```markdown
## Solving KKT Systems

### Matrix Equation
\[
\begin{bmatrix}
H & A^T \\
A & 0
\end{bmatrix}
\begin{bmatrix}
v \\
w
\end{bmatrix}
= -
\begin{bmatrix}
g \\
h
\end{bmatrix}
\]

### Solution Methods

- **LDLᵀ Factorization**

- **Elimination (if \( H \) nonsingular)**
  \[
  AH^{-1}A^T w = h - AH^{-1}g, \quad Hv = -(g + A^T w)
  \]

- **Elimination with Singular \( H \):** Write as
  \[
  \begin{bmatrix}
  H + A^T QA & A^T \\
  A & 0
  \end{bmatrix}
  \begin{bmatrix}
  v \\
  w
  \end{bmatrix}
  = -
  \begin{bmatrix}
  g + A^T Qh \\
  h
  \end{bmatrix}
  \]
  with \( Q \succeq 0 \) for which \( H + A^T QA \succ 0 \), and apply elimination.

---

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