```markdown
### Problems with Generalized Inequalities

**Objective:**
- Minimize \( f_0(x) \)

**Subject to:**
- \( f_i(x) \preceq_{K_i} 0 \), for \( i = 1, \ldots, m \)
- \( h_i(x) = 0 \), for \( i = 1, \ldots, p \)

**Definitions:**
- \( \preceq_{K_i} \) is a generalized inequality on \( \mathbb{R}^{k_i} \).

**Key Concepts:**
- **Lagrange Multiplier:**  
  For \( f_i(x) \preceq_{K_i} 0 \), the Lagrange multiplier is a vector \( \lambda_i \in \mathbb{R}^{k_i} \).

- **Lagrangian:**  
  \( L : \mathbb{R}^n \times \mathbb{R}^{k_1} \times \cdots \times \mathbb{R}^{k_m} \times \mathbb{R}^p \rightarrow \mathbb{R} \) is defined as:
  \[
  L(x, \lambda_1, \ldots, \lambda_m, \nu) = f_0(x) + \sum_{i=1}^{m} \lambda_i^T f_i(x) + \sum_{i=1}^{p} \nu_i h_i(x)
  \]

- **Dual Function:**  
  \( g : \mathbb{R}^{k_1} \times \cdots \times \mathbb{R}^{k_m} \times \mathbb{R}^p \rightarrow \mathbb{R} \) is defined as:
  \[
  g(\lambda_1, \ldots, \lambda_m, \nu) = \inf_{x \in \mathcal{D}} L(x, \lambda_1, \ldots, \lambda_m, \nu)
  \]
```
