Engineering Optimization: Theory and Practice, Fourth Edition

410 Nonlinear Programming III: Constrained Optimization Techniques

6.IfSi= 0 ,find the maximum step lengthλMthat is permissible without violating

anyof the constraints asλM= inm (λk),λk> 0 andkis any integer among 1 to

mother thanj 1 , j 2 ,... , jp. Also find the value ofdf/dλ(λM)=STi∇ f(Xi+

λMSi) If. df/dλ(λM) s zero or negative, take the step length asi λi=λM. On

the other hand, ifdf/dλ(λM) s positive, find the minimizing step lengthi λ∗i

either by interpolation or by any of the methods discussed in Chapter 5, and

takeλi=λ∗i.

7 .Find the new approximation to the minimum as

Xi+ 1 =Xi+λiSi

Ifλi=λMor ifλM≤λ∗i, some new constraints (one or more) become active

atXi+ 1 and hence generate the new matrixNpto include the gradients of all

active constraints evaluated atXi+ 1. Set the new iteration number asi=i+ 1 ,

and go to step 4. Ifλi=λ∗i andλ∗i< λM, no new constraint will be active at

Xi+ 1 and hence the matrixNpremains unaltered. Set the new value ofias

i=i+1, and go to step 3.

Example 7.3

Minimizef (x 1 , x 2 )=x 12 +x^22 − 2 x 1 − 4 x 2

subject to

g 1 (x 1 , x 2 )=x 1 + 4 x 2 − 5 ≤ 0

g 2 (x 1 , x 2 )= 2 x 1 + 3 x 2 − 6 ≤ 0

g 3 (x 1 , x 2 ) =−x 1 ≤ 0

g 4 (x 1 , x 2 ) =−x 2 ≤ 0

starting from the pointX 1 =

{ 1. 0

1. 0

}

.

SOLUTION

Iterationi= 1

Step 3: Sincegj(X 1 ) = 0 forj=1, we havep=1 andj 1 =. 1

Step 4: AsN 1 =[∇g 1 (X 1 )]=

[ 1

4

]

,the projection matrix is given by

P 1 =

[

1 0

0 1

]

−

[

1

4

][

[1 4]

[

1

4

]]− 1

[ 1 4 ]

=

1

17

[

16 − 4

−4 1

]

The search directionS 1 is given by

S 1 = −

1

17

[

16 − 4

−4 1

]{

0

− 2

}

=

{

− 178

2

17

}

=

{

− 0. 4707

0. 1177

}