```markdown
### Example: Semidefinite Programming (with \( F_i \in S^p \))

**Problem Formulation:**
- **Objective:** Minimize \( c^T x \)
- **Subject to:** 
  \[
  F(x) = \sum_{i=1}^{n} x_i F_i + G \preceq 0
  \]

**Key Concepts:**

- **Logarithmic Barrier:**
  \[
  \phi(x) = \log \det(-F(x)^{-1})
  \]

- **Central Path:**
  - \( x^*(t) \) minimizes \( t c^T x - \log \det(-F(x)) \)
  - Satisfies:
    \[
    t c_i - \text{Tr}(F_i F(x^*(t))^{-1}) = 0, \quad i = 1, \ldots, n
    \]

- **Dual Point on Central Path:**
  \[
  Z^*(t) = -(1/t) F(x^*(t))^{-1}
  \]
  - Feasible for:
    \[
    \begin{align*}
    \text{Maximize:} & \quad \text{Tr}(GZ) \\
    \text{Subject to:} & \quad \text{Tr}(F_i Z) + c_i = 0, \quad i = 1, \ldots, n \\
    & \quad Z \succeq 0
    \end{align*}
    \]

- **Duality Gap on Central Path:**
  \[
  c^T x^*(t) - \text{Tr}(GZ^*(t)) = p/t
  \]

---

*Source: A. d’Aspremont, Convex Optimization M2, Slide 30/36.*
```
