```markdown
### Logarithmic Barrier Function

The image describes the logarithmic barrier function:

\[
\phi(x) = -\sum_{i=1}^{m} \log(-f_i(x))
\]

**Domain**:
\[
\text{dom } \phi = \{ x \mid f_1(x) < 0, \ldots, f_m(x) < 0 \}
\]

**Properties**:
- **Convex**: Follows from composition rules.
- **Twice Continuously Differentiable**, with derivatives:

Gradient:
\[
\nabla \phi(x) = \sum_{i=1}^{m} \frac{1}{-f_i(x)} \nabla f_i(x)
\]

Hessian:
\[
\nabla^2 \phi(x) = \sum_{i=1}^{m} \frac{1}{f_i(x)^2} \nabla f_i(x) \nabla f_i(x)^T + \sum_{i=1}^{m} \frac{1}{-f_i(x)} \nabla^2 f_i(x)
\]

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