```markdown
### Least-norm solution of linear equations

**Objective:**
- Minimize \( x^T x \) 
- Subject to \( Ax = b \)

**Dual Function:**
- **Lagrangian:** \( L(x, \nu) = x^T x + \nu^T (Ax - b) \)
- To minimize \( L \) over \( x \), set gradient equal to zero:
  \[
  \nabla_x L(x, \nu) = 2x + A^T \nu = 0 \implies x = -(1/2)A^T \nu
  \]
- Plug in \( x \) in \( L \) to obtain \( g \):
  \[
  g(\nu) = L((-1/2)A^T \nu, \nu) = -\frac{1}{4} \nu^T A A^T \nu - b^T \nu
  \]
  - \( g(\nu) \) is a concave function of \( \nu \)

**Lower Bound Property:**
- \( p^\star \geq -(1/4)\nu^T A A^T \nu - b^T \nu \) for all \( \nu \)

*Note: Content is from A. d’Aspremont's "Convex Optimization M2."*
```
