Old and New Non-Gradient Methods in Engineering Optimization

  • J. Blachut
Part of the International Centre for Mechanical Sciences book series (CISM, volume 425)


Current interest in non-gradient optimization can be attributed to two, distinct developments. The first stems from studying natural processes and the second from fast progress in the computing environment. The first has resulted in new optimization techniques like simulated annealing, tabu search and genetic algorithms whilst the second has made many of the existing zero order methods computationally affordable. This chapter discusses simulated annealing, tabu search, dynamic programming, random methods and other heuristics. A variety of illustrative examples is provided together with practical examples drawn from structural mechanics.


Cost Function Design Variable Simulated Annealing Dynamic Programming Design Space 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aarts, E., and Korst, J. (1989). Simulated Annealing and Boltzman Machines. Chichester - New York: John Wiley and Sons. 272 p.Google Scholar
  2. Aarts, E., and Lenstra, J.K., eds, (1998). Local Search in Combinatorial Optimization. Chichester - New York: John Wiley and Sons. 512 p.Google Scholar
  3. ABAQUS: Hibbit, Karlsson and Sorensen Inc., (1998). ABAQUS. User’s and Theory Manual,Version 5.8, Pawtucket, RI 02860–4847, USA.Google Scholar
  4. Andreev, L.V., and Dyachenko, V.E. (1977). Application of dynamic programming to studying the stability of shells of revolution with finite displacements. Soviet Aeronautics 20: 1–6.ADSGoogle Scholar
  5. Arora, J.S., Elwakeil, O.A., and Chahande, A.I. (1995). Global optimization methods for engineering applications: a review. Structural Optimization 9: 137–159.CrossRefGoogle Scholar
  6. Balling, R.J. (1991). Optimal steel frame design by simulated annealing. Journal of Structural Engineering 117: 1780–1795.CrossRefGoogle Scholar
  7. Balling, R.J. (1996). Application of the simulated annealing algorithm to structural design. In Grierson, D.E., and Hajela, P., eds., Emergent Computing Methods in Engineering Design. Berlin: Springer. 283–293.CrossRefGoogle Scholar
  8. Bellman, R.E. (1957). Dynamic Programming. Princeton: Princeton University Press. New Jersey. Bellman, R.E., and Dreyfuss, S.E. (1962). Applied Dynamic Programming. Princeton: Princeton University Press. New Jersey.Google Scholar
  9. Bennage, W.A., and Dhingra, A.K. (1995a). Single and multiobjective structural optimization in discrete-continuous variables using simulated annealing. International Journal for Numerical Methods in Engineering 38: 2753–2773.CrossRefMATHADSGoogle Scholar
  10. Bennage, W.A., and Dhingra A.K. (1995b). Optimization of truss topology using tabu search. International Journal for Numerical Methods in Engineering 38: 4035–4052.CrossRefMATHADSGoogle Scholar
  11. Blachut, J. (1977). Optimal design of a compressed rod with large deflections by means of dynamic programming. Mechanika Teoretyczna i Stosowana 15: 375–385.MathSciNetMATHGoogle Scholar
  12. Blachut, J. (1988). Search for optimal torispherical end closures under buckling constraints. International Journal of Mechanical Sciences 31: 623–633.CrossRefGoogle Scholar
  13. Blachut, J. (1991). Shape optimization of FRP dome closures under buckling constraints. In Eschenauer, H.A., Matteck, C., and Olhoff, N., eds., Engineering Optimization in Design Processes. Berlin: Springer-Verlag. 155–164.CrossRefGoogle Scholar
  14. Blachut, J. (1992). Influence of meridional shaping on the collapse strength of FRP domes. Engineering Optimization 19: 65–80.CrossRefGoogle Scholar
  15. Blachut, J. (1997). Minimum weight of internally pressurised domes subject to plastic load failure. Thin-Walled Structures 27: 127–146.CrossRefGoogle Scholar
  16. Blachut, J., and Wang, P. (1999). On the performance of barrel-shaped composite shells subjected to hydrostatic pressure. In Toropov, V.V., ed., Engineering Design Optimization. Bradford: MCB University Press. 59–65.Google Scholar
  17. Blachut, J., and Wang, P. (2000). Buckling of barreled shells subjected to external hydrostatic pressure. In Deardorff, A.F., ed., PVP- Vol. 407 Pressure Vessel and Piping Codes and Standards. New York: ASME. 107–114.Google Scholar
  18. Bland, J.A. (1994). A tabu search approach to engineering optimisation. In Rzevsky, G., Adey, R.A., and Russell, D.W., eds., Application of Artificial Intelligence in Engineering IX. Southampton: Computational Mechanics Publ Limited. 423–430.Google Scholar
  19. Bland, J.A. (1995). Discrete-variable optimal structural design using tabu search. Structural Optimization 10: 87–93.Google Scholar
  20. Bland, J.A. (1998). Structural design optimization with reliability constraints using tabu search. Engineering Optimization 30: 55–74.CrossRefGoogle Scholar
  21. Bohachewsky, I.O., Johnson, M.E., and Stein, M.L. (1986). Generalized simulated annealing for function optimization. Technometrics 28: 209–217.CrossRefGoogle Scholar
  22. Botello, S., Marroquin, J.L., Onate, E., and van Horebeek, J. (1999). Solving structural optimization problems with genetic algorithms and simulated annealing: Éii/ernational Journal for Numerical Methods in Engineering 45: 1069–1084.MATHADSGoogle Scholar
  23. Box, M.J. (1965). A new method of constrained optimization and a comparison with other methods. Computer Journal 8: 42–52.MathSciNetCrossRefMATHGoogle Scholar
  24. Bushnell, D. (1976). Bosor 5–program for buckling of elastic-plastic complex shells of revolution including large deflections and creep. Computers and Structures 6: 221–239.CrossRefMATHGoogle Scholar
  25. Bushnell, D. (1977). Bosor4 program for stress, buckling, and vibration of complex shells of revolution. In Structural Mechanics Software Series. Charlottesville: University Press of Virginia. VA. 11–143.Google Scholar
  26. Ceranic, B., Fryer, C., and Baines, R.W. (2000). An application of simulated annealing to the optimum design of reinforced concrete retaining structures. Computers and Structures, to appear.Google Scholar
  27. Chen, G-S., Bruno, R.J., and Salama, M. (1991). Optimal placement of active/passive members in truss structures using simulated annealing. American Institute of Aeronautics and Astronautics Journal 29: 1327–1334.CrossRefGoogle Scholar
  28. Connor, A.M., Seffen, K.A., Parks, G.T., and Clarkson, P.J. (1999). Efficient optimisation of structures using tabu search. In Toropov, V.V., ed., Engineering Design Optimization. Bradford: MCB University Press. 127–133.Google Scholar
  29. Corana, A., Marchesi, M., Martini, C., and Ridella, S. (1987). Minimizing multimodal functions of continuous variables with the simulated annealing algorithm. ACM Transactions on Mathematical Software 13: 262–280.MathSciNetCrossRefMATHGoogle Scholar
  30. Davis, L., ed., 1991. Handbook of Genetic Algorithms. New York: Van Nostrand Reinhold.Google Scholar
  31. Distefano, N. (1975). Dynamic programming and the optimization of two-point boundary value systems. Journal of Mathematical Analysis and Applications 52: 142–150.MathSciNetCrossRefMATHGoogle Scholar
  32. Distefano, N., and Rath, A. (1975). A dynamic programming approach to the optimization of elastic trusses. Journal of Optimization Theory and Applications 15: 13–26.MathSciNetCrossRefMATHGoogle Scholar
  33. Gero, J.S., Sheehan, P.J., and Becker, J.M. (1978). Building design using feedforward non-serial dynamic programming. Engineering Optimization 3: 183–192.CrossRefGoogle Scholar
  34. Gero, J.S., and Kaneshalingam, K. (1978). A method for the optimum design of traditional formwork. Engineering Optimization 3: 249–251.CrossRefGoogle Scholar
  35. Gill, P.E., Murray, W., and Wright, M.H. (1999). Practical Optimization. London: Academic Press. 401 p.Google Scholar
  36. Glass, C.A., and Potts, C.N. (1996). A comparison of local search methods for flow shop scheduling. Annals of Operations Research 63: 489–509.CrossRefMATHGoogle Scholar
  37. Glover, F. (1977). Heuristic search–an approach to the nonconvex optimization problem. Decision Sci. 8: 156–166.CrossRefGoogle Scholar
  38. Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Reading: Addison-Wesley. Massachussets.Google Scholar
  39. Hajela, P., and Lee, E. (1995). Genetic algorithms in truss topological optimization. International Journal of Solids and Structures 32: 3341–3357.MathSciNetCrossRefMATHGoogle Scholar
  40. Hajela, P. (1999). Nongradient methods in multidisciplinary design optimization–status and potential. Journal of Aircraft 36: 255–265.CrossRefGoogle Scholar
  41. Hamada, M., Seguchi, Y., and Tada, Y. (1981). An optimal design of beam with variable cross section. Bulletin of the Japanese Society of Mechanical Engineers 24: 621–627.CrossRefGoogle Scholar
  42. Haupt, R.L., and Haupt, S.E. (1998). Practical Genetic Algorithms. New York: John Wiley. 177 p.MATHGoogle Scholar
  43. Holland, J.H. (1975). Adaptation in Natural and Artificial Systems. Ann Arbor: University of Michigan Press. Michigan.Google Scholar
  44. Howell, G.C., and Doyle, W.S. (1978). Dynamic programming and direct iteration for the optimum design of skeletal towers. Computers and Structures 9: 621–627.CrossRefMATHGoogle Scholar
  45. Hu, N. (1992). Tabu search method with random moves for globally optimal design. International Journal for Numerical Methods in Engineering 35: 1055–1070.CrossRefADSGoogle Scholar
  46. Jendo, S., Marks, W., and Thierauf, G. (1985). Multicriteria optimization in optimum structural design. Large Scale Systems 9: 141–150.MATHGoogle Scholar
  47. Jenkins, W.M. (1991). Towards structural optimization via genetic algorithm. Computers and Structures 40: 1321–1327.CrossRefMATHGoogle Scholar
  48. Kincaid, R.K. (1990). Minimizing distortion and internal forces in truss structures by simulated annealing. In Proceedings of 31st AIAA/ASME/ASCE/AHS/ASC Structural Materials and Dynamics Conference, Long Beach: AIAA. USA. 327–333.Google Scholar
  49. Kincaid, R.K. (1991). Minimizing distortion in truss structures: a comparison of simulated annealing and tabu search. In Proceedings of 32nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Baltimore: AIAA. USA. 424–430.Google Scholar
  50. Kirkpatrick, S., Gelatt Jr, C.D., and Vecchi, M.P. (1983). Optimization by simulated annealing. Science 220: 671–680.MathSciNetCrossRefMATHADSGoogle Scholar
  51. Kneppe, G., and Sattler, H.J. (1982). Modifikation des Boxschen Complex-Verfahrens. Zeitschrift fuer Angewandte Mathematik und Mechanik T 371–373.Google Scholar
  52. Laarhoven van, P.J.M., and Aarts, E.H.L. (1992). Simulated Annealing - Theory and Applications. Dordrecht: Kluwer Academic Publishers. 187 p.Google Scholar
  53. Leite, J.P.B., and Topping, B.H.V. (1999). Parallel simulated annealing for structural optimization. Computers and Structures 73: 545–564.CrossRefMATHGoogle Scholar
  54. Li, Y.H., Richards, E.B., Jiang, Y.J., and Azarmi, N.A.S. (1996). Localised simulated annealing in constraint satisfaction and optimization. In Rayward-Smith, V.J., Osman, I.H., and Reeves, C.R., eds., Modern Heuristic Search Methods. Chichester: John Wiley. 27–39.Google Scholar
  55. Lin, C.Y., and Chen, W.T. (2000). Stochastic multistage algorithms for multimodal structural optimization. Computers and Structures 74: 233–241.CrossRefGoogle Scholar
  56. Lipson, S.L., and Gwin, L.B. (1977). The complex method applied to optimal truss configuration. Computers and Structures 7: 461–468.CrossRefGoogle Scholar
  57. Luchi, M.L., and Poggialini, A. (1980). An interactive optimization procedure applied to the design of gas turbine discs. Computers and Structures 11: 629–637.CrossRefMATHGoogle Scholar
  58. Lundy, M., and Mees, A. (1986). Convergence of an annealing algorithm. Mathematical Programming 34: 111–124.MathSciNetCrossRefMATHGoogle Scholar
  59. Manoharan, S., and Shanmuganathan, S. (1999). A comparison of search mechanisms for structural optimization. Computers and Structures 73: 363–372.CrossRefMATHGoogle Scholar
  60. May, S.A., and Balling, R.J. (1992). A filtered simulated annealing strategy for discrete optimization of a 3D steel frameworks. Structural Optimization 4: 142–148.CrossRefGoogle Scholar
  61. Nagendra, S., Jestin, D., Gurdal, Z., Haftka, R.T., and Watson, L.T. (1996). Improved genetic algorithm for the design of stiffened composite panels. Computers and Structures 58: 543–555.CrossRefMATHGoogle Scholar
  62. Nelder, J.A., and Mead, R. (1965). A simplex method for function minimization. Computer Journal 7: 308–313.CrossRefMATHGoogle Scholar
  63. Nemhauser, G.L. (1966). Introduction to Dynamic Programming. New York: John Wiley. 256 p.Google Scholar
  64. Odland, J. (1981). Theoretical and experimental buckling loads of imperfect spherical shell segments. Journal of Ship Research 25: 201–218.Google Scholar
  65. Osman, C., and Parent, P. (1982). Piecewise smooth dynamic programming application to a casing design for ultradeep drills. Engineering Optimization 5: 249–256.CrossRefGoogle Scholar
  66. Osman, I.H., and Laporte, G. (1996). Metaheuristics: A bibliography. Annals of Operations Research 63: 513–623.MathSciNetCrossRefMATHGoogle Scholar
  67. Palmer, A.C. (1968). Optimal structure design by dynamic programming. Proceedings of the American Society of Civil Engineers 94: 1887–1906.Google Scholar
  68. Palmer, A.C. (1969). Limit analysis of cylindrical shells by dynamic programming. International Journal of Solids and Structures 5: 289–302.CrossRefMATHGoogle Scholar
  69. Palmer, A.C., and Sheppard, D.J. (1970). Optimizing the shape of pin jointed structures. Proceedings of the Institution of Civil Engineers 47: 363–376.CrossRefGoogle Scholar
  70. Parkinson, J.M., and Hutchinson, D. (1972). An investigation into the efficiency of variants on the simplex method. In Lootsma, F.A., ed., Numerical Methods for Non-linear Optimization. New York: Academic Press. 115–135.Google Scholar
  71. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. (1986). Numerical Recipes - The Art of Scientific Computing. Cambridge: Cambridge University Press. UK. 818 p.MATHGoogle Scholar
  72. Raj, P.P., and Durrant, S.O. (1976). Optimum structural design by dynamic programming. Journal of the Structural Division, American Society of Civil Engineers 102: 1575–1589.Google Scholar
  73. Rao, S.S. (1984). Optimization - Theory and Applications. New Dehli: Wiley Eastern Limited. 747 p.MATHGoogle Scholar
  74. Rayward-Smith, V.J., Osman, I.H., Reeves, C.R., and Smith, G.D., eds., (1996). Modern Heuristic Search Methods. Chichester: John Wiley. 294 p.MATHGoogle Scholar
  75. Rechenberg, I. (1973). Evolution Strategy: Optimization of Technical Systems According to Biological Evolution. Stuttgart: Frommann-Holzboog.Google Scholar
  76. Reeves, C.R., ed., (1995). Modern Heuristic Techniques for Combinatorial Problems. London: McGraw-Hill. 320 p.Google Scholar
  77. Salama, M., Bruno, R., Chen, G.S., and Garba, J. (1990). Optimal placement of excitations and sensors by simulated annealing. In Recent Advances in Multidisciplinary Analysis and Optimization, NASA CP-3031, 1441–1458.Google Scholar
  78. Schmidt, H., and Krysik, R. (1991). Towards recommendations for shell stability design by means of numerically determined buckling loads. In Jullien, J.F., ed., Buckling of Shell Structures, on Land, in the Sea and in the Air. London New York: Elsevier Applied Science. 508–519.Google Scholar
  79. Schnack, E., and Sporl, U. (1986). A mechanical dynamic programming algorithm for structure optimization. International Jounal for Numerical Methods in Engineering 23: 1985–2004.MathSciNetCrossRefMATHADSGoogle Scholar
  80. Shim, P.Y., and Manoochehri, S. (1997). Generating optimal configurations in structural design using simulated annealing. International Journal for Numerical Methods in Enginering 40: 1053–1069.CrossRefMATHADSGoogle Scholar
  81. Shim, P.Y., and Manoochehri, S. (1999). A hybrid deterministic/stochastic optimization approach for the shape configuration design of structures. Structural Optimization 17: 113–129.Google Scholar
  82. Siarry, P., and Berthiau, G. (1997). Fitting of tabu search to optimize functions of continuous variables. International Journal for Numerical Methods in Engineering 40: 2449–2457.MathSciNetCrossRefMATHADSGoogle Scholar
  83. Spendley, W., Hext, G.R., and Himsworth, F.R. (1962). Sequential application of simplex designs in optimisation and evolutionary operation. Technometrics 4: 441–461.MathSciNetCrossRefMATHGoogle Scholar
  84. Swann, W.H. (1974). Constrained optimization by direct search. In Gill, P.E., and Murray, W., eds., Numerical Methods for Constrained Optimization. New York: Academic Press. 191–268.Google Scholar
  85. Szewczyk, Z., and Hajela, P. (1993). Neural network approximations in a simulated annealing based optimal structural design. Structural Optimization 5: 159–165.Google Scholar
  86. Thevendran, V. (1982). Dynamic programming in minimum weight design of shells. International Journal for Numerical Methods in Engineering 18: 1349–1360.CrossRefMATHADSGoogle Scholar
  87. Thevendran, V. (1984). Dynamic programming in minimum-weight design of axisymmetric plates. Journal of Optimization Theory and Applications 44: 689–700.MathSciNetCrossRefMATHGoogle Scholar
  88. Topping, B.H.V., Khan, A.I., and de Barros Leite, J.P. (1993). Topological design of truss structures using simulated annealing. In Topping, B.H.V., and Khan,. A.I., eds., Neural Networks and Combinatorial Optimization in Civil and Structural Engineering. Edinburgh: Civil-Comp Press. 151–165.CrossRefGoogle Scholar
  89. Tzan, S.R., and Pantelides, C.P. (1996). Annealing strategy for optimal structural design. Journal of Structural Engineering, Transactions of the American Society of Civil Engineers 122: 815–827.Google Scholar
  90. Vanderbilt, D., and Louie, S.G. (1984). A Monte Carlo simulated annealing approach to optimization over continuous variables. Journal of Computational Physics 56: 259–271.MathSciNetCrossRefMATHADSGoogle Scholar
  91. Vanderplaats, G.N. (1984). Numerical Optimization Techniques for Engineering Design. New York: McGraw-Hill Book Company. 333 p.MATHGoogle Scholar
  92. Vanderplaats, G.N. (1999). Structural design optimization status and direction. Journal of Aircraft 36: 11–20.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • J. Blachut
    • 1
  1. 1.Mechanical EngineeringThe University of LiverpoolLiverpoolUK

Personalised recommendations