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

Abstract

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.

Notation

BBM

branch and bound method

D

set of discrete values for all the discrete variables

Di

set of discrete values for thei-th variable

dij

j-th discrete value for thei-th variable

f

cost function to be minimized

f*

upper bound for the cost function

gi

i-th constraint function

IP

integer programming

ILP

integer linear programming

L

Lagrangian

LP

linear programming

m

total number of constraints

MDLP

mixed-discrete linear programming

MDNLP

mixed-discrete nonlinear programming

nd

number of discrete variables

NLP

nonlinear programming

p

number of equality constraints; acceptance probability used in simulated annealing

qi

number of discrete values for thei-th variable

SLP

sequential linear programming

SQP

sequential quadratic programming

x

design variable vector of dimension n

xiL

smallest allowed value for thei-th variable

xiU

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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