Stable math program

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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 .

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