Optimization and Computational Fluid Dynamics - Department of ...

236 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto
determinate outputs, when it undergoes through certain inputs. Whoever
wants to look into a system, she/he has to, first of all, build a model of it:
the simplest possible representation, yet bearing the most important features
of the system itself. By means of its model, a system can be studied and
improved using mathematical tools, whenever a quantitative relation between
inputs and outputs can be established. Thus, a system can be seen as a black
box acting on inputs to produce outputs, as in the following relation:
⎧
⎨
O =
⎩
O1
...
Om
⎫ ⎧
⎬ ⎨
= f(X)=f
⎭ ⎩
X1
...
Xn
⎫
⎬
⎭
(8.37)
where O is a set of outputs and f is a generic relation linking outputs to
inputs set X.
Optimization is the act of obtaining the best solution under given circumstances,
as Rao states in [45]. From a system point of view, this means one is
searching a maximum, or respectively a minimum, for function f, depending
on the desired goal. Without loss of generality, noting that the maximum of
f coincides with the minimum of its opposite −f, an optimization problem
can be taken as either a minimization or a maximization one.
The existence of optimization methods can be traced back to the beginning
of differential calculus that allows minimization of functionals, in both
unconstrained and constrained domains. But, in real problems, the function f
is unlikely to be a simple analytical expression, in which case the study of the
function by classical mathematical analysis is sufficient. It is rather a usually
unknown relation that might lack continuity, derivativeness, or connectedness.
Differential calculus, therefore, is of little help in such circumstances.
When a relation is unknown, a trial and error methodology is the oldest
practice, and no further contributions to optimization techniques has been
provided until the advent of digital computers, which have made implementation
of optimization procedures a feasible task. From an optimization point of
view inputs and outputs in Eq. (8.37) can be renamed after their conceptual
meanings. Inputs are usually known in literature as design variables, while
outputs, being the goal of an optimization process, are known as objective
functions or simply objectives. In many practical problems, design variables
cannot be chosen arbitrarily, but they have to satisfy specified requirements.
These are called design constraints. Even the objectives could undergo restrictions.
They are called functional constraints. In addition to Eq. (8.37),
these two kind of constraints can be formally expressed as:
g (X) ≤ 0 (8.38)
m (f (X)) ≤ 0 (8.39)
where g and m are two general applications. Equality relations are easily
obtained replacing the symbol “≤” with “=”. Optimization problems can be