zbMATH — the first resource for mathematics

A sufficient condition for lower semicontinuity of solution sets of systems of convex inequalities. (English) Zbl 0562.90088
Let $$f_ i:$$ $$R^ n\times \Lambda \to R$$ be a continuous function, $$i=0,1...m$$, and let $$M\lambda =\{x\in R^ n$$ $$|$$ $$f_ i(x,\lambda)\leq 0$$, $$i=1,...,m\}$$, $$\lambda\in \Lambda$$. It is well-known that in order for the problem $\psi (\lambda)=\inf_{x\in M\lambda}f_ 0(x,\lambda),\quad \lambda \in \Lambda$ to be stable at some $$\lambda^ 0\in \Lambda$$, a regularity (Slater-like) condition needs to be imposed on the set $$M\lambda^ 0$$. In this paper, a condition which is not of the Slater type is shown to yield stability. Let I be any subset of $$\{$$ 1,...,m$$\}$$ and define $$M^ I\lambda =\{x\in R^ n$$ $$|$$ $$f_ i(x,\lambda)\leq 0$$, $$i\in I\}$$ and ch X$$=\{i\in \{1,...,m\}$$ $$|$$ $$f_ i(x,\lambda)=0$$ for all $$x\in X\}$$. The main theorem is: Let $$\lambda^ 0\in \Lambda$$ be such that $$M\lambda^ 0\neq \emptyset$$, and let $$I=ch M\lambda^ 0$$. Suppose $$M^ I\lambda^ 0$$ is an affine subset of $$R^ n$$ such that for each $$\lambda\in \Lambda:$$ (i) $$M^ I\lambda \neq \emptyset$$, and (ii) the dimension of the lineality space of $$M^ I\lambda$$ is greater than or equal to the dimension of $$M^ I\lambda^ 0$$. Then M is lower semicontinuous at $$\lambda^ 0.$$
The verification of the above theorem may be difficult. However, the author shows a special case where verification is easier.
Reviewer: C.B.Garcia

MSC:
 90C31 Sensitivity, stability, parametric optimization
Full Text: