```markdown
### Lagrange Dual Function

**Definition:**  
The Lagrange dual function \( g : \mathbb{R}^m \times \mathbb{R}^p \rightarrow \mathbb{R} \) is defined as:

\[
g(\lambda, \nu) = \inf_{x \in D} L(x, \lambda, \nu)
\]

\[
= \inf_{x \in D} \left( f_0(x) + \sum_{i=1}^{m} \lambda_i f_i(x) + \sum_{i=1}^{p} \nu_i h_i(x) \right)
\]

**Properties:**

- \( g \) is concave and can be \(-\infty\) for some \(\lambda, \nu\).

**Lower Bound Property:**

- If \(\lambda \succeq 0\), then \(g(\lambda, \nu) \leq p^\star\).

**Proof:**

- If \(\tilde{x}\) is feasible and \(\lambda \succeq 0\), then:

\[
f_0(\tilde{x}) \geq L(\tilde{x}, \lambda, \nu) \geq \inf_{x \in D} L(x, \lambda, \nu) = g(\lambda, \nu)
\]

- Minimizing over all feasible \(\tilde{x}\) gives \(p^\star \geq g(\lambda, \nu)\).
```
