Biomedical Engineering Reference
In-Depth Information
where
u
(
b
) is the solution of (
19
). This form allows decoupling the solution of the
state equation and the optimization problem.
A general multi-objective optimization seeks to optimize the components of a
vector-valued objective function mathematically formulated as
D
Minimize F.
b
/
.f
1
.
b
/; f
2
.
b
/; :::; f
m
.
b
//
b
upper
i
b
lower
i
D
b
i
;i
1; :::; n
subject to
(22)
D
g
k
.
b
/
0; k
1; :::; p
where
b
(
b
1
, :::,
b
n
) is the design vector,
b
i
lower
and
b
i
upper
represent the lower
and upper boundary of the
i
th design variable
b
i
,
f
j
(
b
)isthe
j
th objective function
and
g
k
(
b
)the
k
th constraint. Unlike single objective optimization approaches, the
solution to this problem is not a single point, but a family of points known as the
Pareto-optimal set. A Pareto optimal solution is defined as one that is not dominated
by any other solution of the multi-objective optimization problem. Typically, there
are infinitely many Pareto optimal solutions for a multi-objective problem. Thus, it
is often necessary to incorporate user preferences for various objectives in order to
determine a single suitable solution.
Genetic algorithms are well suited to multi-objective optimization problems as
they are fundamentally based on biological processes which are inherently multi-
objective.
D
4.2
Genetic Search
A genetic algorithm (GA) is a stochastic search method based on evolution and
genetics exploiting the concept of survival of the fittest. For a given problem or
design domain there exists a multitude of possible solutions that form a solution
space. In a genetic algorithm, a highly effective search of the solution space is
performed, allowing a population of strings representing possible design vectors
to evolve through basic genetic operators. The goal of these operators is to
progressively reduce the space design driving the process into more promising
regions.
Multi-objective genetic algorithms (MOGAs) were first suggested by Schaffer
[
24
]. Since then several algorithms have been proposed on the basis of an evolution-
ary process searching for Pareto optimal solutions. MOGAs have been successfully
applied to solve various kinds of multi-objective problems as they are not as
much affected by nonlinearities and complex objective functions as mathematical
programming algorithms.
One MOGA strategy consists of transferring multi-objectives to a single objec-
tive by a weighted sum approach. The weighted sum method for multi-objective
optimization problems continues to be used extensively not only to provide multiple
solution points by varying the weights consistently, but also to provide a single
