On Solving the Quadratic Sum-of-Ratios Problems

  • Tatiana V. GruzdevaEmail author
  • Alexander S. Strekalovsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)


This paper addresses the numerical solution of fractional programs with quadratic functions in the ratios. Instead of considering a sum-of-ratios problem directly, we developed an efficient global search algorithm, which is based on two approaches to the problem. The first one adopts a reduction of the fractional minimization problem to the solution of an equation with an optimal value of a parametric d.c. minimization problem. The second approach reduces the original problem to the optimization problem with nonconvex (d.c.) constraints. Hence, the fractional programs can be solved by applying the Global Search Theory of d.c. optimization.

The global search algorithm developed for sum-of-ratios problems was tested on the examples with quadratic functions in the numerators and denominators of the ratios. The numerical experiments demonstrated that the algorithm performs well when solving rather complicated quadratic sum-of-ratios problems with up to 100 variables or 1000 terms in the sum.


Fractional optimization Nonconvex problem Difference of two convex functions Quadratic functions Global search algorithm Computational testing 


  1. 1.
    Ashtiani, A.M., Ferreira, P.A.V.: A branch-and-cut algorithm for a class of sum-of-ratios problems. Appl. Math. Comput. 268, 596–608 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barkova, M.V.: On generating nonconvex optimization test problems. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) MOTOR 2019. LNCS, vol. 11548, pp. 21–33. Springer, Cham (2019). Scholar
  3. 3.
    Bugarin, F., Henrion, D., Lasserre, J.B.: Minimizing the sum of many rational functions. Math. Programm. Comput. 8(1), 83–111 (2015). Scholar
  4. 4.
    Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dur, M., Horst, R., Thoai, N.V.: Solving sum-of-ratios fractional programs using efficient points. Optimization 49(5–6), 447–466 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. J. Glob. Optim. 19(1), 83–102 (2001). Scholar
  7. 7.
    Frenk, J.B.G., Schaible, S.: Fractional programming. In: Hadjisavvas, S.S.N., Komlosi, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity. Series Nonconvex Optimization and Its Applications, vol. 76, pp. 335–386. Springer, Heidelberg (2002). Scholar
  8. 8.
    Gruzdeva, T.V., Enkhbat, R., Tungalag, N.: Fractional programming approach to a cost minimization problem in electricity market. Yugoslav J. Oper. Res. 29(1), 43–50 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gruzdeva, T.V., Strekalovskiy, A.S.: On solving the sum-of-ratios problem. Appl. Math. Comput. 318, 260–269 (2018)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gruzdeva, T., Strekalovsky, A.: An approach to fractional programming via D.C. constraints problem: local search. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 404–417. Springer, Cham (2016). Scholar
  11. 11.
    Gruzdeva, T., Strekalovsky, A.: A D.C. programming approach to fractional problems. In: Battiti, R., Kvasov, D.E., Sergeyev, Y.D. (eds.) LION 2017. LNCS, vol. 10556, pp. 331–337. Springer, Cham (2017). Scholar
  12. 12.
    Gruzdeva, T.V., Strekalovsky, A.S.: On a Solution of Fractional Programs via D.C. Optimization Theory. In: CEUR Workshop Proceedings, OPTIMA-2017, vol. 1987, pp. 246–252 (2017)Google Scholar
  13. 13.
    Hiriart-Urruty, J.B.: Generalized differentiability, duality and optimization for problems dealing with difference of convex finctions. In: Ponstein, J. (ed.) Convexity and Duality in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 256, pp. 37–69. Springer, Berlin (1985). Scholar
  14. 14.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Heidelberg (1996). Scholar
  15. 15.
    Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization. Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  16. 16.
    Kuno, T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. J. Glob. Optim. 22, 155–174 (2002). Scholar
  17. 17.
    Ma, B., Geng, L., Yin, J., Fan, L.: An effective algorithm for globally solving a class of linear fractional programming problem. J. Softw. 8(1), 118–125 (2013)CrossRefGoogle Scholar
  18. 18.
    Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optim. Meth. Softw. 18, 219–229 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Strekalovsky, A.S.: Elements of Nonconvex Optimization. Nauka, Novosibirsk (2003). (in Russian)Google Scholar
  20. 20.
    Strekalovsky, A.S.: Global optimality conditions and exact penalization. Optim. Lett. 13(3), 597–615 (2019). Scholar
  21. 21.
    Strekalovsky, A.S.: On local search in d.c. optimization problems. Appl. Math. Comput. 255, 73–83 (2015)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Strekalovsky, A.S.: On solving optimization problems with hidden nonconvex structures. In: Rassias, T.M., Floudas, C.A., Butenko, S. (eds.) Optimization in Science and Engineering, pp. 465–502. Springer, New York (2014). Scholar
  23. 23.
    Strekalovsky, A.S.: On the merit and penalty functions for the D.C. Optimization. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 452–466. Springer, Cham (2016). Scholar
  24. 24.
    Strekalovsky, A.S.: Global optimality conditions in nonconvex optimization. J. Optim. Theory Appl. 173(3), 770–792 (2017). Scholar
  25. 25.
    Strekalovsky, A.S.: Minimizing sequences in problems with D.C. constraints. Comput. Math. Math. Phys. 45(3), 418–429 (2005)MathSciNetGoogle Scholar
  26. 26.
    Strekalovsky, A.S., Gruzdeva, T.V.: Local Search in Problems with Nonconvex Constraints. Comput. Math. Math. Phys. 47, 381–396 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-Convex Constraints. Sequential and Parallel Algorithms. Springer, New York (2000)CrossRefGoogle Scholar
  28. 28.
    Tuy, H.: Parametric decomposition. Convex Analysis and Global Optimization. SOIA, vol. 110, pp. 283–336. Springer, Cham (2016). Scholar
  29. 29.
    Vicente, L.N., Calamai, P.H., Judice, J.J.: Generation of disjointly constrained bilinear programming test problems. Comput. Optim. Appl. 1(3), 299–306 (1992). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control Theory of SB of RASIrkutskRussia

Personalised recommendations