```markdown
### Lower Bound Property

- **Statement**: If \(\lambda_i \succeq_{K_i^*} 0\), then \(g(\lambda_1, \ldots, \lambda_m, \nu) \leq p^\star\).

- **Proof**: 
  - If \(\tilde{x}\) is feasible and \(\lambda \succeq_{K_i^*} 0\), then:
    \[
    f_0(\tilde{x}) \geq f_0(\tilde{x}) + \sum_{i=1}^{m} \lambda_i^T f_i(\tilde{x}) + \sum_{i=1}^{p} \nu_i h_i(\tilde{x})
    \]
    \[
    \geq \inf_{x \in \mathcal{D}} L(x, \lambda_1, \ldots, \lambda_m, \nu)
    \]
    \[
    = g(\lambda_1, \ldots, \lambda_m, \nu)
    \]
  - Minimizing over all feasible \(\tilde{x}\) gives \(p^\star \geq g(\lambda_1, \ldots, \lambda_m, \nu)\).

### Dual Problem

- **Objective**: Maximize \(g(\lambda_1, \ldots, \lambda_m, \nu)\)
- **Subject to**: \(\lambda_i \succeq_{K_i^*} 0\), for \(i = 1, \ldots, m\)

### Duality

- **Weak Duality**: \(p^\star \geq d^\star\) always
- **Strong Duality**: \(p^\star = d^\star\) for convex problems with constraint qualification (e.g., Slater’s condition: primal problem is strictly feasible)

---

*Slide reference: A. d'Aspremont, Convex Optimization M2, Slide 31/49.*
```
