```markdown
### Image Description

The image contains two sections discussing optimization concepts with mathematical formulations.

1. **Introducing Equality Constraints:**
   - Original problem:
     - Minimize \( f_0(A_0x + b_0) \)
     - Subject to \( f_i(A_ix + b_i) \leq 0, \quad i = 1, \ldots, m \)
   - Equivalent problem:
     - Minimize (over \( x, y_i \)) \( f_0(y_0) \)
     - Subject to:
       - \( f_i(y_i) \leq 0, \quad i = 1, \ldots, m \)
       - \( y_i = A_ix + b_i, \quad i = 0, 1, \ldots, m \)

2. **Introducing Slack Variables for Linear Inequalities:**
   - Original problem:
     - Minimize \( f_0(x) \)
     - Subject to \( a_i^T x \leq b_i, \quad i = 1, \ldots, m \)
   - Equivalent problem:
     - Minimize (over \( x, s \)) \( f_0(x) \)
     - Subject to:
       - \( a_i^T x + s_i = b_i, \quad i = 1, \ldots, m \)
       - \( s_i \geq 0, \quad i = 1, \ldots, m \)

The source is noted as "A. d'Aspremont. Convex Optimization M2." and the slide number is 45/67.
```