Engineering Optimization: Theory and Practice, Fourth Edition

7.10 Sequential Quadratic Programming 427

We assume the matrix [H 1 ] to be the identity matrix and hence the objective function
of Eq. (7.138) becomes

Q(S)= 0. 1 s 1 + 0. 05773 s 2 + 0. 5 s^21 + 0. 5 s 22 (E 5 )

Equation (7.139) givesβ 1 =β 3 = since 0 g 1 =g 3 = and 0 β 2 = 1. 0 sinceg 2 < , and 0
hence the constraints of Eq. (7.138) can be expressed as

g ̃ 1 = − 0. 004254 s 1 − 0. 007069 s 2 ≤ 0 (E 6 )

g ̃ 2 = − 5. 8765 −s 1 ≤ 0 (E 7 )

g ̃ 3 = −s 2 ≤ 0 (E 8 )

We solve this quadratic programming problem [Eqs.(E 5 ) ot (E 8 ) directly with the use]
of the Kuhn–Tucker conditions. The Kuhn–Tucker conditions are given by

∂Q

∂s 1

+

∑^3

j= 1

λj

∂g ̃j

∂s 1

= 0 (E 9 )

∂Q

∂s 2

+

∑^3

j= 1

λj

∂g ̃j

∂s 2

= 0 (E 10 )

λjg ̃j= 0 , j= 1 , 2 , 3 (E 11 )

g ̃j≤ 0 , j= 1 , 2 , 3 (E 12 )

λj≥ 0 , j= 1 , 2 , 3 (E 13 )

Equations(E 9 ) nda (E 10 ) an be expressed, in this case, asc

0. 1 +s 1 − 0. 004254 λ 1 −λ 2 = 0 (E 14 )

0. 05773 +s 2 − 0. 007069 λ 1 −λ 3 = 0 (E 15 )

By considering all possibilities of active constraints, we find that the optimum solution
of the quadratic programming problem [Eqs. (E 5 ) o (Et 8 ) is given by]

s 1 ∗ = − 0. 04791 , s 2 ∗ = 0. 02883 , λ∗ 1 = 21. 2450 , λ∗ 2 = 0 , λ∗ 3 = 0

The new design vector,X, can be expressed as

X=X 1 +αS=

{

11. 8765 − 0. 04791 α

7. 0 + 0. 02883 α

}

whereαcan be found by minimizing the functionφin Eq. (7.141):

φ= 0. 1 ( 11. 8765 − 0. 04791 α)+ 0. 05773 ( 7. 0 + 0. 02883 α)

+ 12. 2450

(

0. 6

11. 8765 − 0. 04791 α

+

0. 3464

7. 0 + 0. 02883 α

− 0. 1

)