2001 | OriginalPaper | Buchkapitel
On the Connectedness of Solution Sets of Parametrized Equations and of Solution Sets in Linear Complementarity Problems
verfasst von : M. Seetharama Gowda, G. S. R. Murthy, T. Parthasarathy
Erschienen in: Complementarity: Applications, Algorithms and Extensions
Verlag: Springer US
Enthalten in: Professional Book Archive
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In this article, we prove, under certain conditions, the connectedness of sets of the form {x: f(x, y) = 0, y ∈ E} where f is a function with x varying over an open set in Rn and the parameter y varying over a topological space. Based on this, we show that the partitioned matrix $$M = \left[ \begin{gathered} A\,\,\,\,\,\,B \hfill \\ B\,\,\,\,\,\,D \hfill \\\end{gathered} \right]$$ is (LCP) connected (i.e., for all q, the solution set of LCP(q, M) is connected) when A ∈ P0 ∩ Q, C = 0, and D is connected. We also show that (a) any nonnegative P0 ∩ Q0-matrix is connected and (b) any matrix M partitioned as above with C and D nonnegative, and A ∈ P0 ∩ Q is connected.