Two characterization theorems in complementarity theory
Two main results are proved in this paper. The first theorem concerns the linear complementarity problem (LCP) \(w=q+Mx\geq 0\), \(x\geq 0\), \(w^ Tx=0\) where q is a given n-vector and M is a given \(n\times n\) metric. It is shown that M is non-degenerate \(\Leftrightarrow\) For each vector q, the problem has a finite number of solutions \(\Leftrightarrow\) For each vector q, each solution to the problem is locally unique. The second theorem concerns the differentiability of a solution to the parametric nonlinear complementarity problem (NCP): \(w=f(x,\lambda)\geq 0\), \(x\geq 0\) and \(w^ Tx=0\) where \(f: R^ n\times R^ m\to R^ n\). A necessary and sufficient condition for x(\(\lambda)\) to be differentiable is stated and proved where x(\(\lambda)\) is a solution of (NCP) for a given \(\lambda \in R^ m\).
scientific article; zbMATH DE number 4049418