Structural optimization

, Volume 8, Issue 2–3, pp 69–85

Methods for optimization of nonlinear problems with discrete variables: A review

  • J. S. Arora
  • M. W. Huang
  • C. C. Hsieh
Review Papers


The methods for discrete-integer-continuous variable nonlinear optimization are reviewed. They are classified into the following six categories: branch and bound, simulated annealing, sequential linearization, penalty functions, Lagrangian relaxation, and other methods. Basic ideas of each method are described and details of some of the algorithms are given. They are transcribed into a step-by-step format for easy implementation into a computer. Under “other methods”, rounding-off, heuristic, cutting-plane, pure discrete, and genetic algorithms are described. For nonlinear problems, none of the methods are guaranteed to produce the global minimizer; however, “good practical” solutions can be obtained.



branch and bound method


set of discrete values for all the discrete variables


set of discrete values for thei-th variable


j-th discrete value for thei-th variable


cost function to be minimized


upper bound for the cost function


i-th constraint function


integer programming


integer linear programming




linear programming


total number of constraints


mixed-discrete linear programming


mixed-discrete nonlinear programming


number of discrete variables


nonlinear programming


number of equality constraints; acceptance probability used in simulated annealing


number of discrete values for thei-th variable


sequential linear programming


sequential quadratic programming


design variable vector of dimension n


smallest allowed value for thei-th variable


largest allowed value for thei-th variable

the gradient operator


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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • J. S. Arora
    • 1
  • M. W. Huang
    • 1
  • C. C. Hsieh
    • 2
  1. 1.Optimal Design Laboratory, College of EngineeringThe University of IowaIowa CityUSA
  2. 2.GM Systems EngineeringTroyUSA

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