```markdown
### Inequality Constrained Minimization

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

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

**Conditions:**
- \( f_i \) are convex and twice continuously differentiable.
- \( A \in \mathbb{R}^{p \times n} \) with \(\text{Rank } A = p\).
- Assume \( p^\star \) is finite and attained.
- Problem is strictly feasible: there exists \( \tilde{x} \) such that:
  - \( \tilde{x} \in \text{dom } f_0 \)
  - \( f_i(\tilde{x}) < 0, \quad i = 1, \ldots, m \)
  - \( A\tilde{x} = b \)

**Conclusion:**
- Strong duality holds and the dual optimum is attained.

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