```markdown
## The Dual Problem

### Lagrange Dual Problem

- **Objective:** Maximize \( g(\lambda, \nu) \)
- **Constraint:** \( \lambda \succeq 0 \)

### Key Points

- Finds the best lower bound on \( p^* \), obtained from the Lagrange dual function.
- It is a convex optimization problem; the optimal value is denoted \( d^* \).
- \( \lambda, \nu \) are dual feasible if \( \lambda \succeq 0 \) and \( (\lambda, \nu) \in \text{dom } g \).
- Often simplified by making the implicit constraint \( (\lambda, \nu) \in \text{dom } g \) explicit.

### Example

**Standard form LP and its dual (page 8):**

- **Primal:** 
  - Minimize \( c^T x \)
  - Subject to \( Ax = b \)
  - \( x \succeq 0 \)

- **Dual:**
  - Maximize \( -b^T \nu \)
  - Subject to \( A^T \nu + c \succeq 0 \)

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