```markdown
### Quadratic Program

#### Primal Problem
- **Assumption**: \( P \in S_{++}^n \)
- **Objective**: Minimize \( x^T P x \)
- **Constraint**: Subject to \( Ax \preceq b \)

#### Dual Function
- \( g(\lambda) = \inf_x \left( x^T P x + \lambda^T (Ax - b) \right) = -\frac{1}{4} \lambda^T A P^{-1} A^T \lambda - b^T \lambda \)

#### Dual Problem
- **Objective**: Maximize \(-\frac{1}{4} \lambda^T A P^{-1} A^T \lambda - b^T \lambda \)
- **Constraint**: Subject to \( \lambda \succeq 0 \)

#### Additional Notes
- From Slater's condition: \( p^* = d^* \) if \( A\tilde{x} \prec b \) for some \( \tilde{x} \)
- In fact, \( p^* = d^* \) always

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