Skip to main content
SpringerLink
Log in

Game Theory Based Evolutionary Algorithms: A Review with Nash Applications in Structural Engineering Optimization Problems

  • Original Paper
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

A general review of game-theory based evolutionary algorithms (EAs) is presented in this study. Nash equilibrium, Stackelberg game and Pareto optimality are considered, as game-theoretical basis of the evolutionary algorithm design, and also, as problems solved by evolutionary computation. Applications of game-theory based EAs in computational engineering are listed, with special emphasis in structural optimization and, particularly, in skeletal structures. Additionally, a set of three problems are solved: reconstruction inverse problem, fully stressed design problem and minimum constrained weight, for discrete sizing of frame skeletal structures. We compare panmictic EAs, Nash EAs using 4 different static domain decompositions, including also a new dynamic domain decomposition. Two frame structural test cases of 55 member size and 105 member size are evaluated with the linear stiffness matrix method. Numerical experiments show the efficiency of the Nash EAs approach, confirmed with statistical significance analysis of results, and enhanced with the dynamic domain decomposition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37

Similar content being viewed by others

References

  1. Adeli H, Cheng NT (1993) Integrated GA for optimization of space Structures. J Aerospace Eng 6(4):315–328

    Google Scholar 

  2. Adeli H, Cheng NT (1994) Augmented lagrangian genetic algorithm for structural optimization. J Aerospace Eng 7(1):104–118

    Google Scholar 

  3. Adeli H, Cheng NT (1994) Concurrent genetic algorithms for optimization of large structures. J Aerospace Eng 7(3):276–296

    Google Scholar 

  4. Adeli H, Kumar S (1995) Concurrent structural optimization on massively parallel supercomputer. J Struct Eng ASCE 121(11):1588–1597

    Google Scholar 

  5. Adeli H, Kumar S (1995) Distributed genetic algorithm for structural optimization. J Aerospace Eng 8(3):156–163

    Google Scholar 

  6. Alba E, Tomassini M (2002) Parallelism and evolutionary algorithms. IEEE T Evolut Comput 6(5):443–462

    Google Scholar 

  7. Alberdi R, Khandelwal K (2015) Comparison of robustness of metaheuristic algorithms for steel frame optimization. Eng Struct 102:40–60

    Google Scholar 

  8. Alemdar N, Sirakaya S (2003) On-line computation of Stackelberg equilibria with synchronous parallel genetic algorithms. J Econ Dyn Control 27(1503):1515

    MathSciNet  MATH  Google Scholar 

  9. Antonio LM, Coello Coello C (2015) A non-cooperative game for faster convergence in cooperative coevolution for multi-objective optimization. In: Proceedings of the IEEE C evol computat, pp 109–116

  10. Arias-Montano A, Coello Coello C, Mezura-Montes E (2012) Multiobjective evolutionary algorithms in aeronautical and aerospace engineering. IEEE T Evolut Comput 16(5):662–694

    MATH  Google Scholar 

  11. Argyris JH (1960) Energy theorems and structural analysis. Butterworth, London

    Google Scholar 

  12. Aubin JP (1979) Mathematical methods of game and economic theory. North-Holland Publishing Co., Amsterdam

    MATH  Google Scholar 

  13. Aumann R (1974) Subjectivity and correlation in randomized strategies. J Math Econ 1:67–96

    MathSciNet  MATH  Google Scholar 

  14. Bader J, Zitzler E (2011) HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol Comput 19(1):45–76

    Google Scholar 

  15. Barbosa HJC (1997) A coevolutionary genetic algorithm for a game approach to structural optimization. In: Back T (ed) Proceedings of the seventh international conference on genetic algorithms. Morgan Kaufmann Publishers, San Mateo, pp 545–552

    Google Scholar 

  16. Barbosa HJC (1999) A coevolutionary genetic algorithm for constrained optimization. In: Proceedings of the 1999 IEEE C evol computat, pp 1605–1611

  17. Barbosa H, Barreto AM (2001) An interactive genetic algorithm with co-evolution of weights for multiohjective prohlems. In: Spector L, Goodman ED, Wu A, Langdon W, Voigt HM, Gen M, Sen S, Dorigo M, Pezeshk S, Canon MH, Burke E (eds) Proceedings of the genetic and evolutionary computation conference (GECCO ‘2001). Morgan Kaufmann Publishers, San Francisco, pp 203–210

    Google Scholar 

  18. Bauso D (2016) Game theory with engineering applications. SIAM

  19. Bergmann G, Hommel G (1988) Improvements of general multiple test procedures for redundant systems of hypotheses. In: Bauer P, Hommel G, Sonnemann E (eds) Multiple hypotheses testing. Springer, New York, pp 100–115

  20. Beume N, Naujoks B, Emmerich M (2007) SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur J Oper Res 181(3):1653–1669

    MATH  Google Scholar 

  21. Boryczka U, Juszczuk P (2013) The differential evolution with the entropy based population size adjustment for the nash equilibria problem. In: Nguyen NT, Badica C, Jędrzejowicz P (eds) Computational collective intelligence technologies and applications—5th international conference. Springer, New York

    Google Scholar 

  22. Boussaid I, Lepagnot J, Siarry P (2013) A survey on optimization metaheuristics. Inf Sci 237:82–117

    MathSciNet  MATH  Google Scholar 

  23. Bränke J, Deb K, Miettinen K, Slowinski R (2008) Multiobjective optimization. Interactive and evolutionary approaches. Theoretical computer science and general issues series 5252. Springer, New York

  24. Burns SA (2002) Recent advances in optimal structural design. Institute of American Society of ASCE-SE

  25. Ceylan H, Bell M (2005) Genetic algorithm solution for the stochastic equilibrium transportation networks under congestion. Transp Res B Meth 39:169–185

    Google Scholar 

  26. Chen H, Wong KP, Nguyen DHM, Chung CY (2006) Analyzing oligopolistic electricity market using coevolutionary computation. IEEE T Power Syst 21(1):143–152

    Google Scholar 

  27. Chen A, Subprasom K, Ji Z (2006) A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem. Optim Eng 7:225–247

    MathSciNet  MATH  Google Scholar 

  28. Coelho RF (2013) Co-evolutionary optimization for multi-objective design under uncertainty. J Mech Des T ASME 135(2):1–8

  29. Coello Coello C (2006) Evolutionary multi-objective optimization: a historical view of the field. IEEE Comput Intell M 1:28–36

    Google Scholar 

  30. Coello Coello C, Sierra MR (2003) A coevolutionary multi-objective evolutionary algorithm. In: Sarker R, Reynolds R, Abbass H, Tan KC, McKay B, Essam D, Gedeon T (eds) Proceedings of IEEE C evol computat, pp 482–489

  31. Coello Coello C, Lamont G, Van Veldhuizen D (2007) Evolutionary algorithms for solving multi-objective problems. In: Goldberg D, Koza J (eds) Genetic and evolutionary computation series, 2nd edn. Springer, New York

    Google Scholar 

  32. Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153:235–256

    MathSciNet  MATH  Google Scholar 

  33. Conceicao Antonio CA, Torres-Marques A, Soeiro A (1995) Optimization of laminated composite structures using a bilevel strategy. Compos Struct 33:193–200

    Google Scholar 

  34. Clune J, Goings S, Punch B, Goodman E (2005) Investigations in meta-GAs: panaceas or pipe dreams? Proceeding GECCO ‘05 proceedings of the 2005 workshops on genetic and evolutionary computation, pp 235–241

  35. D’Amato E, Daniele E, Mallozzi L, Petrone G (2012) Equilibrium strategies via GA to Stackelberg games under multiple follower’s best reply. Int J Intell Syst 27:74–85

    Google Scholar 

  36. Darwin CR (1859) On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. John Murray, London

    Google Scholar 

  37. Deb K (1991) Optimal design of a welded beam via genetic algorithms. AIAA J 29(11):2013–2015

    Google Scholar 

  38. Deb K (2001) Multiobjective optimization using evolutionary algorithms. Wiley, New York

    MATH  Google Scholar 

  39. Deb K, Bandaru S, Greiner D, Gaspar-Cunha A, Celal-Tutum C (2014) An integrated approach to automated innovization for discovering useful design principles: case studies from engineering. Appl Soft Comput 15:42–56

    Google Scholar 

  40. Deb K, Pratap A, Agrawal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm NSGAII. IEEE T Evolut Comput 6(2):182–197

    Google Scholar 

  41. Desideri JA (2012) Cooperation and competition in multidisciplinary optimization application to the aero-structural aircraft wing shape optimization. Comput Optim Appl 52(1):29–68

    MathSciNet  MATH  Google Scholar 

  42. Dugardin F, Yalaoui F, Amodeo L (2010) New multi-objective method to solve reentrant hybrid flow shop scheduling problem. Eur J Oper Res 203(1):22–31

    MathSciNet  MATH  Google Scholar 

  43. Dumitrescu D, Lung RD, Gaskó N, Nagy R (2012) Equilibria detection in non-cooperative game theory—an evolutionary approach. Games ‘12 proceedings of the 4th world congress of the game theory society, pp 1–16

  44. Dumitrescu D, Lung RI, Gaskó N, Dan TM (2010) Evolutionary detection of Aumann equilibrium. GECCO ‘10 Proceedings of the 12th annual conference on genetic and evolutionary computation, pp 827–828

  45. Dumitrescu D, Lung RI, Gasko N (2010) An Evolutionary approach for detecting Aumann equilibrium in congestion games. In: Proceedings of the IEEE international symposium on computational intelligence and informatics, pp 43–46

  46. El Majd BA, Desideri JA, Habbal A (2010) Aerodynamic and structural optimization of a business-jet wingshape by a Nash game and an adapted split of variables. Mec Ind 11(3–4):209–214

    Google Scholar 

  47. Ficici SG, Pollack JB (2000) A game-theoretic approach to the simple coevolutionary algorithm. Parallel problem solving from nature—PPSN VI 6th international conference. Springer, New York

  48. Ficici SG, Pollack JB (2003) A game-theoretic memory mechanism for coevolution. Lect Notes Comput Sci 2723:286–297

    MATH  Google Scholar 

  49. Friedman M (1937) The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J Am Stat Assoc 32:675–701

    MATH  Google Scholar 

  50. Fudenberg D, Tirole J (1998) Game theory. MIT Press, Cambridge

    MATH  Google Scholar 

  51. Galante M (1993) Un Algoritmo genético simple para la optimización de estructuras planas articuladas. Rev Int Metod Numer 9(2):179–199

    Google Scholar 

  52. Galante M (1996) Genetic algorithms as an approach to optimise realworld trusses. Int J Numer Meth Eng 39:361–382

    MathSciNet  MATH  Google Scholar 

  53. Galván B, Greiner D, Periaux J, Sefrioui M, Winter G (2003) Parallel evolutionary computation for solving complex CFD optimization problems: a review and some nozzle applications. In: Matsuno K et al (eds) Parallel computational fluid dynamics: new frontiers and multi-disciplinary applications, North-Holland, pp 573–602

  54. García S, Herrera F (2008) An extension on “Statistical Comparisons of Classifiers over Multiple Data Sets” for all pairwise comparisons. J Mach Learn Res 9:2677–2694

    MATH  Google Scholar 

  55. Goldberg DE (1989) Genetic algorithms for search, optimisation, and machine learning. Addison-Wesley, Reading

    Google Scholar 

  56. Goldberg DE, Samtani MP (1986) Engineering optimization via genetic algorithm. In: Proceedings 9th conference on electronic computation ASCE, New York, pp 471–482

  57. Gonzalez LF, Srinivas K, Seop D, Lee C, Periaux J (2011) Coupling hybrid-game strategies with evolutionary algorithms for multi-objective design problems in aerospace. In: Evolutionary and deterministic methods for design, optimization and control with applications to industrial and societal problems, CIMNE, pp 221–248

  58. Gravel M, Nsakanda AL, Price W (1998) Efficient solutions to the cell-formation problem with multiple routings via a double-loop genetic algorithm. Eur J Oper Res 109:286–298

    MATH  Google Scholar 

  59. Grefenstette JJ (1986) Optimization of control parameters for genetic algorithms. IEEE T Syst Man Cyb 16(1):122–128

    Google Scholar 

  60. Greiner D, Emperador JM, Winter G (2000) Multiobjective optimisation of bar structures by Pareto-GA. European congress on computational methods in applied sciences and engineering ECCOMAS 2000, Barcelona, España

  61. Greiner D, Emperador JM, Winter G, Galván B (2007) Improving computacional mechanics optimum design using helper objectives: an application in frame bar structures. Lect Notes Comput Sci 4403:575–589

    Google Scholar 

  62. Greiner D, Emperador JM, Galvan B, Winter G, Periaux J (2014) Optimum structural design using bio-inspired search methods: a survey and applications. In: Becerra V, Vasile M (eds) Computational Intelligence in aerospace sciences, progress in aerospace sciences 244. American Institute of Aeronautics and Astronautics AIAA, pp 373–414

  63. Greiner D, Emperador JM, Galván B, Winter G (2014) A comparison of minimum constrained weight and fully stressed design problems in discrete cross-section type bar structures. In: Proceedings of the 11th world congress on computational mechanics (WCCM XI) & 5th European conference on computational mechanics (ECCM V), pp 2064–2072

  64. Greiner D, Emperador JM, Galván B, Winter G (2015) Comparing the fully stressed design and the minimum constrained weight solutions in truss structures. In: Magalhaes-Mendes J, Greiner D (eds) Evolutionary algorithms and metaheuristics in civil engineering and construction management. Comput Methods Appl Sci 39:17–34

  65. Greiner D, Galván B, Periaux J, Gauger N, Giannakoglou K, Winter G (2015) Advances in evolutionary and deterministic methods for design, optimization and control in engineering and sciences. Computational methods in applied sciences, vol. 36. Springer, New York

  66. Greiner D, Hajela P (2012) Truss topology optimization for mass and reliability considerations—co-evolutionary multiobjective formulations. Struct Multidiscip O 45:589–613

    MathSciNet  MATH  Google Scholar 

  67. Greiner D, Périaux J, Emperador JM, Galván B, Winter G (2013) A hybrid nash genetic algorithm for reconstruction inverse problems in structural engineering. Report of the Department of Mathematical Information Technology, Series B Scientific Computing, No B 5/2013, University of Jyväskylä, Finland

  68. Greiner D, Periaux J, Emperador JM, Galvan B, Winter G (2015) A study of Nash-evolutionary algorithms for reconstruction inverse problems in structural engineering. Greiner D et al (eds) Advances in evolutionary and deterministic methods for design, optimization and control in engineering and sciences. Computational methods in applied sciences, vol 36. Springer, New York, pp 321–333

  69. Greiner D, Winter G, Emperador JM (2001) Optimising frame structures by different strategies of GA. Finite Elem Anal Des 37(5):381–402

    MATH  Google Scholar 

  70. Greiner D, Winter G, Emperador JM (2004) Single and multi-objective frame optimization by evolutionary algorithms and the auto-adaptive rebirth operator. Comput Method Appl M 37(35):3711–3743

    MATH  Google Scholar 

  71. Greiner D, Winter G, Emperador JM, Galván B (2005) Gray coding in evolutionary multicriteria optimization: application in frame structural optimum desing. Lect Notes Comput Sci 3410:576–591

    MATH  Google Scholar 

  72. Habbal A, Petersson J, Thellner M (2004) Multidisciplinary topology optimization solved as a Nash game. Int J Numer Meth Eng 61:949–963

    MathSciNet  MATH  Google Scholar 

  73. Hajela P (1990) Genetic search—an approach to the nonconvex optimization problem. AIAA J 26(7):1205–1210

    Google Scholar 

  74. Hajela P, Lin CY (1992) Genetic search strategies in multicriterion optimal design. Struct Optim 4:99–107

    Google Scholar 

  75. Herskovits J, Leontiev A, Dias G, Santos G (2000) Contact shape optimization: a bilevel programming approach. Struct Multidiscip O 20:214–221

    Google Scholar 

  76. Hillis WD (1990) Co-evolving parasites improve simulated evolution as an optimization procedure. Phys D 42:228–234

    Google Scholar 

  77. Husbands P, Mill F (1991) Simulated co-evolution as the mechanism for emergent planning and scheduling. In: Proceedings of the 4th international conference on genetic algorithms, pp 264–270

  78. Husbands P (1994) Distributed coevolutionary genetic algorithms for multi-criteria and multi-constraint optimisation. Lect Notes Comput Sci Evol Comput 865:150–165

    Google Scholar 

  79. Iorio AW, Li X (2004) A cooperative coevolutionary multiobjective algorithm using non-dominated sorting. In: Proceedings of GECCO 2004, lecture notes in computer science, vol 3102, pp 537–548

  80. Jarosz P, Burczyński T (2011) Biologically-inspired methods and game theory in multi-criterion decision processes. In: Bouvry P et al (eds) Intelligent decision systems in large-scale distributed environments, series studies in computational intelligence, vol 362, pp 101–124

  81. Jenkins WM (1991) Structural optimization using genetic algorithms. Struct Engr Lond England 69(24):418–422

  82. Jensen M (2004) A new look at solving minimax problems with coevolutionary genetic algorithms. In: Resende M, Pinho de Sousa J (eds) Metaheuristics: computer decision-making. Kluwer Academic Publishers, Norwell, pp 369–384

  83. Kaveh A (2014) Advances in metaheuristic algorithms for optimal design of structures. Springer, New York

    MATH  Google Scholar 

  84. Kaveh A, Kalatjari V (2002) Genetic algorithm for discrete-sizing optimal design of trusses using the force method. Int J Numer Meth Eng 55:55–72

    MATH  Google Scholar 

  85. Kaveh A, Talatahari S (2009) A particle swarm ant colony optimization for truss structures with discrete variables. J Constr Steel Res 65(8):1558–1568

    Google Scholar 

  86. Kaveh A, Talatahari S (2009) Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Comput Struct 87(5–6):267–283

    Google Scholar 

  87. Kaveh A, Talatahari S (2012) Charged system search for optimal design of frame structures. Appl Soft Comput 12:382–393

    Google Scholar 

  88. Keerativuttitumrong N, Chaiyaratana N, Varavithy V (2002) Multi-objective co-operative co-evolutionary genetic algorithm. In: Proceedings of the parallel problem solving from nature VII conference (PPSN). Springer, New York, pp 288–297

  89. Kicinger R, Arciszewski T, De Jong K (2005) Evolutionary Computation and structural design: a survey of the state of the art. Comput Struct 83:1943–1978

    Google Scholar 

  90. Kobelev V (1993) On a game approach to optimal structural design. Struct Optim 6:194–199

    Google Scholar 

  91. Koh A (2011) Differential evolution based bi-level programming algorithm for computing normalized nash equilibrium. In: Gaspar-Cunha A et al (eds) Soft computing in industrial applications. Advances in intelligent and soft computing series, vol 96, pp 7–106

  92. Kostreva MM, Ogryczak W (1999) Linear optimization with multiple equitable criteria. RAIRO Oper Res 33(3):275–297

    MathSciNet  MATH  Google Scholar 

  93. Kostreva MM, Ogryczak W, Wierzbicki A (2004) Equitable aggregations and multiple criteria analysis. Eur J Oper Res 158(2):362–377

    MathSciNet  MATH  Google Scholar 

  94. Lagaros N, Papadrakakis M (2015) Engineering and applied sciences optimization. Computational methods in applied sciences 38. Springer, New York

  95. Lai CC, Doong SH (2004) An optimal material distribution system based on nested genetic algorithm. IEICE T Inf Syst E87D–3:780–784

    Google Scholar 

  96. Lee DS, Gonzalez F, Srinivas K, Periaux J (2009) Multifidelity Nash-Game strategies for reconstruction design in aerospace engineering problems. In: Proceedings of 13th Australian international aerospace conference (AIAC), Melbourne, Australia

  97. Lee DS, Gonzalez F, Periaux J, Srinivas K (2011) Efficient hybrid-game strategies coupled to evolutionary algorithms for robust multidisciplinary design optimization in aerospace engineering. IEEE T Evolut Comput 15(2):133–150

    Google Scholar 

  98. Lee DS, Periaux J, Gonzalez F (2009) UAS mission path planning system (MPPS) using hybrid-Game coupled to multiobjective optimizer (DETC2009-86749). In: Proceedings of 2009 design engineering technical conference and computers and information in engineering conference (ASME IDETC/CIE), San Diego, CA

  99. Léon ER, Pape AL, Costes M, Désidéri JA, Alfano D (2016) Concurrent aerodynamic optimization of rotor blades using a Nash game method. J Am Helicopter Soc 61(2):1–13

    Google Scholar 

  100. Leskinen J, Périaux J (2013) Distributed evolutionary optimization using Nash games and GPUs-applications to CFD design problems. Comput Fluids 80(1):190–201

    MathSciNet  MATH  Google Scholar 

  101. Leskinen J, Wang H, Périaux J (2013) Increasing parallelism of evolutionary algorithms by Nash games in design inverse flow problems. Eng Comput 30(4):581–600

    Google Scholar 

  102. Lewontin RC (1961) Evolution and the theory of games. J Theor Biol 1:382–403

    Google Scholar 

  103. Lewis K, Mistree F (1998) The other side of multidisciplinary design optimization: accommodating a multiobjective, uncertain and non-deterministic world. Eng Optimiz 31(2):161–189

    Google Scholar 

  104. Li H, Ma Y (2015) Discrete optimum design for truss structures by subset simulation algorithm. J Aerosp Eng ASCE 28(4):04014091

    Google Scholar 

  105. Li M, Yang S, Liu X (2015) Pareto or non-pareto: bi-criterion evolution in multi-objective optimization. IEEE T Evolut Comput. doi:10.1109/TEVC.2015.2504730

    Google Scholar 

  106. Li S, Zhang Y, Zhu Q (2005) Nash-optimization enhanced distributed model predictive control applied to the Shell benchmark problem. Inf Sci 170:329–349

    MathSciNet  MATH  Google Scholar 

  107. Li X, Lenaghan SC, Zhang M (2013) Evolutionary game based control for biological systems with applications in drug delivery. J Theor Biol 326:58–69

    MathSciNet  MATH  Google Scholar 

  108. Lin L, Yan F (2013) Nested DE based parameter estimation for multiple vortex ring microburst model. Measurement 46(3):1231–1236

    Google Scholar 

  109. Liu B (1998) Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms. Comput Math Appl 36(7):79–89

    MathSciNet  MATH  Google Scholar 

  110. Liu D, Toropov V, Querin O, Barton D (2011) Bilevel optimization of blended composite wing panels. J Aircr 48(1):107–118

    Google Scholar 

  111. Liu M, Burns SA (2003) Multiple fully stresses designs of steel frame structures with semi-rigid connections. Int J Numer Meth Eng 58:821–838

    MATH  Google Scholar 

  112. Livesley RK (1954) Matrix methods in structural analysis. Pergamon Press, New York

    MATH  Google Scholar 

  113. Longhua M, Yongling Z, Jixin Q (2001) A new hybrid genetic algorithm for global minimax optimization. Proceedings of international conferences on info-tech and info-net, ICII, vol 4, pp 316–322

  114. Loridan P, Morgan J (1989) A theoretical approximation scheme for Stackelberg games. Opt Theory Appl 61(1):95–110

    MATH  Google Scholar 

  115. Lung RI, Dumitrescu D (2008) Computing Nash equilibria by means of evolutionary computation. Int J Comput Commun 3:364–368

    Google Scholar 

  116. Mathieu R, Pittard L, Anandalingam G (1994) Genetic algorithm based approach to bi-level linear programming. Oper Res 28(1):1–21

    MathSciNet  MATH  Google Scholar 

  117. Maxwell JC (1872) On reciprocal figures, frames and diagrams of forces. T Roy Soc Edin 26, plates 1–3: 1–40

  118. Maynard Smith J (1974) The theory of games and the evolution of animal conflicts. J Theor Biol 47:209–221

    MathSciNet  Google Scholar 

  119. Maynard Smith J (1982) Evolution and the theory of games. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  120. Michell AGM (1904) The limits of economy in frame structures. Philos Mag, Section 6; 8(47):589–597

  121. Mirjalili S, Lewis A (2015) Novel performance metrics for robust multi-objective optimization algorithms. Swarm Evol Comput 21:1–23

    Google Scholar 

  122. Moghaddam A, Yalaoui F, Amodeo L (2011) Lorenz versus Pareto dominance in a single machine scheduling problem with rejection, EMO-2011. Lect Notes Comput Sci 6576:520–534

    Google Scholar 

  123. Mueller KM (2000) Sizing of members in the fully stressed design of frame structures. PhD thesis, Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, III

  124. Mueller KM, Burns S (2001) Fully stressed frame structures unobtainable by conventional design methodology. Int J Numer Meth Eng 52:1397–1409

    MATH  Google Scholar 

  125. Mueller KM, Liu M, Burns SA (2002) Fully stresses design of frame structures and multiple load paths. J Struct Eng ASCE 128(6):806–814

    Google Scholar 

  126. Murren P, Khandelwal K (2014) Design-driven harmony search (DDHS) in steel frame optimization. Eng Struct 59:798–808

    Google Scholar 

  127. Nagy R, Suciu M, Dumitrescu D (2012) Lorenz equilibrium: equitability in non cooperative games. Proceedings of the genetic and evolutionary conference GECCO

  128. Nair PB, Keane AJ (2002) Coevolutionary architecture for distributed optimization of complex coupled systems. AIAA J 40(7):1434–1443

    Google Scholar 

  129. Nash JF (1950) Equilibrium points in N-person games. Proc Natl Acad Sci 36:46–49

    MathSciNet  MATH  Google Scholar 

  130. Nash JF (1951) Non-cooperative games. Ann Math 54(2):286–295

    MathSciNet  MATH  Google Scholar 

  131. Obayashi S, Sasaki D (2003) Visualization and data mining of pareto solutions using self-organizing map. Lect Notes Comput Sci 2632:769–809

    MATH  Google Scholar 

  132. Ohsaki M (2006) Local and global searches of approximate optimal designs of regular frames. Int J Numer Meth Eng 67(1):132–147

    MathSciNet  MATH  Google Scholar 

  133. Panait L, Wiegand P, Luke S (2004) A visual demonstration of convergence properties of cooperative coevolution. Lect Notes Comput Sci 3242:892–901

    Google Scholar 

  134. Papadrakakis M, Lagaros N, Tsompanakis Y, Plevris V (2001) Large scale structural optimization: computational methods and optimization algorithms. Arch Comput Method E 8(3):239–301

    MathSciNet  MATH  Google Scholar 

  135. Paredis J (1995) Coevolutionary computation. Artif Life 2(4):355–379

    Google Scholar 

  136. Pareto V (1896) Cours D’ economie politique. Volume I y II, F. Rouge, Lausanne

  137. Parmee IC, Watson AH (1999) Preliminary airframe design using co-evolutionary multi objective genetic algorithms. In: Banzhaf W, Daida J, Eiben AE, Lon MGH, Honavar V, Jakiela M, Smith KE (eds) Proceedings of the genetic and evolutionary computation conference GECCO San Francisco, California, Morgan Kaufmann, vol 2, pp 1657–1665

  138. Pavlidis NG, Parsopoulos KE, Vrahatis MN (2005) Computing Nash equilibria through computational intelligence methods. J Comput Appl Math 175:113–136

    MathSciNet  MATH  Google Scholar 

  139. Periaux J, Chen HQ, Mantel B, Sefrioui M, Sui HT (2001) Combining game theory and genetic algorithms with application to DDM-nozzle optimization problems. Finite Elem Anal Des 37(5):417–429

    MATH  Google Scholar 

  140. Periaux J, Gonzalez F, Lee DS (2012) MOO methods for multidisciplinary design using parallel evolutionary algorithms, game theory and hierarchical topology: theoretical aspects (part 1), VKI lecture series on introduction to optimization and multidisciplinary design in aeronautics and turbomachinery

  141. Periaux J, González F, Lee DSC (2015) Evolutionary optimization and game strategies for advanced multi-disciplinary design. Intelligent systems, control and automation: science and engineering 75. Springer, New York

  142. Periaux J, Greiner D (2016) Efficient parallel Nash genetic algorithm for solving inverse problems in structural engineering. In: Neittaanmäki P et al (eds) Mathematical modeling and optimization of complex structures. Computational methods in applied sciences, vol. 40, Springer,  New York, pp 205–228

  143. Perny P, Spanjaard O, Storme LX (2006) A decision-theoretic approach to robust optimization in multivalued graphs. Ann Oper Res 147(1):317–341

    MathSciNet  MATH  Google Scholar 

  144. Pimpawat C, Chaiyaratana N (2001) Using a co-operative coevolutionary genetic algorithm to solve a three-dimensional container loading problem. In: IEEE congress on evolutionary computation, vol 2. Seoul, Korea, pp 1197–1204

  145. Potter M (1997) The design and analysis of a computational model of cooperative coevolution. PhD thesis, George Mason University, Fairfax, Virginia

  146. Potter M, De Jong KA (1994) A cooperative coevolutionary approach to function optimization. In: Proceedings from the 5th parallel problem solving from nature, Jerusalem, Israel. Springer, New York, pp 530–539

  147. Potter M, De Jong KA (2000) Cooperative coevolution: an architecture for evolving coadapted subcomponents. Evol Comput 8(1):1–29

    Google Scholar 

  148. Rajan S (1995) Sizing, shape and topology design optimization of trusses using genetic algorithms. J Struct Eng ASCE 121(10):1480–1487

    Google Scholar 

  149. Rajeev S, Krishnamoorthty CS (1992) Discrete optimization of structures using genetic algorithms. J Struct Eng ASCE 118(5):1233–1250

    Google Scholar 

  150. Rajeev S, Krishnamoorthy C (1997) Genetic algorithms-based methodologies for design optimization of trusses. J Struct Eng ASCE 123(3):350–358

    Google Scholar 

  151. Ramaswamy A, Ahlawat AS (2005) A review of recent advances in layout optimization of skeletal structures. In: Jagadish KS, Iyengar RN (eds) Recent advances in structural engineering. University Press, Hyderabad, pp 56–89

    Google Scholar 

  152. Rao SS (1987) Game theory approach for multiobjective structural optimization. Comput Struct 25:119–127

    MathSciNet  MATH  Google Scholar 

  153. Rao SS, Venkayya VB, Khot NS (1988) Game theory approach for the integrated design of structures and controls. AIAA J 26(4):463–469

    MathSciNet  MATH  Google Scholar 

  154. Redmond J, Parker G (1996) Actuator placement based on reachable set optimization for expected disturbance. J Optimiz Theory App 90(2):279–300

    MathSciNet  MATH  Google Scholar 

  155. Riechmann T (2001) Genetic algorithm learning and evolutionary games. J Econ Dyn Control 25:1019–1037

    MATH  Google Scholar 

  156. Rosin C, Belew K (1997) New methods for competitive coevolution. Evol Comput 5(1):1–29

    Google Scholar 

  157. Saborido R, Ruiz AB, Luque M (2016) Global WASF-GA: an evolutionary algorithm in multiobjective optimization to approximate the whole pareto optimal front. Evol Comput. doi:10.1162/EVCO_a_00175

    Google Scholar 

  158. Saka MP, Geem ZW (2013) Mathematical and metaheuristic applications in design optimization of steel frame structures: an extensive review. Math Probl Eng Article ID 271031(2013):1–33

    MATH  Google Scholar 

  159. Schoen M, Hoover R, Chinvorarat S, Sinchai S, Schoen G (2009) System identification and robust controller design using genetic algorithms for flexible space structures. J Dyn Syst T ASME 131(3):1–11

    Google Scholar 

  160. Sefrioui M (1998) Algorithmes Evolutionnaires pour le calcul scientifique. Application l’electromagnetisme et la mécanique des fluides numériques. PhD thesis, Université Pierre et Marie Curie, Paris

  161. Sefrioui M, Periaux J (2000) Nash genetic algorithms: examples and applications. In: Proceedings of the IEEE congress on evolutionary computation, pp 509–516

  162. Sefrioui M, Periaux J (2001) Nash genetic algorithms: examples and applications. In: Joly P, Pironneau O, Oñate E, Periaux J (eds) Innovative tools for scientific computation in aeronautical engineering. CIMNE, Barcelona, pp 390–404

    Google Scholar 

  163. Segura C, Coello Coello C, Miranda-Valladares G, Leon C (2013) Using multi-objective evolutionary algorithms for single-objective optimization. 4OR-Q J Oper Res 11(3):201–228

    MathSciNet  MATH  Google Scholar 

  164. Shaffer JD (1984) Sorne experirnents in rnachine leaming using vector evaluated genetic algorithms, Ph.D. Thesis, Nashville, Vanderhilt University

  165. Sim KB, Kim JY, Lee DW (2004) Game theory based co-evolutionary algorithm (GCEA) for solving multiobjective optimization problems. IEICE T Inf Syst E87D-10:2419–2425

    Google Scholar 

  166. Sim KB, Lee DW, Kim JY (2004) Game theory based coevolutionary algorithm: a new computational coevolutionary approach. Int J Control Autom 2(4):463–474

    Google Scholar 

  167. Sinha A, Malo P, Deb K (2013) Efficient evolutionary algorithm for single-objective bilevel optimization. arXiv:1303.3901

  168. Sinha A, Malo P, Frantsev A, Deb K (2014) Finding optimal strategies in a multi-period multi-leader-follower Stackelberg game using an evolutionary algorithm. Comput Oper Res 41:374–385

    MathSciNet  MATH  Google Scholar 

  169. Sinha A, Malo P, Deb K, Korhonen P, Wallenius J (2016) Solving bilevel multicriterion optimization problems with lower level decision uncertainty. IEEE T Evolut Comput 20(2):199–217

    Google Scholar 

  170. Son YS, Baldick R (2004) Hybrid coevolutionary programming for Nash equilibrium search in games with local optima. IEEE T Evolut Comput 8(4):305–315

    Google Scholar 

  171. Stolpe M (2016) Truss optimization with discrete design variables: a critical review. Struct Multidiscip O 53:349–374

    MathSciNet  Google Scholar 

  172. Talbi EG (2013) Metaheuristics for bi-level optimization. Springer, New York

    MATH  Google Scholar 

  173. Talbi EG (2013) A taxonomy of metaheuristics for bilevel optimization. In: Talbi EG (ed) Metaheuristics for bilevel optimization. Studies in computational intelligence series, vol 482, pp 1–39

  174. Tang Z, Bai W, Dong J (2008) Distributed optimization using virtual and real game strategies for multi-criterion aerodynamic design. Sci China Ser E Technol Sci 51(11):1939–1956

    MathSciNet  MATH  Google Scholar 

  175. Tang Z, Desideri JA, Periaux J (2002) Distributed optimization using virtual and real game strategies for aerodynamic design. INRIA research report RR-4543, inria-00072045

  176. Tang Z, Desideri JA, Periaux J (2007) Multi-criteria aerodynamic shape design optimization and inverse problems using control theory and Nash games. J Optimiz Theory App 135(1):599–622

    MATH  Google Scholar 

  177. Tang Z, Périaux J, Desidéri JA (2005) Multicriteria robust design using adjoint methods and game strategies for solving drag optimization problems with uncertainties. In: Proceedings of east west high speed flow fields conference, Beijing, China, pp 487–493

  178. Tang Z, Periaux J, Dong J (2014) Constraints handling in Nash/Adjoint optimization methods for multi-objective aerodynamic design. Comput Method Appl M 271:130–143

    MathSciNet  MATH  Google Scholar 

  179. Turner MJ, Clough RW, Martin HC, Topp LJ (1956) Stiffness and deflection analysis of complex structures. J Aeronaut Sci 23(9):805–823

    MATH  Google Scholar 

  180. Vallée T, Basar T (1999) Off-line computation of Stackelberg solutions with the genetic algorithm. Comput Econ 13:201–209

    MATH  Google Scholar 

  181. Venugopal V, Narendran TT (1992) A genetic algorithm approach to the machine-component grouping problem with multiple objectives. Comput Ind Eng 22(4):469–480

    Google Scholar 

  182. Vincent TL (1983) Game theory as a design tool. J Mech Transm T ASME 105:165–170

    Google Scholar 

  183. Vincent TL, Brown JS (2005) Evolutionary game theory, natural selection, and darwinian dynamics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  184. Von Neumann J, Morgenstern O (1944) The theory of games and economic behavior. Princeton University Press, Princeton

    MATH  Google Scholar 

  185. Von Stackelberg HF (1934) Marktform und Gleichgewicht. Wien & Berlin: Springer VI, 138 S.8

  186. Wang G, Dexter T, Punch W, Goodman ED (1996) Optimization of a GA within a GA for a 2-dimensional layout problem. Proceedings, 1st international conference on evolutionary computation and its applications. Presidium, Russian Academy of Sciences, pp 18–29

  187. Wang G, Goodman ED, Punch WF (1997) On the optimization of a class of blackbox optimization algorithms. In: Proceedings of IEEE international conference on tools for artificial intelligence

  188. Wang G, Wan Z, Wang X, Ly Y (2008) Genetic algorithm based on simplex method for solving linear-quadratic bilevel programming problem. Comput Math Appl 56(10):2550–2555

    MathSciNet  MATH  Google Scholar 

  189. Wang JF (2001) Optimisation Distribuée Multicritère par Algorithmes Génétiques et Théorie des Jeux and Application à la Simulation Numérique de Problèmes d’Hypersustentation en Aérodynamique. PhD thesis, University of Paris 6, Spéc.: Math. App

  190. Wang JF, Periaux J (2001) Computational fluid dynamics for the 21st century, notes on numerical fluid mechanics (NNFM), genetic algorithms and game theory for high lift multi-airfoil design problems in aerodynamics, vol 78, pp 192–207

  191. Weinberg R (1970) Computer simulation of a living cell. Doctoral dissertation vol 31(9), p 5312B. University of Michigan, Ann Arbor (University Microfilms N 71–4766)

  192. Whitley D, Rana S, Heckendorn R (1997) Representation issues in neighborhood search and evolutionary algorithms. In: Quagliarella D, Périaux J, Poloni C, Winter G (eds) Genetic algorithms and evolution strategies in engineering and computer science. Wiley, New York, pp 39–57

    Google Scholar 

  193. Wiegand P (2004) An analysis of cooperative coevolutionary algorithms. phD thesis, George Mason University, USA

  194. Wu Y, Jiang S (2012) Parameters selection method of multiple vortex-ring microburst model based on nested particle swarm optimization. Chin J Electron 40(1):204–208

    Google Scholar 

  195. Yin Y (2000) Genetic algorithm based approach for bilevel programming models. J Transp Eng ASCE 126(2):115–120

    MathSciNet  Google Scholar 

  196. Yu F, Tu F, Pattipati KR (2006) A novel congruent organizational design methodology using group technology and a nested genetic algorithm. IEEE Trans Syst Man Cybern A 36(1):5–18

    Google Scholar 

  197. Zabala G, Nebro A, Luna F, Coello Coello C (2014) A survey of multi-objective metaheuristics applied to structural optimization. Struct Multidiscip O 49:537–558

    MathSciNet  Google Scholar 

  198. Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE T Evolut Comput 11(6):712–731

    Google Scholar 

  199. Zhang YQ, Zheng JB, Gao H, Zeng YH (2001) Nested genetic algorithm for resolving overlapping spectra. Fresen J Anal Chem 371(3):317–322

    Google Scholar 

  200. Zhong W, Su R, Gui L, Fan Z (2016) Topology and sizing optimization of discrete structures using a cooperative coevolutionary genetic algorithm with independent ground structures. Eng Optimiz 48(6):911–932

    Google Scholar 

  201. Zitzler E, Laumanns M, Thiele L (2002) SPEA2: improving the strength pareto evolutionary algorithm for multiobjective optimization. In: Evolutionary methods for design, optimization and control with applications to industrial problems (EUROGEN 2001), Athens, Greece, CIMNE

Download references

Acknowledgments

This research work has been supported through contract CAS12/00400 José Castillejo by Ministerio de Educación, Cultura y Deporte of the Government of Spain and through Ministerio de Economía y Competitivad and FEDER, Grant Contract CTM2014-55014-C3-1-R. The first author also gratefully acknowledges support given at the Mathematical Information Technology Department, University of Jyväskylä (Finland), and in particular to Prof. Pekka Neittaanmäki, during 2016. The second author is grateful to Dr. Dong Seop Chris Lee during his post-doctoral visit at CIMNE/UPC in 2009–2013, and to Prof. Felipe González, for their numerous fruitful discussions on Hybridized Nash EAs; also acknowledgements are given to Prof. Rafael Montenegro and Prof. Blas Galván for their support when visiting CEANI/SIANI (ULPGC) in 2015.

Funding

This study was funded by Ministerio de Educación, Cultura y Deporte of Spain (Grant Contract CAS12/00400) and by Ministerio de Economía y Competitividad and FEDER (Grant Contract CTM2014-55014-C3-1-R).

Author information

Authors and Affiliations

  1. Institute of Intelligent Systems and Numerical Applications in Engineering (SIANI), Universidad de Las Palmas de Gran Canaria (ULPGC), 35017, Las Palmas de Gran Canaria, Spain

    David Greiner, Jose M. Emperador, Blas Galván & Gabriel Winter

  2. Centre Internacional de Metodes Numerics en Enginyeria (CIMNE), Barcelona, Spain

    Jacques Periaux

  3. University of Jyväskylä, Seminaarinkatu 15, 40014, Jyväskylän yliopisto, Finland

    Jacques Periaux

Authors
  1. David Greiner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacques Periaux
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jose M. Emperador
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Blas Galván
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Gabriel Winter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Greiner.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Greiner, D., Periaux, J., Emperador, J.M. et al. Game Theory Based Evolutionary Algorithms: A Review with Nash Applications in Structural Engineering Optimization Problems. Arch Computat Methods Eng 24, 703–750 (2017). https://doi.org/10.1007/s11831-016-9187-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-016-9187-y

Keywords

Search

Navigation