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Solving Constrained Non-linear Integer and Mixed-Integer Global Optimization Problems Using Enhanced Directed Differential Evolution Algorithm

  • Ali Khater Mohamed
  • Ali Wagdy Mohamed
  • Ehab Zaki Elfeky
  • Mohamed Saleh
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 801)

Abstract

This paper proposes an enhanced modified Differential Evolution algorithm (MI-EDDE) to solve global constrained optimization problems that consist of mixed/non-linear integer variables. The MI-EDDE algorithm, which is based on the constraints violation, introduces a new mutation rule that sort all individuals ascendingly due to their constraint violations (cv) value and then the population is divided into three clusters (best, better and worst) with sizes 100p%, (NP-2) * 100p% and 100p% respectively. Where p is the proportion of the partition with respect to the total number of individuals in the population (NP). MI-EDDE selects three random individuals, one of each partition to implement the mutation process. This new mutation scheme shown to enhance the global and local search capabilities and increases the convergence speed. Eighteen test problems with different features are tested to evaluate the performance of MI-EDDE, and a comparison is made with four state-of-the-art evolutionary algorithms. The results show superiority of MI-EDDE to the four algorithms in terms of the quality, efficiency and robustness of the final solutions. Moreover, MI-EDDE shows a superior performance in solving two high dimensional problems and finding better solutions than the known optimal solution.

Keywords

Evolutionary computation Differential evolution Global optimization Novel mutation Handling constraints 

References

  1. 1.
    Mohamed, A.W., Sabry, H.Z.: Constrained optimization based on modified differential evolution algorithm. Inf. Sci. 194, 171–208 (2012)CrossRefGoogle Scholar
  2. 2.
    Mallipeddi, R., Suganthan, P.N., Pan, Q.K., Tasgetiren, M.F.: Differential evolution algorithm with ensemble of parameters and mutation strategies. Appl. Soft Comput. 11(2), 1679–1696 (2011)CrossRefGoogle Scholar
  3. 3.
    Hassanien, A.E., Alamry, E.: Swarm Intelligence: Principles, Advances, and Applications. CRC—Taylor & Francis Group, (2015). ISBN 9781498741064—CAT# K26721Google Scholar
  4. 4.
    Costa, L., Oliveira, P.: Evolutionary algorithms approach to the solution of mixed non-linear programming. Comput. Chem. Eng. 25, 257–266 (2001)CrossRefGoogle Scholar
  5. 5.
    Lin, Y.C., Hwang, K.S., Wang, F.S.: A mixed-coding scheme of evolutionary algorithms to solve mixed-integer nonlinear programming problems. Comput. Math. Appl. 47, 1295–1307 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sandgren, E.: Nonlinear integer and discrete programming in mechanical design. ASME Y. Mech. Des. 112, 223–229 (1990)CrossRefGoogle Scholar
  7. 7.
    Dua, V., Pistikopoulos, E.N.: Optimization techniques for process synthesis and material design under uncertainty. Chem. Eng. Res. Des. 76(3), 408–416 (1998)CrossRefGoogle Scholar
  8. 8.
    Catalão, J.P.S., Pousinho, H.M.I., Mendes, V.M.F.: Mixed-integer nonlinear approach for the optimal scheduling of a head-dependent hydro chain. Electr. Power Syst. Res. 80(8), 935–942 (2010)CrossRefGoogle Scholar
  9. 9.
    Garroppo, R.G., Giordano, S., Nencioni, G., Scutellà, M.G.: Mixed integer non-linear programming models for green network design. Comput. Oper. Res. 40(1), 273–281 (2013)Google Scholar
  10. 10.
    Maldonado, S., Pérez, J., Weber, R., Labbé, M.: Feature selection for support vector machines via mixed integer linear programming. Inf. Sci. 279(20), 163–175 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Çetinkaya, C., Karaoglan, I., Gökçen, H.: Two-stage vehicle routing problem with arc time windows: a mixed integer programming formulation and a heuristic approach. Eur. J. Oper. Res. 230(3), 539–550 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu, P., Whitaker, A., Pistikopoulos, E.N., Li, Z.: A mixed-integer programming approach to strategic planning of chemical centers: a case study in the UK. Comput. Chem. Eng. 35(8), 1359–1373 (2011)CrossRefGoogle Scholar
  13. 13.
    Xu, G., Papageorgiou, L.G.: A mixed integer optimization model for data classification. Comput. Ind. Eng. 56(4), 1205–1215 (2009)CrossRefGoogle Scholar
  14. 14.
    Grossmann, I.E., Sahinidis, N.V. (eds.): Special Issue On Mixed-Integer Programming And Its Application To Engineering, Part I: Optimization Engineering, vol. 3, no. 4. Kluwer Academic Publishers, Netherlands (2002)Google Scholar
  15. 15.
    Grossmann, I.E., Sahinidis, N.V. (eds.): Special Issue on Mixed-integer Programming and its Application to Engineering, Part II: Optimization Engineering, vol. 4, no. 1. Kluwer Academic Publishers, Netherlands (2002)Google Scholar
  16. 16.
    Hsieh Y.C., et al.: Solving nonlinear constrained optimization problems: an immune evolutionary based two-phase approach. Appl. Math. Model. (2015). http://dx.doi.org/10.1016/j.apm.2014.12.019
  17. 17.
    Ng, C.K., Zhang, L.S., Li, D., Tian, W.W.: Discrete filled function method for discrete global optimization. Comput. Optim. Appl. 31(1), 87–115 (2005)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gupta, O.K., Ravindran, A.: Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31(12), 1533–1546 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Borchers, B., Mitchell, J.E.: An improved branch and bound algorithm for mixed integer nonlinear programming. Comput. Oper. Res. 21, 359–367 (1994)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)MathSciNetCrossRefGoogle Scholar
  21. 21.
    DuranMA, G.I.: An outer approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36(3), 307–339 (1986)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fletcher, R., Leyffer, S.: Solving mixed-integer programs by outer approximation. Math. Program. 66(1–3), 327–349 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Quesada, I., Grossmann, I.E.: An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16(10–11), 937–947 (1992)CrossRefGoogle Scholar
  24. 24.
    Westerlund, T., Pettersson, F.: A cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, S131–S136 (1995)CrossRefGoogle Scholar
  25. 25.
    Lee, S., Grossmann, I.E.: New algorithms for nonlinear generalized disjunctive programming. Comput. Chem. Eng. 24, 2125–2142 (2000)CrossRefGoogle Scholar
  26. 26.
    Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5, 186–204 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Abhishek, K., Leyffer, S., Linderoth, J.T.: FilMINT: an outer-approximation-based solver for nonlinear mixed integer programs. INFORMS J. Comput. 22, 555–567 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Num. 22, 1–131 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Liberti, L., Cafieri, S., Tarissan, F.: Reformulations in mathematical programming: a computational approach. In: Abraham, A., Hassanien, A.E., Siarry, P. (eds.) Foundations on computational intelligence, studies in computational intelligence, vol. 203, pp. 153–234. Springer, New York (2009)Google Scholar
  30. 30.
    D’Ambrosio, C., Lodi, A.: Mixed integer nonlinear programming tools: a practical overview. 4OR 9(4), 329–349 (2011) (cit. on p. 13)Google Scholar
  31. 31.
    Trespalacios, F., Grossmann, I.E.: Review of mixed-integer nonlinear and generalized disjunctive programming methods. Chem. Ing. Tec. 86, 991–1012 (2014)CrossRefGoogle Scholar
  32. 32.
    Burer, S., Letchford, A.N.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012)MathSciNetGoogle Scholar
  33. 33.
    Grossmann, I.E.: Review of non-linear mixed integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Cardoso, M.F., Salcedo, R.L., Feyo de Azevedo, S., Barbosa, D.: A simulated annealing approach to the solution of minlp problems. Comput. Chem. Eng. 21(12), 1349–1364 (1997)CrossRefGoogle Scholar
  35. 35.
    Rosen, S.L., Harmonosky, C.M.: An improved simulated annealing simulation optimization method for discrete parameter stochastic systems. Comput. Oper. Res. 32, 343–358 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Glover, F.: Parametric tabu-search for mixed integer programs. Comput. Oper. Res. 33(9), 2449–2494 (2006)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Hua, Z., Huang, F.: A variable-grouping based genetic algorithm for large-scale integer programming. Inf. Sci. 176(19), 2869–2885 (2006)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Kesen, S.E., Das, S.K., Güngör, Z.: A genetic algorithm based heuristic for scheduling of virtual manufacturing cells (VMCs). Comput. Oper. Res. 37(6), 1148–1156 (2010)CrossRefGoogle Scholar
  39. 39.
    Turkkan, N.: Discrete optimization of structures using a floating-point genetic algorithm. In: Annual Conference of the Canadian Society for Civil Engineering, Moncton, Canada, 4–7 June 2003Google Scholar
  40. 40.
    Yokota, T., Gen, M., Li, Y.X.: Genetic algorithm for non-linear mixed integer programming problems and its applications. Comput. Ind. Eng. 30, 905–917 (1996)CrossRefGoogle Scholar
  41. 41.
    Wasanapradit, T., Mukdasanit, N., Chaiyaratana, N., Srinophakun, T.: Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean J. Chem. Eng. 28(1), 32–40 (2011)CrossRefGoogle Scholar
  42. 42.
    Deep, K., Singh, K.P., Kansal, M.L., Mohan, C.: A real coded genetic algorithm for solving integer and mixed integer optimization problems. Appl. Math. Comput. 212(2), 505–518 (2009)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Cai, J., Thierauf, G.: Evolution strategies for solving discrete optimization problems. Adv. Eng. Softw. 25, 177–183 (1996)CrossRefGoogle Scholar
  44. 44.
    Costa, L., Oliveira, P.: Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Comput. Chem. Eng. 25(2–3), 257–266Google Scholar
  45. 45.
    Cao, Y.J., Jiang, L., Wu, Q.H.: An evolutionary programming approach to mixed-variable optimization problems. App. Math. Model. 24, 931–942 (2000)CrossRefGoogle Scholar
  46. 46.
    Mohan, C., Nguyen, H.T.: A controlled random search technique incorporating the simulating annealing concept for solving integer and mixed integer global optimization problems. Comput. Optim. Appl. 14, 103–132 (1999)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Woon, S.F., Rehbock, V.: A critical review of discrete filled function methods in solving nonlinear discrete optimization problems. Appl. Math. Comput. 217(1), 25–41 (2010)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Yongjian, Y., Yumei, L.: A new discrete filled function algorithm for discrete global optimization. J. Comput. Appl. Math. 202(2), 280–291 (2007)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Socha, K.: ACO for continuous and Mixed-Variable Optimization. Ant Colony, Optimization and Swarm Intelligence. Springer, Berlin, Heidelberg, pp. 25–36 (2004)Google Scholar
  50. 50.
    Schlüter, M., Egea, J.A., Banga, J.R.: Extended ant colony optimization for non-convex mixed integer nonlinear programming. Comput. Oper. Res. 36(7), 2217–2229 (2009)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Yiqing, L., Xigang, Y., Yongjian, L.: An improved PSO algorithm for solving non-convex NLP/MINLP problems with equality constraints. Comput. Chem. Eng. 31(3), 153–162 (2007)CrossRefGoogle Scholar
  52. 52.
    Yue, T., Guan-zheng, T., Shu-guang, D.: Hybrid particle swarm optimization with chaotic search for solving integer and mixed integer programming problems. J. Cent. S. Univ. 21, 2731–2742 (2014)CrossRefGoogle Scholar
  53. 53.
    Gao, Y., Ren, Z., Gao, Y.: Modified differential evolution algorithm of constrained nonlinear mixed integer programming problems. Inf. Technol. J. 10(11), 2068–2075 (2011)CrossRefGoogle Scholar
  54. 54.
    Lin, Y.C., Hwang, K.S., Wang, F.S.: A mixed-coding scheme of evolutionary algorithms to solve mixed-integer nonlinear programming problems. Comput. Math. Appl. 47(8–9), 1295–1307 (2004)Google Scholar
  55. 55.
    Li, H., Zhang, L.: A discrete hybrid differential evolution algorithm for solving integer programming problems. Eng. Optim. 46(9), 1238–1268 (2014)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Mohamed, A.W.: An improved differential evolution algorithm with triangular mutation for global numerical optimization. Comput. Ind. Eng. 85, 359–375 (2015)CrossRefGoogle Scholar
  57. 57.
    Storn, R., Price, K.: Differential evolution—a simple and efficient adaptive scheme for global optimization over continuous spaces. Technical Report TR-95–012, ICSI http://www.icsi.berkeley.edu/~storn/litera.html (1995)
  58. 58.
    Storn, R., Price, K.: Differential Evolution- a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Engelbrecht, A.P.: Fundamentals of Computational Swarm Intelligence. John Wiley & Sons Ltd (2005)Google Scholar
  60. 60.
    Das, S., Suganthan, P.N.: Differential evolution: a survey of the state-of-the-art. IEEE Trans. Evol. Comput. 15(1), 4–31 (2011)CrossRefGoogle Scholar
  61. 61.
    Deb, K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186, 311–338 (2000)CrossRefGoogle Scholar
  62. 62.
    Venkatraman, S., Yen, G.G.: A generic framework for constrained optimization using genetic algorithms. IEEE Trans. Evol. Comput. 9(4), 424–435 (2005)CrossRefGoogle Scholar
  63. 63.
    Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: A Practical Approach to Global Optimization, 1st edn. Springer-Verlag, New York (2005)zbMATHGoogle Scholar
  64. 64.
    Fan, H.Y., Lampinen, J.: A trigonometric mutation operation to differential evolution. J. Glob. Optim. 27(1), 105–129 (2003)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Mohamed, A.W., Suganthan, P.N.: Real-parameter unconstrained optimization based on enhanced fitness-adaptive differential evolution algorithm with novel mutation. Soft. Comput. (2017).  https://doi.org/10.1007/s00500-017-2777-2
  66. 66.
    Mohamed, A.W., Sabry, H.Z., Farhat, A.: Advanced differential evolution algorithm for global numerical optimization. In: Proceedings of the IEEE International Conference on Computer Applications and Industrial Electronics (ICCAIE 2011), Penang, Malaysia, pp. 156–161 (2011)Google Scholar
  67. 67.
    Mohamed, A.W.: A novel differential evolution algorithm for solving constrained engineering optimization problems. J. Intell. Manuf. (2017)Google Scholar
  68. 68.
    Mohamed, A.W., Almazyad, A.S.: Differential evolution with novel mutation and adaptive crossover strategies for solving large scale global optimization problems. Appl. Comput. Intell. Soft Comput. 2017, 18 (2017).  https://doi.org/10.1155/2017/7974218CrossRefGoogle Scholar
  69. 69.
    Wang, Y., Cai, Z., Zhang, Q.: Differential evolution with composite trial vector generation strategies and control parameters. IEEE Trans. Evol. Comput. 15(1), 55–66 (2011)CrossRefGoogle Scholar
  70. 70.
    Zhang, J.Q., Sanderson, A.C.: JADE: adaptive differential evolution with optional external archive. IEEE Trans. Evol. Comput. 13(5), 945–958 (2009)CrossRefGoogle Scholar
  71. 71.
    Mohamed, A.W., Mohamed, A.K.: Adaptive guided differential evolution algorithm with novel mutation for numerical optimization. Int. J. Mach. Learn. Cyber. (2017).  https://doi.org/10.1007/s13042-017-0711-7
  72. 72.
    Mohamed A.K., Mohamed A.W., Elfeky E.Z., Saleh M. (2018) Enhancing AGDE algorithm using population size reduction for global numerical optimization. In: Hassanien, A., Tolba, M., Elhoseny, M., Mostafa, M. (eds.) The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2018). AMLTA 2018. Advances in Intelligent Systems and Computing, vol. 723. Springer, ChamGoogle Scholar
  73. 73.
    Mezura-Montes, E., Coello, C.A.C.: A simple multimembered evolution strategy to solve constrained optimization problems. IEEE Trans. Evol. Comput. 9(1), 1–17 (2005)CrossRefGoogle Scholar
  74. 74.
    Mohamed, A.W.: An efficient modified differential evolution algorithm for solving constrained non-linear integer and mixed-integer global optimization problems. Int. J. Mach. Learn. Cybernet. 8, 989 (2017)CrossRefGoogle Scholar
  75. 75.
    Lampinen, J., Zelinka, I.: Mixed integer-discrete-continuous optimization by differential evolution, part 1: the optimization method. In: Ošmera, P. (ed.) Proceedings of MENDEL’99, 5th International Mendel Conference on Soft Computing, Brno, Czech Republic, 9–12 June 1999Google Scholar
  76. 76.
    Omran, M.G.H., Engelbrecht, A.P.: Differential evolution for integer programming problems. In: 2007 IEEE Congress on Evolutionary Computation, pp. 2237–2242, Sept 2007Google Scholar
  77. 77.
    Li, Y., Gen, M.: Nonlinear mixed integer programming problems using genetic algorithm and penalty function. In: IEEE International Conference on Systems, Man, and Cybernetics, vol. 4, pp. 2677–2682 (1996)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ali Khater Mohamed
    • 1
  • Ali Wagdy Mohamed
    • 2
  • Ehab Zaki Elfeky
    • 3
  • Mohamed Saleh
    • 3
  1. 1.Department of Business AdministrationCollege of Sciences and Humanities, Majmaah UniversityMajmaahSaudi Arabia
  2. 2.Operations Research DepartmentInstitute of Statistical Studies and Research, Cairo UniversityGizaEgypt
  3. 3.Faculty of Computers and Information, Decision Support DepartmentCairo UniversityGizaEgypt

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