Engineering Optimization: Theory and Practice, Fourth Edition

302 Nonlinear Programming II: Unconstrained Optimization Techniques

are satisfied. The pointX∗is guaranteed to be a relative minimum if the Hessian matrix

is positive definite, that is,

JX∗=[J]X∗=

[

∂^2 f

∂xi∂xj

(X∗)

]

=positive definite (6.3)

Equations (6.2) and (6.3) can be used to identify the optimum point during numerical

computations. However, if the function is not differentiable, Eqs. (6.2) and (6.3) cannot

be applied to identify the optimum point. For example, consider the function

f (x)=

{

ax for x≥ 0

−bx for x≤ 0

wherea>0 andb>0. The graph of this function is shown in Fig. 6.1. It can be

seen that this function is not differentiable at the minimum point,x∗= , and hence 0

Eqs. (6.2) and (6.3) are not applicable in identifyingx∗. In all such cases, the commonly

understood notion of a minimum, namely,f (X∗) < f ( X)for allX, can be used only

to identify a minimum point. The following example illustrates the formulation of a

typical analysis problem as an unconstrained minimization problem.

Example 6.1 A cantilever beam is subjected to an end forceP 0 and an end moment

M 0 as shown in Fig. 6.2a.By using a one-finite-element model indicated in Fig. 6.2b,

the transverse displacement,w(x), can be expressed as [6.1]

w(x)= {N 1 (x) N 2 (x) N 3 (x) N 4 (x)}











u 1

u 2

u 3

u 4











(E 1 )

whereNi(x) are calledshape functionsand are given by

N 1 (x)= 2 α^3 − 3 α^2 + 1 (E 2 )

N 2 (x) =(α^3 − 2 α^2 + α)l (E 3 )

N 3 (x) =− 2 α^3 + 3 α^2 (E 4 )

N 4 (x) =(α^3 −α^2 )l (E 5 )

α=x/ l, andu 1 , u 2 , u 3 , nda u 4 are the end displacements (or slopes) of the beam.

The deflection of the beam at pointAcan be found by minimizing the potential energy

Figure 6.1 Function is not differentiable at mini-

mum point.