```markdown
## Equality Constrained Minimization

### Problem Formulation
- **Objective**: Minimize \( f(x) \)
- **Subject to**: \( Ax = b \)

### Assumptions
- \( f \) is convex and twice continuously differentiable.
- \( A \in \mathbb{R}^{p \times n} \) with \( \text{Rank } A = p \).
- \( p^\star \) is finite and attained.

### Optimality Conditions
- \( x^\star \) is optimal if and only if there exists a \( \nu^\star \) such that:
  - \( \nabla f(x^\star) + A^T \nu^\star = 0 \)
  - \( Ax^\star = b \)

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