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

Chapter

## Abstract

In this article, we prove, under certain conditions, the connectedness of sets of the form { is (LCP) connected (i.e., for all

*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$$M = \left[ \begin{gathered} A\,\,\,\,\,\,B \hfill \\ B\,\,\,\,\,\,D \hfill \\\end{gathered} \right]$$

*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}∩*is connected.***Q**## Keywords

Linear complementarity problem solution sets connectedness weak univalence## Preview

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