```markdown
### Barrier Method

**Algorithm Steps:**
1. **Initialize:** Given strictly feasible \( x \), set \( t := t^{(0)} > 0 \), \( \mu > 1 \), tolerance \( \epsilon > 0 \).
2. **Repeat:**
   - **Centering Step:** Compute \( x^*(t) \) by minimizing \( tf_0 + \phi \), subject to \( Ax = b \).
   - **Update:** Set \( x := x^*(t) \).
   - **Stopping Criterion:** Quit if \( \left(\sum_i \theta_i\right)/t < \epsilon \).
   - **Increase \( t \):** Set \( t := \mu t \).

**Notes:**
- The only difference is the duality gap \( m/t \) on the central path is replaced by \( \sum_i \theta_i/t \).
- **Number of Outer Iterations:**
  \[
  \left\lceil \frac{\log((\sum_i \theta_i)/( \epsilon t^{(0)}))}{\log \mu} \right\rceil
  \]
- **Complexity Analysis:** Self-concordance applies to SDP, SOCP.

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