ 
g(\lambda, \nu, \mu) = 
\begin{cases} 
-\sum_{i=1}^{n}(b_i^T \nu_i + d_i \mu_i \lambda_i) & \text{if } \sum_{i=1}^{m}(A_i^T \nu_i + \mu_i c_i) = c, \\
& \|\nu_i\|_2 \leq \lambda_i, \, \mu = \lambda \\
-\infty & \text{otherwise}
\end{cases}
\]

which leads to the dual problem:

\[
\begin{align*}
\text{maximize} & \quad -\sum_{i=1}^{n}(b_i^T \nu_i + d_i \lambda_i) \\
\text{subject to} & \quad \sum_{i=1}^{m}(A_i^T \nu_i + \lambda_i c_i) = c \\
& \quad \|\nu_i\|_2 \leq \lambda_i, \quad i = 1, \ldots, m.
\end{align*}
\]

which is again an SOCP.

*Source: A. d'Aspremont, Convex Optimization M2.*
```
