```markdown
**Title:** Newton iterations per centering step: from self-concordance theory

**Main Formula:**
\[ 
\# \text{Newton iterations} \leq \frac{\mu t f_0(x) + \phi(x) - \mu t f_0(x^+) - \phi(x^+)}{\gamma} + c 
\]

**Key Points:**
- **Bound on Effort:** Computing \(x^+ = x^\star(\mu t)\) starting at \(x = x^\star(t)\).
- **Constants:** \(\gamma, c\) are constants depending only on Newton algorithm parameters.
- **From Duality:** Using \(\lambda = \lambda^\star(t)\), \(\nu = \nu^\star(t)\):

  \[
  \mu t f_0(x) + \phi(x) - \mu t f_0(x^+) - \phi(x^+)
  \]

  \[
  = \mu t f_0(x) - \mu t f_0(x^+) + \sum_{i=1}^{m} \log(-\mu t \lambda_i f_i(x^+)) - m \log \mu
  \]

  \[
  \leq \mu t f_0(x) - \mu t f_0(x^+) - \mu t \sum_{i=1}^{m} \lambda_i f_i(x^+) - m - m \log \mu
  \]

  \[
  \leq \mu t f_0(x) - \mu t g(\lambda, \nu) - m - m \log \mu
  \]

  \[
  = m(\mu - 1 - \log \mu)
  \]

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