10 Sequential Parameter Optimization for Mixed-Discrete Problems
347
recombining the strategy parameters and computes the mean of both parental
vectors. The mutation of the offspring relies on operators acting differently on real,
integer and discrete variables, all respecting the requirements for a mutation strategy
in the search spaces: accessibility, feasibility, symmetry, similarity, scalability, and
maximal entropy [31].
The MIES achieves this by adding normal distributed noise to real-valued
variables. For integer variables, the distribution is based on the difference of two
geometrical distributions. Categorical variables are simply re-sampled (uniform
randomly) with some probability p [31].
In the next step of the iteration, the λ offspring individuals are ranked on the
basis of the objective function. The μ best individuals out of the union of the λ
offspring individuals and the μ parental individuals are selected. The generational
loop is repeated until the number of evaluation exceeds the budget.
10.5 Case Study: Optimization of Composite Multi-Layered
Plate
In this section, a mixed-discrete problem based on a real-word application is
discussed. The problem consists in the design optimization of a composite multilayered plate. Our idea is to demonstrate the difficulties that researchers and
practitioners encounter when facing black-box, time-consuming, mixed-discrete
problems under uncertainty through an illustrative example. A performance comparison between the MIES and a general purpose optimizer is also given.
10.5.1 Overview
The objective of the optimization problem is to find the materials (represented
by categorical variables) and lamination angles (continuous variables) in order to
minimize the bending of a loaded plate composed of five layers.
The available materials and their properties are reported in Table 10.1. As one
can see, the list includes orthotropic and isotropic materials. In case of orthotropic
materials, the stiffness of the material is crucially affected by the lamination
angle. Contrarily, isotropic materials have equal in-plane and out-of-plane Young’s
modulus. Hence, the lamination angle of isotropic materials does not affect the
material behaviour. The use of both types of materials considerably increases the
difficulty of the problem from the modeling perspective: The importance of the
continuous parameters depends on the value of the categorical variables.
The plate has been loaded by a lifting load, which is linearly distributed along
the length of the plate applied on the nodes in the centreline. An encastre at the root
of the plate has been enforced as shown in Fig. 10.2. With the intent to reproduce
347
recombining the strategy parameters and computes the mean of both parental
vectors. The mutation of the offspring relies on operators acting differently on real,
integer and discrete variables, all respecting the requirements for a mutation strategy
in the search spaces: accessibility, feasibility, symmetry, similarity, scalability, and
maximal entropy [31].
The MIES achieves this by adding normal distributed noise to real-valued
variables. For integer variables, the distribution is based on the difference of two
geometrical distributions. Categorical variables are simply re-sampled (uniform
randomly) with some probability p [31].
In the next step of the iteration, the λ offspring individuals are ranked on the
basis of the objective function. The μ best individuals out of the union of the λ
offspring individuals and the μ parental individuals are selected. The generational
loop is repeated until the number of evaluation exceeds the budget.
10.5 Case Study: Optimization of Composite Multi-Layered
Plate
In this section, a mixed-discrete problem based on a real-word application is
discussed. The problem consists in the design optimization of a composite multilayered plate. Our idea is to demonstrate the difficulties that researchers and
practitioners encounter when facing black-box, time-consuming, mixed-discrete
problems under uncertainty through an illustrative example. A performance comparison between the MIES and a general purpose optimizer is also given.
10.5.1 Overview
The objective of the optimization problem is to find the materials (represented
by categorical variables) and lamination angles (continuous variables) in order to
minimize the bending of a loaded plate composed of five layers.
The available materials and their properties are reported in Table 10.1. As one
can see, the list includes orthotropic and isotropic materials. In case of orthotropic
materials, the stiffness of the material is crucially affected by the lamination
angle. Contrarily, isotropic materials have equal in-plane and out-of-plane Young’s
modulus. Hence, the lamination angle of isotropic materials does not affect the
material behaviour. The use of both types of materials considerably increases the
difficulty of the problem from the modeling perspective: The importance of the
continuous parameters depends on the value of the categorical variables.
The plate has been loaded by a lifting load, which is linearly distributed along
the length of the plate applied on the nodes in the centreline. An encastre at the root
of the plate has been enforced as shown in Fig. 10.2. With the intent to reproduce