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Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 6, pp 1585–1603 | Cite as

Applying Gröbner Basis Method to Multiparametric Polynomial Nonlinear Programming

  • Benyamin M.-Alizadeh
  • Abdolali Basiri
  • Sajjad RahmanyEmail author
Original Paper
  • 45 Downloads

Abstract

In this paper, we present first a new algorithm based on Gröbner basis to analyze and/or solve a convex polynomial nonlinear programming problem that is a convex nonlinear programming in which the objective and constraints are algebraic polynomials. The main efficiency of our algorithm is that there is no need to compute feasible points to find the optimum value. Next, we generalize our results to analyze and/or solvemultiparametric convex polynomial nonlinear programming problems. The mainproperty of this method is that it preserves the parametric scheme of theproblem until the end of the algorithm. Even the output of our algorithm depends onparameters and so one can present the optimum value and optimizer points as afunction on parameters. To show the ability of this algorithm, we will state two applied examples: the problem of minimizing the expense of forage in a chicken farm with unpredictable price of forage and the number of chickens, and an optimal control problem in model predictive control. The presented algorithms in this paper are all implemented in the Maple software.

Keywords

Polynomial nonlinear programming Multiparametric polynomial nonlinear programming Eigenvalue system Gröbner basis Comprehensive Gröbner system 

Mathematics Subject Classification

13P25 13P10 

Notes

Acknowledgements

The authors are thankful to Dr. Amir Hashemi for his useful comments onpreparing this paper. They would also like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

References

  1. 1.
    Abbasi Molai, A., Khorram, E.: Linear programming problem with interval coefficients and an interpretation for its constraints. Iran. J. Sci. Technol. 32(4), 369–390 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Becker, T., Weispfenning, V.: Gröbner Bases, a Computational Approach to Commutative Algebra. Springer, Berlin (1993)zbMATHGoogle Scholar
  3. 3.
    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bemporad, A., Filippi, C.: An algorithm for approximate multiparametric convex programming. Comput. Optim. Appl. 35(1), 87–108 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bertsimas, D., Perakis, D., Tayur, S.: A new algebraic geometry algorithm for integer programming. Manag. Sci. 46(7), 999–1008 (2000)CrossRefGoogle Scholar
  6. 6.
    Blanco, V., Puerto, J.: Partial Gröbner bases for multiobjective integer linear optimization. SIAM J. Discret. Math. 23(2), 571–595 (2009)CrossRefGoogle Scholar
  7. 7.
    Blanco, V., Puerto, J.: Some algebraic methods for solving multiobjective polynomial integer programs. J. Symbol. Comput. 46(5), 511–533 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Conti, P., Traverso, C.: Buchberger algorithm and integer programming. Appl. Algebra Algebr. Algorithms Error Correct. Codes, 130–139 (1991)Google Scholar
  9. 9.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, Berlin (2005)zbMATHGoogle Scholar
  10. 10.
    Dehghani Darmian, M., Hashemi, A.: Parametric FGLM algorithm. J. Symbol. Comput. 82, 38–56 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Domínguez, L., Narciso, D., Pistikopoulos, E.: Recent advances in multiparametric nonlinear programming. J. Comput. Chem. Eng. 34, 707–716 (2010)CrossRefGoogle Scholar
  12. 12.
    Dua, V.: Mixed integer polynomial programming. J. Comput. Chem. Eng. 72, 387–394 (2015)CrossRefGoogle Scholar
  13. 13.
    Eustáquio, R.G., Karas, E.W., Ribeiro, A.A.: Constraint Qualifications for Non-linear Programming. Federal University of Paraná, Paraná (2008)Google Scholar
  14. 14.
    Faugère, J.-C.: A New Efficient Algorithm for Computing Gröbner Bases Without Reduction to Zero \((F_5)\). ISSAC’02, pp. 75–83. ACM Press, New York (2002)Google Scholar
  15. 15.
    Fotiou, I.A., Rostalski, P.A., Parrilo, P.A., Morari, M.: Parametric optimization and optimal control using algebraic geometry methods. Int. J. Control 79(11), 1340–1358 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fotiou, I.A., Rostalski, P., Sturmfels, B., Morari, M.: An algebraic geometry approach to nonlinear parametric optimization in control. American Control Conference, pp. 3618–3623 (2006)Google Scholar
  17. 17.
    Gao, S., Guan, Y., Volny IV, F.: A New Incremental Algorithm for Computing Groebner Bases. ISSAC’10, pp. 13–19. ACM Press, New York (2010)zbMATHGoogle Scholar
  18. 18.
    Gao, S., Volny, F., Wang, M.: A New Algorithm for Computing Gröbner Bases. Cryptology ePrint Archive, Report 2010/641. http://eprint.iacr.org/ (2010)
  19. 19.
    Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern. http://www.singular.uni-kl.de (2011)
  20. 20.
    Hägglöf, K., Lindberg, P., Svensson, L.: Computing global minima to polynomial optimization problems using Gröbner bases. J. Glob. Optim. 7(2), 115–125 (1995)CrossRefGoogle Scholar
  21. 21.
    Hashemi, A., Dehghani Darmian, M., M.-Alizadeh, B.: Applying Buchberger’s criteria on Montes DISPGB algorithm. Bull. Iran. Math. Soc. 38(3), 715–724 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hashemi, A., M.-Alizadeh, B., Dehghani Darmian, M.: Computing comprehensive Gröbner systems: a comparison of two methods. Comput. Sci. J. Moldova 25(3(75)), 278–302 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jiménez, J.M., Ucha, J.M.: Finding multiplies solutions for non-linear integer programming. Electron. Notes Discret. Math. 68, 125–130 (2018)CrossRefGoogle Scholar
  24. 24.
    Kapur, D., Sun, Y., Wang, D.: A New Algorithm for Computing Comprehensive Gröbner Systems. ISSAC’10, pp. 29–36. ACM Press, New York (2010)zbMATHGoogle Scholar
  25. 25.
    Kapur, D., Sun, Y., Wang, D.: An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial system. J. Symbol. Comput. 49, 27–44 (2013)CrossRefGoogle Scholar
  26. 26.
    Leverenz, J., Xu, M., Wiecek, M.: Multiparametric optimization for multidisciplinary engineering design. J. Struct. Multidiscip. Optim. 54, 1–16 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Möller, H.M., Tenberg, R.: Multivariate polynomial system solving using intersections of eigenspaces. J. Symbol. Comput. 32, 513–531 (2001)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Montes, A.: A new algorithm for discussing Gröbner bases with parameters. J. Symbol. Comput. 33(1–2), 183–208 (2002)CrossRefGoogle Scholar
  29. 29.
    Narciso, D.: Developments in Nonlinear Multiparametric Programming and Control. PhD Thesis, London (2009)Google Scholar
  30. 30.
    Pistikopoulos, E., Georgiadis, M., Dua, V.: Multi-parametric Programming: Theory, Algorithms, and Applications, vol. 1. Wiley, Chichester (2007)CrossRefGoogle Scholar
  31. 31.
    Pistikopoulos, E., Georgiadis, M., Dua, V.: Multi-parametric Programming: Theory, Algorithms, and Applications, vol. 2. Wiley, Chichester (2007)CrossRefGoogle Scholar
  32. 32.
    Shaocheng, T.: Interval number and fuzzy number linear programmings. J. Fuzzy Sets Syst. 66(3), 301–306 (1994)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sturmfels, B.: Solving systems of polynomial equations. AMS (2002)Google Scholar
  34. 34.
    Sun, W., Yuan, Y.-X.: Optimization Theory and Methods: Non-linear Programming. Springer, Berlin (2006)Google Scholar
  35. 35.
    Weispfenning, V.: Comprehensive Gröbner bases. J. Symbol. Comput. 14(1), 1–29 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computer SciencesDamghan UniversityDamghanIran

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