Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 237
classified in several ways that depend on different aspects of the problems
themselves. In [45], the following classifications are highlighted.
A first classification can be based on:
1. Existence of constraints. As stressed earlier, problems can be classified
constrained or unconstrained. Constraint handling is not a trivial task for
most optimization techniques.
2. Nature of design variables. f can be a function of a primitive set of variables
depending on further parameters, thus becoming trajectory optimization
problems [24].
3. Physical structure of the problem. Depending on the structure of the problem,
optimal control theory can be applied, where a global cost functional
is minimized to obtain the desired solution.
4. Nature of the relations involved. When known or at least well guessed, the
nature of the equations governing the model of the system under study
can address the choice to the most efficient among a set of optimization
methods. Linear, quadratic geometric and nonlinear programming are examples.
5. Permissible values of design variables. Design variables can be real value
or discrete.
6. Deterministic nature of the variables. The deterministic or stochastic nature
of the parameters is a criterion to classify optimization problems.
In particular, the concept of Robust Design or Robust Optimization has
recently gained popularity [32].
7. Separability of the functions. A problem is considered separable if f functions
can be considered a combination of functions of single design variables
f1(X1), f2(X2),..., f2(Xn) andf becomes:
f(X)=
n�
fi(Xi) .
The advantage of such a feature is that in nonlinear problems, nonlinearities
are mathematically independent [22].
8. Number of objective functions. Depending on the number of objective functions,
the problem can be single- or multi-objective. This is an outstanding
distinction, since in multi-objective optimizations, objectives are usually
conflicting. No single optimum exists, but rather a set of designs the decision
maker has to choose from. This is one of the motivations that have led
to the birth of Evolutionary Multi-Objective Optimization (EMOO) [8].
Depending on the characteristics of optimization problems, many techniques
have been developed to solve them. These can be roughly divided into two
categories.
i=1
1. Traditional mathematical programming techniques. They require a certain
knowledge of the relation between objectives and design variables, and
they are usually best suited for single-objective optimizations.