A Global Optimization Algorithm for the Solution of Tri-Level Mixed-Integer Quadratic Programming Problems

  • Styliani Avraamidou
  • Efstratios N. PistikopoulosEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


A novel algorithm for the global solution of a class of tri-level mixed-integer quadratic optimization problems containing both integer and continuous variables at all three optimization levels is presented. The class of problems we consider assumes that the quadratic terms in the objective function of the second level optimization problem do not contain any third level variables. To our knowledge, no other solution algorithm can tackle the class of problems considered in this work. Based on multi-parametric theory and our earlier results for tri-level linear programming problems, the main idea of the presented algorithm is to recast the lower levels of the tri-level optimization problem as multi-parametric programming problems, in which the optimization variables (continuous and integer) of all the upper level problems, are considered as parameters at the lower levels. The resulting parametric solutions are then substituted into the corresponding higher-level problems sequentially. Computational studies are presented to asses the efficiency and performance of the presented algorithm.


Tri-level optimization Multi-parametric programming Mixed-integer optimization 


  1. 1.
    Alguacil, N., Delgadillo, A., Arroyo, J.: A trilevel programming approach for electric grid defense planning. Comput. Oper. Res. 41(1), 282–290 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Avraamidou, S., Pistikopoulos, E.N.: B-POP: Bi-level parametric optimization toolbox. Comput. Chem. Eng. 122, 193–202 (2018)CrossRefGoogle Scholar
  3. 3.
    Avraamidou, S., Pistikopoulos, E.N.: A Multi-Parametric optimization approach for bilevel mixed-integer linear and quadratic programming problems. Comput. Chem. Eng. 122, 98–113 (2019)CrossRefGoogle Scholar
  4. 4.
    Avraamidou, S., Pistikopoulos, E.N.: Multi-parametric global optimization approach for tri-level mixed-integer linear optimization problems. J. Global Optim. (2018)Google Scholar
  5. 5.
    Blair, C.: The computational complexity of multi-level linear programs. Ann. Oper. Res. 34(1), 13–19 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brown, G., Carlyle, M., Salmerón, J., Wood, K.: Defending critical infrastructure. Interfaces 36(6), 530–544 (2006)CrossRefGoogle Scholar
  7. 7.
    Chen, B., Wang, J., Wang, L., He, Y., Wang, Z.: Robust optimization for transmission expansion planning: minimax cost vs. minimax regret. IEEE Trans. Power Syst. 29(6), 3069–3077 (2014)CrossRefGoogle Scholar
  8. 8.
    Dua, V., Bozinis, N., Pistikopoulos, E.: A multiparametric programming approach for mixed-integer quadratic engineering problems. Comput. Chem. Eng. 26(4–5), 715–733 (2002)CrossRefGoogle Scholar
  9. 9.
    Faisca, N.P., Saraiva, P.M., Rustem, B., Pistikopoulos, E.N.: A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems. Comput. Manag. Sci. 6, 377–397 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Han, J., Zhang, G., Hu, Y., Lu, J.: A solution to bi/tri-level programming problems using particle swarm optimization. Inf. Sci. 370–371, 519–537 (2016)CrossRefGoogle Scholar
  11. 11.
    Kassa, A., Kassa, S.: A branch-and-bound multi-parametric programming approach for non-convex multilevel optimization with polyhedral constraints. J. Global Optim. 64(4), 745–764 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Misener, R., Floudas, C.: Antigone: algorithms for continuous/integer global optimization of nonlinear equations. J. Global Optim. 59(2–3), 503–526 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Moreira, A., Street, A., Arroyo, J.: An adjustable robust optimization approach for contingency-constrained transmission expansion planning. IEEE Trans. Power Syst. 30(4), 2013–2022 (2015)CrossRefGoogle Scholar
  14. 14.
    Ning, C., You, F.: Data-driven adaptive nested robust optimization: general modeling framework and efficient computational algorithm for decision making under uncertainty. AIChE J. 63, 3790–3817 (2017)CrossRefGoogle Scholar
  15. 15.
    Oberdieck, R., Diangelakis, N., Nascu, I., Papathanasiou, M., Sun, M., Avraamidou, S., Pistikopoulos, E.: On multi-parametric programming and its applications in process systems engineering. Chem. Eng. Res. Design 116, 61–82 (2016)CrossRefGoogle Scholar
  16. 16.
    Oberdieck, R., Diangelakis, N., Papathanasiou, M., Nascu, I., Pistikopoulos, E.: Pop-parametric optimization toolbox. Ind. Eng. Chem. Res. 55(33), 8979–8991 (2016)CrossRefGoogle Scholar
  17. 17.
    Oberdieck, R., Pistikopoulos, E.: Explicit hybrid model-predictive control: the exact solution. Automatica 58, 152–159 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Oberdieck, R., Diangelakis, N.A., Avraamidou, S., Pistikopoulos, E.N.: On unbounded and binary parameters in multi-parametric programming: Applications to mixed-integer bilevel optimization and duality theory. J. Glob. Optim. 69(3), 587–606 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sahinidis, N.: BARON 17.8.9: Global Optimization of Mixed-Integer Nonlinear Programs, User’s ManualGoogle Scholar
  20. 20.
    Sakawa, M., Matsui, T.: Interactive fuzzy stochastic multi-level 0–1 programming using tabu search and probability maximization. Expert Syst. Appl. 41(6), 2957–2963 (2014)CrossRefGoogle Scholar
  21. 21.
    Sakawa, M., Nishizaki, I., Hitaka, M.: Interactive fuzzy programming for multi-level 0–1 programming problems through genetic algorithms. Eur. J. Oper. Res. 114(3), 580–588 (1999)CrossRefGoogle Scholar
  22. 22.
    Woldemariam, A., Kassa, S.: Systematic evolutionary algorithm for general multilevel Stackelberg problems with bounded decision variables (SEAMSP). Ann. Oper. Res. (2015)Google Scholar
  23. 23.
    Xu, X., Meng, Z., Shen, R.: A tri-level programming model based on conditional value-at-risk for three-stage supply chain management. Comput. Ind. Eng. 66(2), 470–475 (2013)CrossRefGoogle Scholar
  24. 24.
    Yao, Y., Edmunds, T., Papageorgiou, D., Alvarez, R.: Trilevel optimization in power network defense. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 37(4), 712–718 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Texas A&M Energy Institute, Texas A&M UniversityCollege StationUSA

Personalised recommendations