```markdown
### Feasibility Problems

#### Feasibility Problem A in \( x \in \mathbb{R}^n \):

- \( f_i(x) < 0, \quad i = 1, \ldots, m, \)
- \( h_i(x) = 0, \quad i = 1, \ldots, p \)

#### Feasibility Problem B in \( \lambda \in \mathbb{R}^m, \nu \in \mathbb{R}^p \):

- \( \lambda \succeq 0, \quad \lambda \neq 0, \quad g(\lambda, \nu) \geq 0 \)

Where:
\[ g(\lambda, \nu) = \inf_x \left( \sum_{i=1}^m \lambda_i f_i(x) + \sum_{i=1}^p \nu_i h_i(x) \right) \]

- **Feasibility Problem B is Convex**: \( g \) is concave, even if problem A is not.
- **A and B are Always Weak Alternatives**: At most one is feasible.
  - **Proof**: Assume \( \tilde{x} \) satisfies A, \( \lambda, \nu \) satisfy B
    \[
    0 \leq g(\lambda, \nu) \leq \sum_{i=1}^m \lambda_i f_i(\tilde{x}) + \sum_{i=1}^p \nu_i h_i(\tilde{x}) < 0
    \]
- **A and B are Strong Alternatives**: If exactly one of the two is feasible (can prove infeasibility of A by producing solution of B and vice-versa).

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