Engineering Optimization: Theory and Practice, Fourth Edition

14.8 Multilevel Optimization 757

Y(l)≤Y≤Y(u)

Z(l)k ≤Zk≤Z(u)k , k= 1 , 2 ,... , K (14.97)

Similarly, the objective functionf (X)can be expressed as

f (X)=

∑K

k= 1

f(k)(Y,Zk) (14.98)

wheref(k)(Y,Zk) enotes the contribution of thed kth subsystem to the overall objective
function. Using Eqs. (14.95) to (14.98), the two-level approach can be stated as follows.

First-level Problem. Tentatively fix the values ofYatY∗so that the problem of
Eqs. (14.89) to (14.92) [or Eqs. (14.95) to (14.98)] can be restated (decomposed) asK
independent optimization problems as follows:

FindZkwhich minimizesf(k)(Y,Zk)

subjectto

g(k)(Y,Zk)≤ 0

h(k)(Y,Zk)= 0 (14.99)

Z(l)k ≤Zk≤Z(u)k ; k= 1 , 2 ,... , K

It can be seen that the first-level problem seeks to find the minimum of the function

f (Y,Z)=

∑K

k= 1

f(k)(Y,Zk) (14.100)

for the (tentatively) fixed vectorY∗.

Second-level Problem. The following problem is solved in this stage:

Find a newY∗which minimizesf(Y)=

∑K

k= 1

f(k)(Y,Z∗k)

subjectto

Y(l)≤Y≤Y(u) (14.101)

whereZ∗k, k = 1 , 2 ,... , K, are the optimal solutions of the first-level problems. An
additional constraint to ensure a finite value off (Y∗) s also to be included whilei
solving the problem of Eqs. (14.101). Once the problem is solved and a newY∗found,
weproceed to solve the first-level problems. This process is to be continued until
convergence is achieved. The iterative process can be summarized as follows:

1.Start with an initial coordination vector,Y∗.

2 .Solve theKfirst-level optimization problems, stated in Eqs. (14.99), and find

the optimal vectorsZ∗k(k = 1 , 2 ,... , K).