```markdown
## LP and SOCP as SDP

### LP and equivalent SDP

- **LP**: 
  - Minimize: \( c^T x \)
  - Subject to: \( Ax \preceq b \)

- **SDP**: 
  - Minimize: \( c^T x \)
  - Subject to: \( \text{diag}(Ax - b) \preceq 0 \)

(Note: different interpretation of generalized inequality \( \preceq \))

### SOCP and equivalent SDP

- **SOCP**: 
  - Minimize: \( f^T x \)
  - Subject to: \( \|A_i x + b_i\|_2 \leq c_i^T x + d_i, \quad i = 1, \ldots, m \)

- **SDP**: 
  - Minimize: \( f^T x \)
  - Subject to: 
    \[
    \begin{bmatrix}
    (c_i^T x + d_i)I & A_i x + b_i \\
    (A_i x + b_i)^T & c_i^T x + d_i
    \end{bmatrix} \succeq 0, \quad i = 1, \ldots, m
    \]

---

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