```markdown
### Implicit Constraints

#### LP with Box Constraints: Primal and Dual Problem

- **Primal Problem:**
  - Minimize: \( c^T x \)
  - Subject to: 
    - \( Ax = b \)
    - \( -1 \preceq x \preceq 1 \)

- **Dual Problem:**
  - Maximize: \( -b^T \nu - 1^T \lambda_1 - 1^T \lambda_2 \)
  - Subject to:
    - \( c + A^T \nu + \lambda_1 - \lambda_2 = 0 \)
    - \( \lambda_1 \succeq 0, \lambda_2 \succeq 0 \)

#### Reformulation with Box Constraints Made Implicit

- Minimize: 
  \[
  f_0(x) = 
  \begin{cases} 
  c^T x & \text{if } -1 \preceq x \preceq 1 \\
  \infty & \text{otherwise}
  \end{cases}
  \]
- Subject to: \( Ax = b \)

#### Dual Function

- \( g(\nu) = \inf_{-1 \preceq x \preceq 1} (c^T x + \nu^T (Ax - b)) \)
- Simplifies to: \( -b^T \nu - \|A^T \nu + c\|_1 \)

#### Dual Problem

- Maximize: \( -b^T \nu - \|A^T \nu + c\|_1 \)

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*Source: A. d’Aspremont. Convex Optimization M2. Slide 29/49.*
```