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Solution of Fractional Quadratic Programs on the Simplex and Application to the Eigenvalue Complementarity Problem

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Abstract

In this paper, we introduce an implementation of Dinkelbach’s algorithm for computing a global maximum of a fractional linear quadratic program (FLQP) on the simplex that employs an efficient block principal pivoting algorithm in each iteration. A new sequential FLQP algorithm is introduced for computing a stationary point (SP) of a fractional quadratic program (FQP) on the simplex. Global convergence for this algorithm is established. This sequential algorithm is recommended for the solution of the symmetric eigenvalue complementarity problem (EiCP), as this problem is equivalent to the computation of an SP of an FQP on the simplex. Computational experience reported in this paper indicates that the implementation of Dinkelbach’s method for the FLQP and the sequential FLQP algorithm are quite efficient in practice. An extension of the sequential FLQP algorithm for solving the nonsymmetric EiCP is also introduced. Since this method solves a special variational inequality (VI) problem in each iteration, it can be considered as a sequential VI algorithm. Although the convergence of this algorithm has yet to be established, preliminary computational experience indicates that the sequential VI algorithm is quite a promising technique for the solution of the nonsymmetric EiCP.

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Notes

  1. The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The research of Valentina Sessa benefited from the support of the FMJH Program PGMO and from the support of EDF. The research of Joaquim J. Júdice was partially supported in the scope of R&D Unit UID/EEA/50008/2019, financed by the applicable financial framework (FCT/MEC) through national funds and when applicable co-funded by FEDER-PT2020 Partnership Agreement.

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Authors and Affiliations

  1. Instituto de Telecomunicações, FCTUC - DEEC Polo II Universidade de Coimbra, Coimbra, 3030-290, Portugal

    Joaquim Júdice

  2. MINES ParisTech, CMA Centre de Mathématiques Appliquées, PSL Research University, Sophia Antipolis, France

    Valentina Sessa

  3. The Kyoto College of Graduate Studies for Informatics, Kyoto, Japan

    Masao Fukushima

Authors
  1. Joaquim Júdice
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  2. Valentina Sessa
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  3. Masao Fukushima
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Communicated by Massimo Pappalardo.

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Júdice, J., Sessa, V. & Fukushima, M. Solution of Fractional Quadratic Programs on the Simplex and Application to the Eigenvalue Complementarity Problem. J Optim Theory Appl 193, 545–573 (2022). https://doi.org/10.1007/s10957-022-02019-w

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