```markdown
### Standard form LP

**Objective:**
- Minimize \( c^T x \)
- Subject to \( Ax = b \), \( x \succeq 0 \)

**Dual Function:**

- **Lagrangian:**
  \[
  L(x, \lambda, \nu) = c^T x + \nu^T (Ax - b) - \lambda^T x = -b^T \nu + (c + A^T \nu - \lambda)^T x
  \]

- **Properties:**
  - \( L \) is linear in \( x \), hence:
    \[
    g(\lambda, \nu) = \inf_x L(x, \lambda, \nu) = 
    \begin{cases} 
    -b^T \nu & \text{if } A^T \nu - \lambda + c = 0 \\
    -\infty & \text{otherwise}
    \end{cases}
    \]
  - \( g \) is linear on affine domain \(\{(\lambda, \nu) \mid A^T \nu - \lambda + c = 0\}\), hence concave

- **Lower bound property:**
  \[
  p^\star \geq -b^T \nu \text{ if } A^T \nu + c \succeq 0
  \]

**Source:**
- A. d'Aspremont, Convex Optimization M2, Slide 8/49
```
