1.5 Classification of Optimization Problems 15
1.5.2 Classification Based on the Nature of the Design Variables
Based on the nature of design variables encountered, optimization problems can be
classified into two broad categories. In the first category, the problem is to find values
to a set of design parameters that make some prescribed function of these parameters
minimum subject to certain constraints. For example, the problem of minimum-weight
design of a prismatic beam shown in Fig. 1.8asubject to a limitation on the maximum
deflection can be stated as follows:
FindX=
{
b
d
}
which minimizes
f (X)=ρlbd
(1.4)
subject to the constraints
δtip(X)≤δmax
b≥ 0
d≥ 0
whereρis the density andδtipis the tip deflection of the beam. Such problems are
calledparameterorstatic optimization problems. In the second category of problems,
the objective is to find a set of design parameters, which are all continuous functions
of some other parameter, that minimizes an objective function subject to a set of
constraints. If the cross-sectional dimensions of the rectangular beam are allowed to
vary along its length as shown in Fig. 1.8b, the optimization problem can be stated as
FindX(t)=
{
b(t)
d(t)
}
which minimizes
f[X(t)]=ρ
∫l
0
b(t) d(t) dt (1.5)
subject to the constraints
δtip[ X(t)]≤δmax, 0 ≤t≤l
b(t)≥ 0 , 0 ≤t≤l
d(t)≥ 0 , 0 ≤t≤l
Figure 1.8 Cantilever beam under concentrated load.
