```markdown
### Sum of Infeasibilities Phase I Method

The slide describes a method for addressing infeasible problems by minimizing the sum of infeasibilities. The optimization problem is formulated as:

- **Objective**: Minimize \( 1^T s \)
- **Subject to**:
  - \( s \succeq 0 \)
  - \( f_i(x) \leq s_i \) for \( i = 1, \ldots, m \)
  - \( Ax = b \)

This approach aims to produce a solution that satisfies more inequalities compared to the basic Phase I method.

### Example

An example is provided with an infeasible set of 100 linear inequalities in 50 variables. Two histograms are shown:

- **Left**: Basic Phase I solution, satisfying 39 inequalities.
- **Right**: Sum of infeasibilities Phase I solution, satisfying 79 inequalities.

Both histograms plot the distribution of \( b_i - a_i^T x \) values, with the y-axis labeled "number" and the x-axis labeled \( b_i - a_i^T x \).

- **Left Histogram**: Most values are concentrated near zero.
- **Right Histogram**: Shows a similar concentration near zero, but with more inequalities satisfied.

**Source**: A. d'Aspremont, Convex Optimization M2, Slide 19/36.
```
