```markdown
- **Unconstrained Problem**: \( x \) is optimal if and only if:
  \[
  x \in \text{dom } f_0, \quad \nabla f_0(x) = 0
  \]

- **Equality Constrained Problem**: Minimize \( f_0(x) \) subject to \( Ax = b \).
  \( x \) is optimal if and only if there exists a \( \nu \) such that:
  \[
  x \in \text{dom } f_0, \quad Ax = b, \quad \nabla f_0(x) + A^T \nu = 0
  \]

- **Minimization over Nonnegative Orthant**: Minimize \( f_0(x) \) subject to \( x \succeq 0 \).
  \( x \) is optimal if and only if:
  \[
  x \in \text{dom } f_0, \quad x \succeq 0, \quad
  \begin{cases} 
  \nabla f_0(x)_i \geq 0 & x_i = 0 \\
  \nabla f_0(x)_i = 0 & x_i > 0 
  \end{cases}
  \]

- **Source**: A. d’Aspremont, Convex Optimization M2, Slide 43/67.
```
