On the quadratic eigenvalue complementarity problem

We introduce several new results on the Quadratic Eigenvalue Complementarity Problem (QEiCP), focusing on the nonsymmetric case, i,e, without making symmetry assumptions on the matrices defining the problem. First we establish a new sufficient condition for existence of solutions of this problem, which is somewhat more manageable than previously existent ones. This condition works through the introduction of auxiliary variables which leads to the reduction of QEiCP to an Eigenvalue Complementarity Problem (EiCP) in higher dimension. Hence, this reduction suggests a new strategy for solving QEiCP, which is also analyzed in the paper. We also present an upper bound for the number of solutions of QEiCP and exhibit some examples of instances of QEiCP whose solution set has large cardinality, without attaining though the just mentioned upper bound. We also investigate the numerical solution of the QEiCP by solving a Variational Inequality Problem (VIP) on the 2n-dimensional simplex, which is equivalent to a 2n-dimensional EiCP. Some numerical experiments with a projection method for solving this VIP are reported, illustrating the value of this methodology in practice. ∗Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Portugal, [email protected]. The work of this author was partially supported by CMA/FCT/UNL, under the project PEstOE/MAT/UI0297/2011. †Instituto de Matématica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, RJ, CEP 22460320, Brazil, [email protected]. The work of this author was partially supported by CNPq grant no. 301280/86. ‡Instituto de Telecomunicações, Portugal, [email protected].