```markdown
### Minimizing Spectral Radius of Nonnegative Matrix

**Perron-Frobenius eigenvalue \(\lambda_{pf}(A)\):**
- Exists for (elementwise) positive \(A \in \mathbb{R}^{n \times n}\).
- A real, positive eigenvalue of \(A\), equal to spectral radius \(\max_i |\lambda_i(A)|\).
- Determines asymptotic growth (decay) rate of \(A^k\): \(A^k \sim \lambda_{pf}^k\) as \(k \to \infty\).
- Alternative characterization: \(\lambda_{pf}(A) = \inf \{\lambda \mid Av \succeq \lambda v \text{ for some } v \succ 0\}\).

**Minimizing Spectral Radius of Matrix of Posynomials:**
- Minimize \(\lambda_{pf}(A(x))\), where the elements \(A(x)_{ij}\) are posynomials of \(x\).
- Equivalent geometric program:
  - Minimize \(\lambda\)
  - Subject to \(\sum_{j=1}^n A(x)_{ij} v_j / (\lambda v_i) \leq 1\), for \(i = 1, \ldots, n\).

**Variables:** \(\lambda, v, x\)

*Source: A. d'Aspremont. Convex Optimization M2. Page 62/67.*
```