Computer Aided Engineering Design

350 COMPUTER AIDED ENGINEERING DESIGN

Fig. 12.8 Examples of unconstrained optimization with two variables

Consider, for example, an over-bridge design wherein it may be desired to minimize the overall
deflection (strain energy) subject to a stipulated amount of material volume. We may further impose
that the stress levels in bridge members do not exceed the yield limit of the material. If the member
cross-sections are chosen as design variables, then the latter cannot assume negative values. In other
words, some problems may require the variables to be bounded. In this section and the following we
discuss multi-variable optimization with equality and inequality constraints.
Consider first, minimizing a function f(X) in n variables X≡ [x 1 ,x 2 ,... , xn]T with mequality
constraintsgi (X) = 0, i= 1,... , m, where m≤n. For m>n, the problem is overdetermined and there
may not exist a solution. A method can be of direct substitution wherein by solving the mequality
constraints, any set of mvariables may be expressed in terms of the remaining n–m variables. The
problem then becomes unconstrained in n–m variables and can be solved using the criteria discussed
in section 12.3.1. Unfortunately, this method poses difficulties if the constraints are nonlinear in that
there is no straighforward way to eliminate the constraints.
The method of Lagrange multipliers works by introducing a variable λi for each of the m constraints
such that the total number of variables to be determined becomes n + m. An augmented Lagrangian
L is constructed such that

LL( , ,... , xx 12 xnm, , 1 2,... ) ( , ) = ( ) + f i=1 g( )

m

λλ λ ≡ XXΛΛ ΣΣλiiX(12.13)

(a) (b)

(c) (d)

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