```markdown
### Slater's Constraint Qualification

**Strong duality holds for a convex problem:**

- **Objective:** Minimize \( f_0(x) \)
- **Constraints:** 
  - \( f_i(x) \leq 0, \quad i = 1, \ldots, m \)
  - \( Ax = b \)

**Condition for Strong Duality:**

- The problem is **strictly feasible**, i.e., there exists \( x \in \text{int} \, \mathcal{D} \) such that:
  - \( f_i(x) < 0, \quad i = 1, \ldots, m \)
  - \( Ax = b \)

**Additional Notes:**

- Guarantees dual optimum is attained (if \( p^* > -\infty \)).
- Can be refined by replacing \(\text{int} \, \mathcal{D}\) with \(\text{reint} \, \mathcal{D}\) (interior relative to affine hull); linear inequalities need not be strict.
- Other types of constraint qualifications exist.

*Source: A. d’Aspremont, Convex Optimization M2.*
```
