Graphics Programs Reference
In-Depth Information
the layout of the members is given,sothat the design variables are the cross-sectional
areas of the members. Here the design is dominatedbyinequality constraints that
consist of prescribed upper limits on the stresses and possibly the displacements.
The majority of available methods are designed for unconstrained optimization ,
where no restrictions are placed on the design variables. In these problems the min-
ima, if they exit, arestationary points (points where gradient vector of F ( x ) vanishes).
In the more difficult problem of constrained optimization the minima are usually lo-
catedwhere the F ( x )surface meets the constraints. There arespecial algorithmsfor
constrained optimization, but theyare not easilyaccessible due to their complexity
and specialization. One way to tackle aproblemwith constraints istouse an uncon-
strained optimizationalgorithm, but modify the meritfunction so that any violation
of constraints isheavilypenalized.
Consider the problem of minimizing F ( x ) where the design variables aresubject
to the constraints
g i ( x )
=
0,
i
=
1
,
2
,...,
M
(10.1a)
=
,
,...,
h j ( x )
0,
j
1
2
N
(10.1b)
We choose the newmeritfunctionbe
F ( x )
=
F ( x )
+ λ
P ( x )
(10.2a)
where
M
N
max 0
h j ( x ) 2
[ g i ( x )] 2
=
+
,
P ( x )
(10.2b)
i
=
1
j
=
1
is the penalty function and
λ
is amultiplier. The function max( a
,
b ) returns the larger
of a and b . It isevidentthat P ( x )
0 if noconstraints are violated. Violation of a
constraint imposes apenaltyproportional to the squareoftheviolation. Hence the
minimizationalgorithm tendstoavoid the violations, the degree of avoidance being
dependenton the magnitudeof
=
λ
λ
issmall, optimizationwill proceed fasterbe-
cause there is more “space”inwhich the procedurecan operate, but there may be
significant violation of constraints. On the other hand, a large
. If
can result inapoorly
conditionedprocedure, but the constraints will betightly enforced. It is advisable to
run the optimizationprogram with
λ
that ison the small side. If the results showun-
acceptable constraint violation, increase
λ
λ
and run the program again,starting with
the results of the previous run.
An optimizationprocedure may also become ill-conditionedwhen the con-
straints have widelydifferent magnitudes. This problem can be alleviatedby scaling
the offending constraints; that is, multiplying the constraintequations by suitable
constants.
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