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### Examples

#### Diet Problem:
- **Objective:** Choose quantities \( x_1, \ldots, x_n \) of \( n \) foods.
- **Constraints:**
  - One unit of food \( j \) costs \( c_j \), contains amount \( a_{ij} \) of nutrient \( i \).
  - A healthy diet requires nutrient \( i \) in quantity at least \( b_i \).

- **Optimization Problem:**
  - **Minimize:** \( c^T x \)
  - **Subject to:** 
    - \( Ax \succeq b \)
    - \( x \succeq 0 \)

#### Piecewise-Linear Minimization:
- **Objective:** Minimize \( \max_{i=1,\ldots,m}(a_i^T x + b_i) \)

- **Equivalent to a Linear Program (LP):**
  - **Minimize:** \( t \)
  - **Subject to:** 
    - \( a_i^T x + b_i \leq t \), \( i = 1, \ldots, m \)

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*Source: A. d’Aspremont. Convex Optimization M2. Slide 50/67*
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