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

Author

Listed:
  • Joaquim Júdice

    (FCTUC - DEEC Polo II Universidade de Coimbra)

  • Valentina Sessa

    (PSL Research University)

  • Masao Fukushima

    (The Kyoto College of Graduate Studies for Informatics)

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.

Suggested Citation

  • Joaquim Júdice & Valentina Sessa & Masao Fukushima, 2022. "Solution of Fractional Quadratic Programs on the Simplex and Application to the Eigenvalue Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 545-573, June.
    • Handle: RePEc:spr:joptap:v:193:y:2022:i:1:d:10.1007_s10957-022-02019-w
    DOI: 10.1007/s10957-022-02019-w
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    References listed on IDEAS

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    1. Luís Fernandes & Joaquim Júdice & Hanif Sherali & Masao Fukushima, 2014. "On the computation of all eigenvalues for the eigenvalue complementarity problem," Journal of Global Optimization, Springer, vol. 59(2), pages 307-326, July.
    2. Brás, Carmo P. & Fischer, Andreas & Júdice, Joaquim J. & Schönefeld, Klaus & Seifert, Sarah, 2017. "A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 36-48.
    3. Siegfried Schaible, 1976. "Fractional Programming. II, On Dinkelbach's Algorithm," Management Science, INFORMS, vol. 22(8), pages 868-873, April.
    4. Schaible, Siegfried, 1981. "Fractional programming: Applications and algorithms," European Journal of Operational Research, Elsevier, vol. 7(2), pages 111-120, June.
    5. Masao Fukushima & Joaquim Júdice & Welington Oliveira & Valentina Sessa, 2020. "A sequential partial linearization algorithm for the symmetric eigenvalue complementarity problem," Computational Optimization and Applications, Springer, vol. 77(3), pages 711-728, December.
    6. Werner Dinkelbach, 1967. "On Nonlinear Fractional Programming," Management Science, INFORMS, vol. 13(7), pages 492-498, March.
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