```markdown
## Weak and Strong Duality

### Weak Duality: \(d^* \leq p^*\)
- **Always holds** (for convex and nonconvex problems).
- Can be used to find nontrivial lower bounds for difficult problems.
  - Example: Solving the SDP
    \[
    \begin{align*}
    \text{maximize} & \quad -1^T \nu \\
    \text{subject to} & \quad W + \text{diag}(\nu) \succeq 0
    \end{align*}
    \]
  - Provides a lower bound for the two-way partitioning problem on page 10.

### Strong Duality: \(d^* = p^*\)
- **Does not hold in general.**
- **Usually holds** for convex problems.
- Conditions that guarantee strong duality in convex problems are called **constraint qualifications**.

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