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### Complexity analysis via self-concordance

**Key Points:**

- **Assumptions:**
  - Sublevel sets (of \( f_0 \), on the feasible set) are bounded.
  - \( t f_0 + \phi \) is self-concordant with closed sublevel sets.

- **Second Condition:**
  - Holds for LP (Linear Programming), QP (Quadratic Programming), QCQP (Quadratically Constrained Quadratic Programming).
  - May require reformulating the problem, e.g.,
    \[
    \text{minimize } \sum_{i=1}^n x_i \log x_i \quad \text{subject to } Fx \preceq g \quad \rightarrow \quad \text{minimize } \sum_{i=1}^n x_i \log x_i \quad \text{subject to } Fx \preceq g, \quad x \succeq 0
    \]

- **Notes:**
  - Needed for complexity analysis; the barrier method works even when the self-concordance assumption does not apply.

**Source:**
A. d’Aspremont. Convex Optimization M2. Slide 21/36
```
