```markdown
### Dual Points on Central Path

#### Mathematical Conditions
- **Condition for \( x = x^\star(t) \)**:
  \[
  t \nabla f_0(x) + \sum_{i=1}^{m} \frac{1}{f_i(x)} \nabla f_i(x) + A^T w = 0, \quad Ax = b
  \]

#### Lagrangian Minimization
- **\( x^\star(t) \) minimizes the Lagrangian**:
  \[
  L(x, \lambda^\star(t), \nu^\star(t)) = f_0(x) + \sum_{i=1}^{m} \lambda_i^\star(t) f_i(x) + \nu^\star(t)^T (Ax - b)
  \]

#### Definitions
- **\(\lambda_i^\star(t)\)**:
  \[
  \lambda_i^\star(t) = 1/(-t f_i(x^\star(t)))
  \]
- **\(\nu^\star(t)\)**:
  \[
  \nu^\star(t) = w/t
  \]

#### Intuitive Confirmation
- **As \( t \to \infty \)**:
  \[
  f_0(x^\star(t)) \to p^\star
  \]
- **Inequality**:
  \[
  p^\star \geq g(\lambda^\star(t), \nu^\star(t)) = L(x^\star(t), \lambda^\star(t), \nu^\star(t)) = f_0(x^\star(t)) - m/t
  \]

---

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