```markdown
### Duality: SOCP

The slide discusses deriving the dual of a Second-Order Cone Program (SOCP) using two methods:

1. **Lagrange Dual Method:**
   - Introduce new variables \( y_i \in \mathbb{R}^{n_i} \) and \( t_i \in \mathbb{R} \).
   - Use the equalities \( y_i = A_ix + b_i \) and \( t_i = c_i^T x + d_i \).
   - Derive the Lagrange dual.

2. **Conic Dual Method:**
   - Start from the conic formulation of the SOCP.
   - Use the fact that the second-order cone is self-dual:
     \[
     t \geq \|x\| \iff tv + x^T y \geq 0, \text{ for all } v, y \text{ such that } v \geq \|y\|
     \]
   - The condition \( x^T y \leq tv \) is a simple Cauchy-Schwarz inequality.

**Author:** A. d'Aspremont. Convex Optimization M2.  
**Slide:** 34/49
```
