On the Connectedness of Solution Sets of Parametrized Equations and of Solution Sets in Linear Complementarity Problems

Abstract

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 R ⁿ 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 ∈ P ₀ ∩ Q, C = 0, and D is connected. We also show that (a) any nonnegative P ₀ ∩ Q ₀-matrix is connected and (b) any matrix M partitioned as above with C and D nonnegative, and A ∈ P ₀ ∩ Q is connected.