```markdown
**Title:** Duality: SOCP

**Content:**

- **Lagrangian Function**: 
  \[
  L(x, u_i, v_i) = \left( c - \sum_i (A_i^T u_i + c_i v_i) \right)^T x - \sum_i (b_i^T u_i + d_i v_i)
  \]

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

- **Conic Dual**:
  \[
  \text{maximize } -\sum_{i=1}^n (b_i^T u_i + d_i v_i)
  \]

  \[
  \text{subject to } \sum_{i=1}^m (A_i^T u_i + v_i c_i) = c
  \]

  \[
  (v_i, u_i) \succeq_{K_i^*} 0, \quad i = 1, \ldots, m.
  \]

**Footer:** A. d’Aspremont. Convex Optimization M2. Page 39/49
```
