```markdown
## A Nonconvex Problem with Strong Duality

### Problem Formulation
- **Objective**: Minimize \( x^T A x + 2b^T x \)
- **Constraint**: \( x^T x \leq 1 \)
- **Nonconvex**: If \( A \npreceq 0 \)

### Dual Function
- **Expression**: \( g(\lambda) = \inf_x (x^T (A + \lambda I)x + 2b^T x - \lambda) \)
- **Properties**:
  - Unbounded below if \( A + \lambda I \npreceq 0 \) or if \( A + \lambda I \succeq 0 \) and \( b \notin \mathcal{R}(A + \lambda I) \)
  - Minimized by \( x = -(A + \lambda I)^\dagger b \) otherwise: \( g(\lambda) = -b^T (A + \lambda I)^\dagger b - \lambda \)

### Dual Problem and Equivalent SDP
- **Maximize**: \( -b^T (A + \lambda I)^\dagger b - \lambda \)
- **Subject to**: 
  - \( A + \lambda I \succeq 0 \)
  - \( b \in \mathcal{R}(A + \lambda I) \)

- **Alternative Formulation**:
  - **Maximize**: \( -t - \lambda \)
  - **Subject to**:
    \[
    \begin{bmatrix}
    A + \lambda I & b \\
    b^T & t
    \end{bmatrix}
    \succeq 0
    \]

### Notes
- Strong duality holds even though the primal problem is not convex (not easy to show).

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