```markdown
## Convergence Analysis

**Number of Outer (Centering) Iterations:**
- Exactly:  
  \[
  \left\lceil \frac{\log(m/(e t^{(0)}))}{\log \mu} \right\rceil
  \]
- Plus the initial centering step (to compute \( x^*(t^{(0)}) \))

**Centering Problem:**
- Minimize \( t f_0(x) + \phi(x) \)
- See convergence analysis of Newton's method

**Conditions:**
- \( t f_0 + \phi \) must have closed sublevel sets for \( t \geq t^{(0)} \)
- Classical analysis requires strong convexity, Lipschitz condition
- Analysis via self-concordance requires self-concordance of \( t f_0 + \phi \)

*A. d’Aspremont. Convex Optimization M2.*  
*Page 14/36*
```