```markdown
### Properties (without proof)

For \( y \succ_K 0 \),

- \( \nabla \psi(y) \succeq_{K^*} 0 \),
- \( y^T \nabla \psi(y) = \theta \).

#### Nonnegative Orthant \( \mathbb{R}^n_+ \):

- \( \psi(y) = \sum_{i=1}^n \log y_i \)
- \( \nabla \psi(y) = (1/y_1, \ldots, 1/y_n) \)
- \( y^T \nabla \psi(y) = n \)

#### Positive Semidefinite Cone \( \mathcal{S}^n_+ \):

- \( \psi(Y) = \log \det Y \)
- \( \nabla \psi(Y) = Y^{-1} \)
- \( \text{Tr}(Y \nabla \psi(Y)) = n \)

#### Second-Order Cone \( K = \{ y \in \mathbb{R}^{n+1} \mid (y_1^2 + \cdots + y_n^2)^{1/2} \leq y_{n+1} \} \):

- \( \psi(y) = \frac{2}{y_{n+1}^2 - y_1^2 - \cdots - y_n^2} \)
- Gradient vector:
  \[
  \begin{bmatrix}
  -y_1 \\
  \vdots \\
  -y_n \\
  y_{n+1}
  \end{bmatrix}
  \]
- \( y^T \nabla \psi(y) = 2 \)

---

#### Reference

A. d'Aspremont. Convex Optimization M2. Slide 27/36.
```
