8 An Introduction to Many-Objective Evolutionary Optimization
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money. In these cases it is impossible or inefficient to evaluate all required points to
find the Pareto front.
Expensive evaluation is not an exclusive problem which is only applicable
to many-objective optimization. Even a single-objective optimization task would
consider expensive evaluation as a challenge. A popular solution is using surrogate
model/function which is applicable on single-, multi-, and many-objective optimization problems. The surrogate model is a prediction and simplification of the real
model, based on some early samples. Further explanation of surrogate models is
available in the other chapters, while a specific application of surrogate modelling
in multi- and many-objective optimization is available in Sect. 8.5.
8.4.1.3 Visualization Challenge
When the number of objectives exceeds 3, visualization becomes a problem. The
regular visualization using scatterplot is limited to view a three-dimensional system
projected into a plane (two dimensions). Some recent researches attempt to take
it further to view four-dimensional systems by using a projection onto a threedimensional space [6]; however, it does not change the fact that visualization will
be limited. Some methods to visualize the objective space in a higher dimension are
presented in Sect. 8.4.3.3.
8.4.2 Algorithms Designed for Many-Objective Optimization
Problems
Due to the challenges posed by many-objective problems, MOEAs for multiobjective problems are difficult to use. Researchers around the world devise new
EAs specifically designed for many-objective problems. The algorithms are called
MOEAs or EMOAs.
8.4.2.1 MOEA/D
MOEA/D is the MOEA based on decomposition proposed by Zhang and Li [46].
Decomposition means transforming the many-objective problem into finite number
μ of SOPs by using some aggregation methods. Zhang and Li use the weighted
sum and the Tchebycheff method to decompose the many-objective problem. The
Tchebycheff method (see Fig. 8.16) has the advantage of being able to find points
in the non-convex region of the Pareto front [46].
Each subproblem will lead to a single point in the Pareto front, so in the end μ
solutions will be obtained. The algorithm limits mating only between T number
of neighbors with equal probabilities (equiprobable selection). Diversity is not
287
money. In these cases it is impossible or inefficient to evaluate all required points to
find the Pareto front.
Expensive evaluation is not an exclusive problem which is only applicable
to many-objective optimization. Even a single-objective optimization task would
consider expensive evaluation as a challenge. A popular solution is using surrogate
model/function which is applicable on single-, multi-, and many-objective optimization problems. The surrogate model is a prediction and simplification of the real
model, based on some early samples. Further explanation of surrogate models is
available in the other chapters, while a specific application of surrogate modelling
in multi- and many-objective optimization is available in Sect. 8.5.
8.4.1.3 Visualization Challenge
When the number of objectives exceeds 3, visualization becomes a problem. The
regular visualization using scatterplot is limited to view a three-dimensional system
projected into a plane (two dimensions). Some recent researches attempt to take
it further to view four-dimensional systems by using a projection onto a threedimensional space [6]; however, it does not change the fact that visualization will
be limited. Some methods to visualize the objective space in a higher dimension are
presented in Sect. 8.4.3.3.
8.4.2 Algorithms Designed for Many-Objective Optimization
Problems
Due to the challenges posed by many-objective problems, MOEAs for multiobjective problems are difficult to use. Researchers around the world devise new
EAs specifically designed for many-objective problems. The algorithms are called
MOEAs or EMOAs.
8.4.2.1 MOEA/D
MOEA/D is the MOEA based on decomposition proposed by Zhang and Li [46].
Decomposition means transforming the many-objective problem into finite number
μ of SOPs by using some aggregation methods. Zhang and Li use the weighted
sum and the Tchebycheff method to decompose the many-objective problem. The
Tchebycheff method (see Fig. 8.16) has the advantage of being able to find points
in the non-convex region of the Pareto front [46].
Each subproblem will lead to a single point in the Pareto front, so in the end μ
solutions will be obtained. The algorithm limits mating only between T number
of neighbors with equal probabilities (equiprobable selection). Diversity is not