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

- **Equation**: \( x = x^\star(t) \) if there exists \( w \in \mathbb{R}^p \),

  \[
  t \nabla f_0(x) + \sum_{i=1}^{m} Df_i(x)^T \nabla \psi_i(-f_i(x)) + A^T w = 0
  \]

  - \( Df_i(x) \in \mathbb{R}^{k_i \times n} \) is the derivative matrix of \( f_i \).

- **Lagrangian Minimization**: Therefore, \( x^\star(t) \) minimizes the Lagrangian \( L(x, \lambda^\star(t), \nu^\star(t)) \), where

  \[
  \lambda_i^\star(t) = \frac{1}{t} \nabla \psi_i(-f_i(x^\star(t))), \quad \nu^\star(t) = \frac{w}{t}
  \]

- **Properties**: From properties of \( \psi_i \): \( \lambda_i^\star(t) \succeq \mathcal{K}_i^\star 0 \), with duality gap

  \[
  f_0(x^\star(t)) - g(\lambda^\star(t), \nu^\star(t)) = \frac{1}{t} \sum_{i=1}^{m} \theta_i
  \]

- **Source**: A. d'Aspremont. Convex Optimization M2. Slide 29/36.
```
