# Stable math program

Roughly, one whose solution does not change much under perturbation. For the inequality case, we have the following

Stability conditions:
1. $LaTeX: \textstyle \{x \in X: g(x) \le b\}$ is bounded for some $LaTeX: \textstyle b > 0.$
2. $LaTeX: \textstyle \mbox{cl} \{x \in X: g(x) < 0\} = \{x \in X: g(x) \le 0\}.$

The first stability condition pertains to upper semi-continuity and the second, called the closure condition, pertains to lower semi-continuity.

The conditions are not only sufficient to ensure the respective semi-continuity, but they are necessary when:
1. $LaTeX: \textstyle \{x \in X: g(x) \le 0\}$ is bounded.
2. $LaTeX: \textstyle \{x \in X: g(x) < 0 \} \ne \emptyset .$