```markdown
### Optimization Problem in Standard Form

**Objective:**
- Minimize \( f_0(x) \)

**Subject to:**
- \( f_i(x) \leq 0, \quad i = 1, \ldots, m \)
- \( h_i(x) = 0, \quad i = 1, \ldots, p \)

**Definitions:**
- \( x \in \mathbb{R}^n \) is the optimization variable.
- \( f_0 : \mathbb{R}^n \rightarrow \mathbb{R} \) is the objective or cost function.
- \( f_i : \mathbb{R}^n \rightarrow \mathbb{R}, \quad i = 1, \ldots, m \), are the inequality constraint functions.
- \( h_i : \mathbb{R}^n \rightarrow \mathbb{R} \) are the equality constraint functions.

**Optimal Value:**
- \( p^\star = \inf \{ f_0(x) \mid f_i(x) \leq 0, \quad i = 1, \ldots, m, \quad h_i(x) = 0, \quad i = 1, \ldots, p \} \)

**Special Cases:**
- \( p^\star = \infty \) if the problem is infeasible (no \( x \) satisfies the constraints).
- \( p^\star = -\infty \) if the problem is unbounded below.

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