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

The slide describes how a Second-Order Cone Program (SOCP) can be expressed as a conic form problem. 

**Problem Formulation:**

- **Objective:** Minimize \( c^T x \)
- **Constraints:** 
  \[
  -(c_i^T x + d_i, A_i x + b_i) \preceq_{K_i} 0, \quad i = 1, \ldots, m.
  \]

**Lagrangian:**

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

**Condition:**

For \((v_i, u_i) \succeq_{K_i^*} 0\) (which is also \( v_i \geq \|u_i\| \)).

**Footer:** A. d’Aspremont. Convex Optimization M2. Slide 38/49.
```
