Mathematical Programming Computation

, Volume 8, Issue 1, pp 83–111 | Cite as

Minimizing the sum of many rational functions

  • Florian BugarinEmail author
  • Didier Henrion
  • Jean Bernard Lasserre
Full Length Paper


We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of variables can be relatively large (10 to 100) provided some kind of sparsity is present. We describe a formulation of the rational optimization problem as a generalized moment problem and its hierarchy of convex semidefinite relaxations. Under some conditions we prove that the sequence of optimal values converges to the globally optimal value. We show how public-domain software can be used to model and solve such problems. Finally, we also compare with the epigraph approach and the BARON software.


Rational optimization Global optimization Semidefinite relaxations Sparsity 

Mathematics Subject Classification

46N10 65K05 90C22 90C26 



We are grateful to Michel Devy, Jean-José Orteu, Tomáš Pajdla, Thierry Sentenac and Rekha Thomas for insightful discussions on applications of real algebraic geometry and SDP in computer vision, and to Josh Taylor for his feedback on the example of Sect. 4.2.


  1. 1.
    Ali, M.M., Khompatraporn, C., Zabinsky, Z.B.: A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J. Global Optim. 31(4), 635–672 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Benson, H.P.: Global optimization algorithm for the nonlinear sum of ratios problem. J. Optim. Theory Appl. 112(1), 1–29 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Benson, H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problems. J. Global Optim. 22, 343–364 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Benson, S.J., Ye, Y.: DSDP5 user guide—software for semidefinite programming. Mathematics and Computer Science Division, Argonne National Laboratory (2005)Google Scholar
  5. 5.
    Bersini, H., Dorigo, M., Langerman, S., Seront, G., Gambardella, L.: Results of the first international contest on evolutionary optimisation. IEEE Intl. Conf. Evolutionary Computation, Nagoya (1996)Google Scholar
  6. 6.
    Borchers, B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11, 613–623 (1999)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Curto, R.E., Fialkow, L.A.: Recursiveness, positivity, and truncated moment problems. Houston J. Math. 17, 603–635 (1991)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Czyzyk, J., Mesnier, M., Moré, J.: The NEOS server. IEEE J. Comput. Sci. Eng. 5(3), 68–75 (1998)CrossRefGoogle Scholar
  9. 9.
    Freund, R.W., Jarre, F.: Solving the sum-of-ratios problem by an interior-point method. J. Global Optim. 19(1), 83–102 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Hartley, R., Kahl, F., Olsson, C., Seo, Y.: Verifying global minima for L2 minimization problems in multiple view geometry. Int. J. Comput. Vis. 101(2), 288–304 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Henrion, D., Lasserre, J.B.: GloptiPoly: global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Softw. 29, 165–194 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Henrion, D., Lasserre, J.B., Löfberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24, 761–779 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Jibetean, D., de Klerk, E.: Global optimization of rational functions: a semidefinite programming approach. Math. Program. 106, 93–109 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Kahl, F., Agarwal, S., Chandraker, M., Kriegman, D., Belongie, S.: Practical global optimization for multiview geometry. Int. J. Comput. Vis. 79(3), 271–284 (2008)CrossRefGoogle Scholar
  15. 15.
    Kahl, F., Henrion, D.: Globally optimal estimates for geometric reconstruction problems. IEEE Intl. Conf. Computer Vision, Beijing (2005)CrossRefGoogle Scholar
  16. 16.
    Kuno, T.: A branch-and-bound algorithm for maximizing the sum of several linear ratios. J. Global Optim. 22(1), 155–174 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Lasserre, J.B.: Convergent SDP relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17, 822–843 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Lasserre, J.B.: A semidefinite programming approach to the generalized problem of moments. Math. Program. 112, 65–92 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Lasserre, J.B.: Moments, Positive Polynomials and their Applications. Imperial College Press, London (2010)zbMATHGoogle Scholar
  20. 20.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. Emerg. Appl. Algebr. Geom. 149, 157–270 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Löfberg, J.: Yalmip : A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei (2004)Google Scholar
  22. 22.
    Pujol, J.C.F., Poli, R.: A highly efficient function optimization with genetic programming. Genetic and Evolutionary Computation Conference, Seattle (2004)Google Scholar
  23. 23.
    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969–984 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Schaible, S., Shi, J.: Fractional programming: the sum-of-ratios case. Optim. Methods Softw. 18(2), 219–229 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Schweighofer, M.: Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 17(3), 805–825 (2005)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 12, 625–653 (1999)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Toh, K.C., Todd, M.J., Tutuncu, R.H.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Ser. B 95, 189–217 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Waki, S., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problemswith structured sparsity. SIAM J. Optim. 17, 218–242 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M., Sugimoto, H.: SparsePOP: a sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Softw. 35, 1–13 (2008)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Wang, C.F., Liu, S.Y., Shen, P.P.: Global optimization for sum of geometric fractional functions. Appl. Math. Comput. 216, 2263–2270 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Wu, W.-Y., Sheu, R.-L., Ilker, S.: Birbil. Solving the sum-of-ratios problem by a stochastic search algorithm. J. Global Optim. 42(1), 91–109 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Yamashita, M., Fujisawa, K., Nakata, K., Nakata, M., Fukuda, M., Kobayashi, K., Goto, K.: A high-performance software package for semidefinite programs: SDPA 7. Research Report B-460 Dept. of Mathematical and Computing Science, Tokyo Institute of Technology (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and The Mathematical Programming Society 2015

Authors and Affiliations

  • Florian Bugarin
    • 1
    Email author
  • Didier Henrion
    • 2
    • 3
    • 4
  • Jean Bernard Lasserre
    • 2
    • 3
    • 5
  1. 1.Université de Toulouse; UPS, ICA (Institut Clément Ader)ToulouseFrance
  2. 2.CNRS, LAASToulouseFrance
  3. 3.Université de Toulouse, LAASToulouseFrance
  4. 4.Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic
  5. 5.Institut de Mathématiques de ToulouseUniversité de ToulouseToulouseFrance

Personalised recommendations