Abstract
The Quadratic Eigenvalue Complementarity Problem (QEiCP) is an extension of the Eigenvalue Complementarity Problem (EiCP) that has been introduced recently. Similar to the EiCP, the QEiCP always has a solution under reasonable hypotheses on the matrices included in its definition. This has been established in a previous paper by reducing a QEiCP of dimension n to a special 2n-order EiCP. In this paper we propose an enumerative algorithm for solving the QEiCP by exploiting this equivalence with an EiCP. The algorithm seeks a global minimum of a special Nonlinear Programming Problem (NLP) with a known global optimal value. The algorithm is shown to perform very well in practice but in some cases terminates with only an approximate optimal solution to NLP. Hence, we propose a hybrid method that combines the enumerative method with a fast and local semi-smooth method to overcome the latter drawback. This algorithm is also shown to be useful for computing a positive eigenvalue for an EiCP under similar assumptions. Computational experience is reported to demonstrate the efficacy and efficiency of the hybrid enumerative method for solving the QEiCP.
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Alfredo N. Iusem was partially supported by CNPq grant no. 301280/86.
Joaquim J. Judice was supported in the scope of R, D Unit 50008, financed by the applicable financial framework (FCT/MEC through national funds and when applicable co-funded by FEDER PT2020 partnership agreement).
Valentina Sessa was supported by CNPq grant no. 150606/2014-1.
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Iusem, A.N., Júdice, J.J., Sessa, V. et al. On the numerical solution of the quadratic eigenvalue complementarity problem. Numer Algor 72, 721–747 (2016). https://doi.org/10.1007/s11075-015-0064-9
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DOI: https://doi.org/10.1007/s11075-015-0064-9