```markdown
### Generalized Inequality Constraints

**Convex problem with generalized inequality constraints:**

- **Objective:**
  \[
  \text{minimize } f_0(x)
  \]
- **Subject to:**
  \[
  f_i(x) \preceq_{K_i} 0, \quad i = 1, \ldots, m
  \]
  \[
  Ax = b
  \]

- **Properties:**
  - \( f_0 : \mathbb{R}^n \rightarrow \mathbb{R} \) is convex.
  - \( f_i : \mathbb{R}^n \rightarrow \mathbb{R}^{k_i} \), \( K_i \)-convex with respect to proper cone \( K_i \).
  - Shares properties with standard convex problems (convex feasible set, local optimum is global, etc.).

**Conic form problem:**

- **Special case with affine objective and constraints:**

  - **Objective:**
    \[
    \text{minimize } c^T x
    \]
  - **Subject to:**
    \[
    Fx + g \preceq_K 0
    \]
    \[
    Ax = b
    \]

- **Extension:**
  - Extends linear programming (\( K = \mathbb{R}^m_+ \)) to nonpolyhedral cones.

**Source:**
- A. d’Aspremont. Convex Optimization M2. Page 63/67.
```
