```markdown
### Semidefinite Program

#### Primal SDP \((F_i, G \in \mathcal{S}^k)\)

- **Objective**: Minimize \(c^T x\)
- **Subject to**: \(x_1 F_1 + \cdots + x_n F_n \succeq G\)

#### Key Points

- **Lagrange Multiplier**: Matrix \(Z \in \mathcal{S}^k\)
- **Lagrangian**: 
  \[
  L(x, Z) = c^T x + \text{Tr}\left(Z(x_1 F_1 + \cdots + x_n F_n - G)\right)
  \]
- **Dual Function**:
  \[
  g(Z) = \inf_x L(x, Z) = 
  \begin{cases} 
  -\text{Tr}(GZ) & \text{if } \text{Tr}(F_i Z) + c_i = 0, \, i = 1, \ldots, n \\
  -\infty & \text{otherwise}
  \end{cases}
  \]

#### Dual SDP

- **Objective**: Maximize \(-\text{Tr}(GZ)\)
- **Subject to**: 
  - \(Z \succeq 0\)
  - \(\text{Tr}(F_i Z) + c_i = 0, \, i = 1, \ldots, n\)

#### Optimality Condition

- \(p^* = d^*\) if primal SDP is strictly feasible \((\exists x \text{ with } x_1 F_1 + \cdots + x_n F_n \prec G)\)

---

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