Closure condition

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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}
</p>
<pre>      -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.
</pre>
<p>

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 \}.


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