\(Q\)-matrices and boundedness of solutions to linear complementarity problems
For a given matrix \(M\in \mathbb{R}^{n\times n}\) and a vector \(q\in \mathbb{R}^n\), the linear complementarity problem is that of finding \(w\), \(z\in \mathbb{R}^n\) such that \(w= Mz+ q\), \(w\geq 0\), \(z\geq 0\), \(w^T z= 0\). It has been previously established by \textit{O. L. Mangasarian} [Math. Program. Study 18, 153-166 (1982; Zbl 0487.90088)] that, if \(M\) is copositive plus, then the solution set of the complementarity problem is nonempty and bounded for each \(q\in \mathbb{R}^n\) iff \(M\) is a \(Q\)-matrix. The author has shown that this result also holds if \(M\) belongs to any of the classes of \(L_2\)-matrices, \(P_0\)-matrices, nonnegative matrices, and \(z\)-matrices.
scientific article; zbMATH DE number 698710
scientific article