```markdown
### Introducing New Variables and Equality Constraints

#### Optimization Problem

- **Objective:** Minimize \( f_0(Ax + b) \)

#### Key Points

- **Dual Function:** The dual function is constant: 
  \[
  g = \inf_x L(x) = \inf_x f_0(Ax + b) = p^\star
  \]
- **Observation:** Strong duality exists, but the dual is quite useless.

#### Reformulated Problem and Its Dual

- **Reformulated Problem:**
  - Minimize \( f_0(y) \)
  - Subject to \( Ax + b - y = 0 \)

- **Dual Problem:**
  - Maximize \( b^T \nu - f_0^*(\nu) \)
  - Subject to \( A^T \nu = 0 \)

#### Dual Function

- **Expression:**
  \[
  g(\nu) = \inf_{x, y} (f_0(y) - \nu^T y + \nu^T Ax + b^T \nu)
  \]
  - Equals:
    \[
    \begin{cases} 
    -f_0^*(\nu) + b^T \nu & \text{if } A^T \nu = 0 \\
    -\infty & \text{otherwise}
    \end{cases}
    \]

*Slide reference: A. d’Aspremont. Convex Optimization M2. Slide 27/49*
```