14.8 Multilevel Optimization 759We treat the bars 1 and 2 as subsystems 1 and 2, respectively, withyas the coordination
variable(Y= {y})andz 1 andz 2 as the subsystem variables(Z 1 = {z 1 } nda Z 2 = {z 2 }).
By fixing the value ofyaty∗, we formulate the first-level problems as follows.
Subproblem 1.
Findz 1 which minimizes
f(^1 )(y∗, z 1 ) = 76 , 500 z 1√
(y∗)^2 + 63 (E 1 )subjectto
g 1 (y∗, z 1 )=( 1428. 5714 × 10 −^6 )
√
(y∗)^2 + 63
y∗z 1− 1 ≤ 0 (E 2 )
0 ≤z 1 ≤ 0. 1 (E 3 )Subproblem 2.
Findz 2 which minimizes
f(^2 )(y∗, z 2 ) = 76 , 500 z 2√
(y∗)^2 + 1 (E 4 )subjectto
g 2 (y∗, z 2 )=( 8571. 4285 × 10 −^6 )
√
(y∗)^2 + 1
y∗z 2− 1 ≤ 0 (E 5 )
0 ≤z 2 ≤ 0. 1 (E 6 )We can see that to minimizef(^1 )we need to makez 1 as small as possible without
violating the constraints of Eqs. (E 2 ) and (E 3 ). This gives the solution of subproblem
1 ,z∗ 1 (which makesg 1 active) as
z∗ 1 =( 1428. 5714 × 10 −^6 )
√
(y∗)^2 + 63
y∗(E 7 )
Similarly, the solution of subproblem 2,z∗ 2 (which makesg 2 active) can be expressed
as
z∗ 2 =( 8571. 4285 × 10 −^6 )
√
(y∗)^2 + 1
y∗(E 8 )
Now we state the second-level problem as follows:
Findywhich minimizesf=f(^1 )(y, z∗ 1 )+f(^2 )(y, z∗ 2 )subjectto
1 ≤y≤ 6 (E 9 )Using Eqs. (E 7 ) and (E 8 ), this problem can be restated as (usingyfory∗):