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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

90C31 Sensitivity, stability, parametric optimization
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