```markdown
### Initial point and sublevel set

Algorithms in this chapter require a starting point \( x^{(0)} \) such that:

- \( x^{(0)} \in \text{dom } f \)
- Sublevel set \( S = \{ x \mid f(x) \leq f(x^{(0)}) \} \) is closed

The second condition is hard to verify, except when all sublevel sets are closed:

- Equivalent to the condition that \(\text{epi } f\) is closed
- True if \(\text{dom } f = \mathbb{R}^n\)
- True if \( f(x) \to \infty \) as \( x \to \text{bd dom } f \)

Examples of differentiable functions with closed sublevel sets:

\[ f(x) = \log \left( \sum_{i=1}^{m} \exp(a_i^T x + b_i) \right), \quad f(x) = -\sum_{i=1}^{m} \log(b_i - a_i^T x) \]

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