```markdown
**Title:** Polynomial-time complexity of barrier method

- For \( \mu = 1 + 1/\sqrt{m} \):
  \[
  N = O\left( \sqrt{m} \log \left( \frac{m/t^{(0)}}{\epsilon} \right) \right)
  \]

- Number of Newton iterations for fixed gap reduction is \( O(\sqrt{m}) \).

- Multiply with the cost of one Newton iteration (a polynomial function of problem dimensions) to get a bound on the number of flops.

- This choice of \( \mu \) optimizes worst-case complexity; in practice, we choose \( \mu \) fixed (\( \mu = 10, \ldots, 20 \)).

**Source:** A. d’Aspremont. Convex Optimization M2. Slide 24/36.
```
