Journal of Heuristics

, Volume 22, Issue 2, pp 147–179 | Cite as

PSO-based and SA-based metaheuristics for bilinear programming problems: an application to the pooling problem



Bilinear programming problems (BLP) are subsets of nonconvex quadratic programs and can be classified as strongly NP-Hard. The exact methods to solve the BLPs are inefficient for large instances and only a few heuristic methods exist. In this study, we propose two metaheuristic methods, one is based on particle swarm optimization (PSO) and the other is based on simulated annealing (SA). Both of the proposed approaches take advantage of the bilinear structure of the problem. For the PSO-based method, a search variable, which is selected among the variable sets causing bilinearity, is subjected to particle swarm optimization. The SA-based procedure incorporates a variable neighborhood scheme. The pooling problem, which has several application areas in chemical industry and formulated as a BLP, is selected as a test bed to analyze the performances. Extensive experiments are conducted and they indicate the success of the proposed solution methods.


Bilinear programming Metaheuristics Pooling problem Particle swarm optimization Simulated annealing 


  1. Aarts, E., Korst, J., Michiels, W.: Simulated annealing. In: Search Methodologies, pp. 187–210. Springer (2005)Google Scholar
  2. Adhya, N., Tawarmalani, M., Sahinidis, N.V.: A lagrangian approach to the pooling problem. Ind. Eng. Chem. Res. 38(5), 1956–1972 (1999)CrossRefGoogle Scholar
  3. Al-Khayyal, F.A.: Linear, quadratic, and bilinear programming approaches to the linear complementarity problem. Eur. J. Oper. Res. 24(2), 216–227 (1986)MathSciNetCrossRefMATHGoogle Scholar
  4. Al-Khayyal, F.A.: Jointly constrained bilinear programs and related problems: An overview. Comput. Math. Appl. 19(11), 53–62 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. Al-Khayyal, F.A.: Generalized bilinear programming: Part i. models, applications and linear programming relaxation. Eur. J. Oper. Res. 60(3), 306–314 (1992)CrossRefMATHGoogle Scholar
  6. Alarie, S., Audet, C., Jaumard, B., Savard, G.: Concavity cuts for disjoint bilinear programming. Math. Program. 90(2), 373–398 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. Alfaki, M.: Models and solution methods for the pooling problem. Ph.D. thesis, The University of Bergen (2012)Google Scholar
  8. Alfaki, M., Haugland, D.: A cost minimization heuristic for the pooling problem. Ann. Oper. Res., 1–15 (2013a)Google Scholar
  9. Alfaki, M., Haugland, D.: Strong formulations for the pooling problem. J. Glob. Optim. 56(3), 897–916 (2013b)MathSciNetCrossRefMATHGoogle Scholar
  10. Ali, M.M., Törn, A., Viitanen, S.: A direct search variant of the simulated annealing algorithm for optimization involving continuous variables. Comput. Oper. Res. 29(1), 87–102 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. Almutairi, H., Elhedhli, S.: A new lagrangean approach to the pooling problem. J. Glob. Optim. 45(2), 237–257 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. Audet, C., Hansen, P., Jaumard, B., Savard, G.: A symmetrical linear maxmin approach to disjoint bilinear programming. Math. Program. 85(3), 573–592 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. Audet, C., Brimberg, J., Hansen, P., Digabel, S.L., Mladenović, N.: Pooling problem: Alternate formulations and solution methods. Manag. Sci. 50(6), 761–776 (2004)CrossRefMATHGoogle Scholar
  14. Audet, C., Hansen, P., Le Digabel, S.: Exact solution of three nonconvex quadratic programming problems. Springer, New York (2004)CrossRefMATHGoogle Scholar
  15. Ben-Tal, A., Eiger, G., Gershovitz, V.: Global minimization by reducing the duality gap. Math. Program. 63(1–3), 193–212 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. Bohachevsky, I.O., Johnson, M.E., Stein, M.L.: Generalized simulated annealing for function optimization. Technometrics 28(3), 209–217 (1986)CrossRefMATHGoogle Scholar
  17. Byrd, R.H., Nocedal, J., Waltz, R.A.: Knitro: an integrated package for nonlinear optimization. In: Large-Scale Nonlinear Optimization, pp. 35–39. Springer (2006)Google Scholar
  18. Černỳ, V.: Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. J. Optim. Theory Appl. 45(1), 41–51 (1985)MathSciNetCrossRefMATHGoogle Scholar
  19. Chatterjee, A., Siarry, P.: Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization. Comput. Oper. Res. 33(3), 859–871 (2006)CrossRefMATHGoogle Scholar
  20. Corana, A., Marchesi, M., Martini, C., Ridella, S.: Minimizing multimodal functions of continuous variables with the “simulated annealing” algorithm. ACM Trans. Math. Softw. 13(3), 262–280 (1987)MathSciNetCrossRefMATHGoogle Scholar
  21. Dekkers, A., Aarts, E.: Global optimization and simulated annealing. Math. Program. 50(1–3), 367–393 (1991)MathSciNetCrossRefMATHGoogle Scholar
  22. Eberhart, R.C., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, vol. 1, pp. 39–43. New York (1995)Google Scholar
  23. Eberhart, R.C., Shi, Y.: Comparison between genetic algorithms and particle swarm optimization. In: Evolutionary Programming VII, pp. 611–616. Springer (1998)Google Scholar
  24. Erbeyoğlu, G.: Metaheuristic approaches to the pooling problem. Master’s thesis, Boğaziçi University (2013)Google Scholar
  25. Evans, D.H.: Modular design—a special case in nonlinear programming. Oper. Res. 11(4), 637–647 (1963)CrossRefMATHGoogle Scholar
  26. Floudas, C.A., Aggarwal, A.: A decomposition strategy for global optimum search in the pooling problem. ORSA J. Comput. 2(3), 225–235 (1990)CrossRefMATHGoogle Scholar
  27. Foulds, L.R., Haugland, D., Jörnsten, K.: A bilinear approach to the pooling problem. Optimization 24(1–2), 165–180 (1992)MathSciNetCrossRefMATHGoogle Scholar
  28. Frimannslund, L., El Ghami, M., Alfaki, M., Haugland, D.: Solving the pooling problem with lmi relaxations. Models and Solution Methods for the Pooling Problem (2012)Google Scholar
  29. Gounaris, C.E., Misener, R., Floudas, C.A.: Computational comparison of piecewise-linear relaxations for pooling problems. Ind. Eng. Chem. Res. 48(12), 5742–5766 (2009)CrossRefGoogle Scholar
  30. Greenberg, H.J.: Analyzing the pooling problem. ORSA J. Comput. 7(2), 205–217 (1995)CrossRefMATHGoogle Scholar
  31. Haverly, C.A.: Studies of the behavior of recursion for the pooling problem. ACM SIGMAP Bull. 25, 19–28 (1978)CrossRefGoogle Scholar
  32. Henderson, D., Jacobson, S.H., Johnson, A.W.: The theory and practice of simulated annealing. In: Handbook of Metaheuristics, pp. 287–319. Springer (2003)Google Scholar
  33. Hollander, M., Wolfe, D.A.: Nonparametric Statistical Methods, 2nd edn. Wiley, New York (1999)MATHGoogle Scholar
  34. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of 1995 IEEE International Conference on Neural Networks, pp.1942–1948 (1995)Google Scholar
  35. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)MathSciNetCrossRefMATHGoogle Scholar
  36. Lasdon, L., Waren, A., Sarkar, S., Palacios, F.: Solving the pooling problem using generalized reduced gradient and successive linear programming algorithms. ACM Sigmap Bulletin 27, 9–15 (1979)CrossRefGoogle Scholar
  37. Liberti, L., Pantelides, C.C.: An exact reformulation algorithm for large nonconvex nlps involving bilinear terms. J. Glob. Optim. 36(2), 161–189 (2006)MathSciNetCrossRefMATHGoogle Scholar
  38. Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)MathSciNetCrossRefMATHGoogle Scholar
  39. Misener, R., Floudas, C.A.: Advances for the pooling problem: modeling, global optimization, and computational studies. Appl. Comput. Math. 8(1), 3–22 (2009)MathSciNetMATHGoogle Scholar
  40. Poli, R., Kennedy, J., Blackwell, T.: Particle swarm optimization. Swarm Intell. 1(1), 33–57 (2007)CrossRefGoogle Scholar
  41. Reeves, C.: Modern Heuristic Techniques for Combinatorial Problems. Advanced Topics in Computer Science Series. Halsted Press, Ultimo (1993)MATHGoogle Scholar
  42. Romeijn, H.E., Smith, R.L.: Simulated annealing for constrained global optimization. J. Glob. Optim. 5(2), 101–126 (1994)MathSciNetCrossRefMATHGoogle Scholar
  43. Schutte, J.F., Groenwold, A.A.: A study of global optimization using particle swarms. J. Glob. Optim. 31(1), 93–108 (2005)MathSciNetCrossRefMATHGoogle Scholar
  44. Sherali, H.D., Alameddine, A.: A new reformulation-linearization technique for bilinear programming problems. J. Glob. Optim. 2(4), 379–410 (1992)MathSciNetCrossRefMATHGoogle Scholar
  45. Shi, Y., Eberhart, R.C.: A modified particle swarm optimizer. In: The 1998 IEEE International Conference on Evolutionary Computation Proceedings, 1998. IEEE World Congress on Computational Intelligence, pp. 69–73. IEEE (1998)Google Scholar
  46. Su, Y., Geunes, J.: Multi-period price promotions in a single-supplier, multi-retailer supply chain under asymmetric demand information. Ann. Oper. Res. 211(1), 447–472 (2013)MathSciNetCrossRefMATHGoogle Scholar
  47. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, vol. 65. Springer (2002)Google Scholar
  48. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)MathSciNetCrossRefMATHGoogle Scholar
  49. Visweswaran, V., Floudast, C.: A global optimization algorithm (gop) for certain classes of nonconvex nlps - ii. application of theory and test problems. Comput. Chem. Eng. 14(12), 1419–1434 (1990)CrossRefGoogle Scholar
  50. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetCrossRefMATHGoogle Scholar
  51. Wicaksono, D.S., Karimi, I.: Piecewise milp under and overestimators for global optimization of bilinear programs. AIChE J. 54(4), 991–1008 (2008)CrossRefGoogle Scholar
  52. Zomaya, A.Y., Kazman, R.: Simulated annealing techniques. In: Algorithms and Theory of Computation Handbook, pp. 33–33. Chapman & Hall/CRC, Boca Raton (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Industrial EngineeringBoğaziçi UniversityIstanbulTurkey

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