Biomedical Engineering Reference
In-Depth Information
optimize and find an optimum, or at least a feasible practical solution, different
optimization strategies have been developed. However, whichever optimization
strategy is adopted to tackle the particular problem, a primary step requires proper
mathematical formulation of a given problem. The most basic optimization
problem can be expressed in the form of
U
ð
p
Þ!
extr for p
2
E
n
with p
T
¼½
p
1
p
2
... p
n
:
ð
3
:
365
Þ
Whether a maximum or a minimum is sought does not influence the choice of
the optimization method, since problems can be formulated either way. Generally,
due to convention, the focus is laid upon the minimum. Then (
3.365
) can be
formulated as follows. A parameter vector p* [ E
n
is sought such that, for all
p [ E
n
, there holds U
ð
p
Þ
U
ð
p
Þ:
The choice of the values of the variables x
i
in technical applications is often
restricted to maintain physically reasonable solutions, and the variables can thus not
be chosen arbitrarily. If such restrictions exist, optimization is referred to as con-
straint optimization. Constraints on the variables restrict them from taking on any
arbitrary value and thus limit the domain of possible values into bounds. Depending
on the type of constraint, it may either reduce the possible space of the overall
solution (inequality constraint) or even downsize the dimensionality of the problem
(equality constraint). Comprising constraints, equation (
3.365
) may take the form
p
2
Q
n
U
ð
p
Þ!
extr
for
ð
3
:
366
Þ
subject to:
Q
¼f
h
i
ð
p
Þ¼
0
;
i
¼
1
;
...
;
n
h
g
j
ð
p
Þ ¼
0
;
j
¼
1
;
...
;
n
g
l
p
i
u
;
i
¼
1
;
...
;
n
g
where Q denotes the valid parameter domain with a set of equality and inequality
constraints h
i
(p) and g
i
(p), and explicit variable bounds l and u restricting each
parameter individually.
If, in addition, more than a single objective is to be optimized, multi-objective
optimization is required and all objective functions are optimized simultaneously.
Here, the single objectives can be independent or in mutual conflict. In the case of
several conflicting objectives the optimization is referred to as a true multi-
objective problem. Multi-objective optimization has its roots in welfare economics
at the beginning of the 20th century and is linked to the economists V. Pareto
(Vilfredo Pareto, *1848-1923) and F. Edgeworth (Francis Edgeworth, *1845-
1926).
A description of a multi-objective problem can be stated as follows
