On the symmetric quadratic eigenvalue complementarity problem

In this paper, the solution of the symmetric Quadratic Eigenvalue Complementarity Problem (QEiCP) is addressed. The QEiCP has a solution provided the so-called co-regular and co-hyperbolic properties hold and is said to be symmetric if all the matrices involved in its definition are symmetric. We show that under the two conditions stated above the symmetric QEiCP can be reduced to the problem of computing a stationary point of an appropriate nonlinear program (NLP). We also investigate the reduction of the QEiCP to a simpler Eigenvalue Complementarity Problem (EiCP). This transformation enables us to show that the co-regular and co-hyperbolic properties are not necessary for the existence of a solution to the QEiCP. Furthermore the QEiCP is shown to be equivalent to the problem of finding a stationary point of a Quadratic Fractional Program (QFP) under special conditions on the matrices of the QEiCP. The use of the so-called Spectral Projected-Gradient (SPG) algorithm for dealing with the programs NLP and QFP is also investigated. Some considerations about the implementation of this algorithm are discussed. Computational experience is included to highlight the efficiency of the algorithm for finding a solution of the QEiCP by exploring the nonlinear programs mentioned above.