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D. Irawan and B. Naujoks
8.4 Many-Objective Optimization
8.4.1 Challenges in Many-Objective Optimization
First, recall that in many-objective optimization problems, at least four objective
functions are considered (Sect. 8.1). There are several complications when dealing
with many-objective problems. First, methods applicable on single-objective problems can only be used if the original many-objective problem is transformed into a
single-objective by using aggregation methods mentioned before (see Sect. 8.3.1.1).
Furthermore, even methods applicable on multi-objective problems may have bad
performance or are even not feasible to use in many-objective cases. The reason for
the latter problem usually stems from the so-called curse of dimensionality. In other
cases it is due to the problems being too expensive to evaluate.
8.4.1.1 Curse of Dimensionality
Curse of dimensionality is a term which expresses how an increased dimension
leads to an extremely high increase of cost or deterioration of solution quality. In
the previous section, it has been shown that the number of dimensions d is the
exponential factor for runtime. Hence increasing d leads to an exponential increase
in the runtime.
Curse of dimensionality can be mitigated by reducing the problem size. The
methods to reduce the size can simply be done by ignoring some objectives (similar
to what was done in lexicographic method mentioned in Sect. 8.3.1.1) or using
model reduction methods such as principal component analysis (PCA) [39].
Another problem with dimensionality in many-objective optimization is the
loss of pressure to find the Pareto front. With increasing dimensionality, nondominatedness is easier to achieve [29]. With low pressure, the population will
converge to the Pareto front only slowly.
8.4.1.2 Expensive Evaluation
Expensive evaluation means that the evaluations have very high cost. However,
“cost” here is not limited to financial cost. Sometimes, it is indeed the monetary cost
that is expensive, but it can also be other kinds of costs. The cost can be evaluation
time, manpower or computational power required, etc. For example, when the
function evaluated is actually a result of a simulation, each of the simulation can
take several minutes, hours, or even days. If a single simulation takes a long time
to finish, the whole optimization process—which requires several simulations to be
run—will also consume a lot of time to finish. Another example is when to evaluate
a design, a prototype must be manufactured. This implies costs in both time and
D. Irawan and B. Naujoks
8.4 Many-Objective Optimization
8.4.1 Challenges in Many-Objective Optimization
First, recall that in many-objective optimization problems, at least four objective
functions are considered (Sect. 8.1). There are several complications when dealing
with many-objective problems. First, methods applicable on single-objective problems can only be used if the original many-objective problem is transformed into a
single-objective by using aggregation methods mentioned before (see Sect. 8.3.1.1).
Furthermore, even methods applicable on multi-objective problems may have bad
performance or are even not feasible to use in many-objective cases. The reason for
the latter problem usually stems from the so-called curse of dimensionality. In other
cases it is due to the problems being too expensive to evaluate.
8.4.1.1 Curse of Dimensionality
Curse of dimensionality is a term which expresses how an increased dimension
leads to an extremely high increase of cost or deterioration of solution quality. In
the previous section, it has been shown that the number of dimensions d is the
exponential factor for runtime. Hence increasing d leads to an exponential increase
in the runtime.
Curse of dimensionality can be mitigated by reducing the problem size. The
methods to reduce the size can simply be done by ignoring some objectives (similar
to what was done in lexicographic method mentioned in Sect. 8.3.1.1) or using
model reduction methods such as principal component analysis (PCA) [39].
Another problem with dimensionality in many-objective optimization is the
loss of pressure to find the Pareto front. With increasing dimensionality, nondominatedness is easier to achieve [29]. With low pressure, the population will
converge to the Pareto front only slowly.
8.4.1.2 Expensive Evaluation
Expensive evaluation means that the evaluations have very high cost. However,
“cost” here is not limited to financial cost. Sometimes, it is indeed the monetary cost
that is expensive, but it can also be other kinds of costs. The cost can be evaluation
time, manpower or computational power required, etc. For example, when the
function evaluated is actually a result of a simulation, each of the simulation can
take several minutes, hours, or even days. If a single simulation takes a long time
to finish, the whole optimization process—which requires several simulations to be
run—will also consume a lot of time to finish. Another example is when to evaluate
a design, a prototype must be manufactured. This implies costs in both time and