Engineering Optimization: Theory and Practice, Fourth Edition

304 Nonlinear Programming II: Unconstrained Optimization Techniques

Equation (E 6 ) can be rewritten as

F=

1

2

∫ 1

0

EI

(

d^2 w

dx^2

) 2

l dα−P 0 u 3 −M 0 u 4

=

EI l

2

∫ 1

0

[

6 u^3

l^2

(− 2 α+ 1 )+

2 u 4

l

( 3 α− 1 )

] 2

dα−P 0 u 3 −M 0 u 4

=

EI

l^3

( 6 u^23 + 2 u^24 l^2 − 6 u 3 u 4 l)−P 0 u 3 −M 0 u 4 (E 9 )

Byusing the relationsu 3 =x 1 , u 4 l=x 2 , P 0 l^3 /EI = 1 , andM 0 l^2 /EI =2,and intro-

ducing the notationf=F l^3 /EI ,Eq. (E 9 ) can be expressed as

f= 6 x 12 − 6 x 1 x 2 + 2 x 22 −x 1 − 2 x 2 (E 10 )

Thus the optimization problem is to determinex 1 andx 2 , which minimize the function

fgiven by Eq. (E 10 ).

6.1.1 Classification of Unconstrained Minimization Methods

Several methods are available for solving an unconstrained minimization problem.

These methods can be classified into two broad categories as direct search methods

and descent methods as indicated in Table 6.1. The direct search methods require only

the objective function values but not the partial derivatives of the function in finding

the minimum and hence are often called thenongradient methods. The direct search

methods are also known aszeroth-order methodssince they use zeroth-order derivatives

of the function. These methods are most suitable for simple problems involving a

relatively small number of variables. These methods are, in general, less efficient than

the descent methods. The descent techniques require, in addition to the function values,

the first and in some cases the second derivatives of the objective function. Since

more information about the function being minimized is used (through the use of

derivatives), descent methods are generally more efficient than direct search techniques.

The descent methods are known asgradient methods. Among the gradient methods,

Table 6.1 Unconstrained Minimization Methods

Direct search methodsa Descent methodsb

Random search method Steepest descent (Cauchy) method

Grid search method Fletcher–Reeves method

Univariate method Newton’s method

Pattern search methods Marquardt method

Powell’s method Quasi-Newton methods

Davidon–Fletcher–Powell method

Broyden–Fletcher–Goldfarb–Shanno method

Simplex method

aDo not require the derivatives of the function.

bRequire the derivatives of the function.