```markdown
### Generalized Inequalities

#### Optimization Problem
- **Objective:** Minimize \( f_0(x) \)
- **Subject to:**
  - \( f_i(x) \preceq_{K_i} 0 \), for \( i = 1, \ldots, m \)
  - \( Ax = b \)

#### Assumptions
- \( f_0 \) is convex.
- \( f_i: \mathbb{R}^n \rightarrow \mathbb{R}^{k_i} \) are convex with respect to proper cones \( K_i \in \mathbb{R}^{k_i} \).
- \( f_i \) are twice continuously differentiable.
- \( A \in \mathbb{R}^{p \times n} \) with \(\text{Rank } A = p\).
- \( p^\star \) is finite and attained.
- The problem is strictly feasible; hence strong duality holds, and the dual optimum is attained.

#### Examples of Interest
- Second-Order Cone Programming (SOCP)
- Semidefinite Programming (SDP)

*Source: A. d'Aspremont. Convex Optimization M2.*
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