```markdown
### Semidefinite Program (SDP)

**Objective:**
- Minimize \( c^T x \)

**Subject to:**
- \( x_1 F_1 + x_2 F_2 + \cdots + x_n F_n + G \succeq 0 \)
- \( Ax = b \)

**With:**
- \( F_i, G \in \mathbb{S}^k \)

**Key Points:**
- The inequality constraint is called a linear matrix inequality (LMI).
- Includes problems with multiple LMI constraints:
  - Example: 
    - \( x_1 \hat{F}_1 + \cdots + x_n \hat{F}_n + \hat{G} \succeq 0 \)
    - \( x_1 \tilde{F}_1 + \cdots + x_n \tilde{F}_n + \tilde{G} \succeq 0 \)

**Equivalence:**
- These are equivalent to a single LMI:
  \[
  x_1 \begin{bmatrix} \hat{F}_1 & 0 \\ 0 & \tilde{F}_1 \end{bmatrix} + x_2 \begin{bmatrix} \hat{F}_2 & 0 \\ 0 & \tilde{F}_2 \end{bmatrix} + \cdots + x_n \begin{bmatrix} \hat{F}_n & 0 \\ 0 & \tilde{F}_n \end{bmatrix} + \begin{bmatrix} \hat{G} & 0 \\ 0 & \tilde{G} \end{bmatrix} \succeq 0
  \]

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