```markdown
## Logarithmic Barrier

### Reformulation of (1) via Indicator Function:

- **Objective:**
  \[
  \text{minimize} \quad f_0(x) + \sum_{i=1}^{m} I_{-}(f_i(x))
  \]
  \[
  \text{subject to} \quad Ax = b
  \]

- **Where:**  
  \( I_{-}(u) = 0 \) if \( u \leq 0 \),  
  \( I_{-}(u) = \infty \) otherwise (indicator function of \( \mathbb{R}_{-} \))

### Approximation via Logarithmic Barrier:

- **Objective:**
  \[
  \text{minimize} \quad f_0(x) - (1/t) \sum_{i=1}^{m} \log(-f_i(x))
  \]
  \[
  \text{subject to} \quad Ax = b
  \]

### Key Points:

- An equality constrained problem
- For \( t > 0 \), \(- (1/t) \log(-u)\) is a smooth approximation of \( I_{-} \)
- Approximation improves as \( t \to \infty \)

### Plot:

- A graph on the right illustrates the behavior of the barrier function as a smooth approximation, with curves converging as \( t \) increases.
```
