14.8 Multilevel Optimization 755
If the vectorSis normalized by dividing its components bysn+ 1 , Eq. (14.86) gives
λ=pand hence Eq. (14.85) gives the desired sensitivity derivatives as
∂x 1
∂p
..
.
∂xn
∂p
=
1
sn+ 1
S (14.87)
Thus the sensitivity of the objective function with respect topcan be computed as
df (X)
dp
= ∇f (X)T
S
sn+ 1
(14.88)
Note that unlike the previous method, this method does not require the values ofλ∗
and the second derivatives off andgj to find the sensitivity derivatives. Also, if
sensitivities with respect to several problem parametersp 1 , p 2 ,... are required, all we
need to do is to add them to the design vectorXin Eq. (14.82).
14.8 Multilevel Optimization
14.8.1 Basic Idea
The design of practical systems involving a large number of elements or subsys-
tems with multiple-load conditions involves excessive number of design variables and
constraints. The optimization problem becomes unmanageably large, and the solution
process becomes too costly and can easily saturate even the largest computers avail-
able. In such cases the optimization problem can be broken into a series of smaller
problems using different strategies. The multilevel optimization is a decomposition
technique in which the problem is reformulated as several smaller subproblems (one
for each subsystem) and a coordination problem (at system level) to preserve the cou-
pling among the subproblems (subsystems). Such approaches have been used in linear
and dynamic programming also. In linear programming, the decomposition method (see
Section 4.4) involves a number of independent linear subproblems coupled by limita-
tions on the shared resources. When an individual subsystem is solved, the cost of the
shared resources is added to its objective function. By a proper variation of the costs
of the shared resources, the proposed optimal strategies of the various subproblems
are sent to the master program, which, in turn, is optimized so that the overall cost is
minimized. In dynamic programming, the problem is treated in stages with an optimal
policy determined in each stage (see Chapter 9). This approach is particularly useful
when the problem has a serial structure.
For nonlinear design optimization problems, several decomposition methods
have been proposed [14.14–14.16]. In the following section we consider a two-level
approach in which the system is decomposed into a number of smaller subproblems,
each with its own goals and constraints. The individual subsystem optimization
problems are solved independently in the first level and the coordinated problem
is solved in the second level. The approach is known as themodel-coordination
method.
