 # Closure condition

The conditinos that the closure of a nonempty strict interior be equal to the level set: $LaTeX: \mbox{cl}( \{x \in X: g(x) < 0 \} = \{x \in X: g(x) \le 0\}$. Here is an example where the closure condition fails: $LaTeX: g(x) = \left\{ \begin{array}{cl}

 -1-x, & x < -1 \\ 0, & -1 \le x < 0 \\ -1 + (x-1)^2, & 0 \le x < 2 \\ x-2, & 2 \le x. \end{array} \right.

$

Note that $LaTeX: g$ is continuous and quasiconvex. Its level set is $LaTeX: \{ x: g(x) \le 0 \} = [-1, 2]$, but the strict interior, is $LaTeX: (0, 2)$, so its closure is only $LaTeX: [0, 2]$. We lose the flat portion, $LaTeX: [-1, 0)$.

This is important in the use of interior point methods and in stability. Here are some functions that satisfy the closure condition:

When equality constraints are present, there are two forms of extension of the closure condition: to consider the relative strict interior, and to consider "feasible sequences". The first generally assumes $LaTeX: h$ is affine, so the closure condition becomes: $LaTeX: \mbox{cl}( \{ x: Ax=b, \; g(x) < 0 \}) = \{ x: Ax=b, g(x) \le 0\}.$
The second defines a family of maps: $LaTeX: I_i - = \{ x \in X: g(x) < 0, \; h_k(x) = 0 \; \mbox{ for } k \neq i, \; h_i(x) < 0 \}$

and $LaTeX: I_i+ = \{ x \in X : g(x) < 0, \; h_k(x) = 0 \; \mbox{ for } k \neq i, \; h_i(x) > 0 \}.$

Then, the closure condition is that $LaTeX: I_i-$ and $LaTeX: I_i+$ are not empty, and $LaTeX: \mbox{cl}(I_i-) = \{ x \in X : g(x) \le 0, \; h_k(x) = 0 \; \mbox{ for } k \neq i, \; h_i(x) \le 0 \}$

and $LaTeX: \mbox{cl}(I_i+) = \{ x \in X : g(x) \le 0, \; h_k(x) = 0 \; \mbox{ for } k \neq i, \; h_i(x) \ge 0 \}.$