```markdown
### Barrier Method

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

**Notes:**
- Terminates with \( f_0(x) - p^* \leq \epsilon \) (stopping criterion follows from \( f_0(x^*(t)) - p^* \leq m/t \)).
- Centering is usually done using Newton's method, starting at current \( x \).
- Choice of \( \mu \) involves a trade-off: large \( \mu \) means fewer outer iterations, more inner (Newton) iterations; typical values: \( \mu = 10\text{--}20 \).
- Several heuristics for choice of \( t^{(0)} \).

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