426 Nonlinear Programming III: Constrained Optimization Techniques
withλj={
|λj| j, = 1 , 2 ,... , m+pin first iteration
max{|λj|,^12 (λ ̃j,|λj| n subsequent iterations)}i(7.142)
andλ ̃j=λjof the previous iteration. The one-dimensional step lengthα∗can be found
by any of the methods discussed in Chapter 5.
OnceXj+ 1 is found from Eq. (7.140), for the next iteration the Hessian matrix [H]
is updated to improve the quadratic approximation in Eq. (7.138). Usually, a modified
BFGS formula, given below, is used for this purpose [7.12]:[Hi+ 1 ]=[Hi]−[Hi]PiPTi[Hi]
PTi[Hi]Pi+
γγT
PTiPi(7.143)
Pi=Xi+ 1 −Xi (7.144)γ=θQi+ ( 1 −θ)[Hi]Pi (7.145)Qi= ∇xL ̃(Xi+ 1 ,λi+ 1 ) −∇xL ̃(Xi,λi) (7.146)θ=
1. 0 ifPTiQi≥ 0. 2 PTi[Hi]Pi
0. 8 PTi[Hi]Pi
PTi[Hi]Pi−PTiQiifPTiQi< 0. 2 PTi[Hi]Pi(7.147)
whereL ̃ is given by Eq. (7.137) and the constants 0.2 and 0.8 in Eq. (7.147) can be
changed, based on numerical experience.Example 7.5 Find the solution of the problem (see Problem 1.31):
Minimizef (X)= 0. 1 x 1 + 0. 05773 x 2 (E 1 )subjectto
g 1 (X)=0. 6
x 1+
0. 3464
x 2− 0. 1 ≤ 0 (E 2 )
g 2 (X)= 6 −x 1 ≤ 0 (E 3 )
g 3 (X)= 7 −x 2 ≤ 0 (E 4 )usingthe sequential quadratic programming technique.SOLUTION Let the starting point beX 1 = 1 ( 1. 8765 , 7. 0 )Twithg 1 (X 1 )=g 3 (X 1 )=
0 ,g 2 (X 1 ) =− 5 .8765, andf (X 1 ) = 1. 5 917. The gradients of the objective and con-
straint functions atX 1 are given by∇f(X 1 )={
0. 1
0. 05773
}
, ∇g 1 (X 1 )=
− 0. 6
x 12
− 0. 3464
x 22
X 1=
{
− 0. 004254
− 0. 007069
}
∇g 2 (X 1 )={
− 1
0
}
, ∇g 3 (X 1 )=