```markdown
### Logarithmic Barrier and Central Path

**Logarithmic Barrier:**
- For \( f_1(x) \preceq_{K_1} 0, \ldots, f_m(x) \preceq_{K_m} 0 \):
  \[
  \phi(x) = -\sum_{i=1}^{m} \psi_i(-f_i(x)), \quad \text{dom } \phi = \{ x \mid f_i(x) \prec_{K_i} 0, \, i = 1, \ldots, m \}
  \]

- \(\psi_i\) is a generalized logarithm for \(K_i\), with degree \(\theta_i\).
- \(\phi\) is convex, twice continuously differentiable.

**Central Path:**
- \(\{ x^\star(t) \mid t > 0 \}\) where \(x^\star(t)\) solves:
  \[
  \text{minimize } \, t f_0(x) + \phi(x)
  \]
  \[
  \text{subject to } \, Ax = b
  \]

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