14 Practical Aspects of Optimization
14.1 Introduction
Although the mathematical techniques described in Chapters 3 to 13 can be used
to solve all engineering optimization problems, the use of engineering judgment and
approximations help in reducing the computational effort involved. In this chapter we
consider several types of approximation techniques that can speed up the analysis time
without introducing too much error [14.1].
These techniques are especially useful in finite element analysis-based optimiza-
tion procedures. The practical computation of the derivatives of static displacements,
stresses, eigenvalues, eigenvectors, and transient response of mechanical and
structural systems is presented. The concept of decomposition, which permits the
solution of a large optimization problem through a set of smaller, coordinated sub-
problems is presented. The use of parallel processing and computation in the solution
of large-scale optimization problems is discussed. Many real-life engineering systems
involve simultaneous optimization of multiple-objective functions under a specified
set of constraints. Several multiobjective optimization techniques are summarized in
this chapter.
14.2 Reduction of Size of an Optimization Problem
14.2.1 Reduced Basis Technique
In the optimum design of certain practical systems involving a large number of (n)
design variables, some feasible design vectorsX 1 ,X 2 ,... ,Xrmay be available to start
with. These design vectors may have been suggested by experienced designers or may
be available from the design of similar systems in the past. We can reduce the size of
the optimization problem by expressing the design vector X as a linear combination of
the available feasible design vectors as
X=c 1 X 1 +c 2 X 2 + · · · +crXr (14.1)
wherec 1 , c 2 ,... , crare the unknown constants. Then the optimization problem can
be solved usingc 1 , c 2 ,... , cras design variables. This problem will have a much
smaller number of unknowns sincer ≪ n. In Eq. (14.1), the feasible design vec-
torsX 1 ,X 2 ,... ,Xrserve as the basis vectors. It can be seen that ifc 1 =c 2 = · · · =
cr= 1 /r, thenXdenotes the average of the basis vectors.
Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S. Rao 737
Copyright © 2009 by John Wiley & Sons, Inc.
