Appendix A: Characteristics of the text-case and implementation
Variable-size design space Goldstein function
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5 continuous variables, 4 discrete variables, 2 dimensional variables
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8 sub-problems
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648 equivalent continuous problems
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1 constraint
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Initial data set size: 104 samples ( e.g. 2 samples per dimension of each sub-problem)
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Number of infilled samples: 104
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10 repetitions
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Compared methods:
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Independent mixed-variable BO of each sub-problem (IO) with both CS and LV kernels
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SOMVSP (BA) with CS and LV kernels and values of a of 2 and 3
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Variable-size design space kernel BO with CS and LV kernels with both SPW and DVW approaches
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Acquisition function: EI under EV constraints
Variable-size design space Rosenbrock function
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8 continuous variables, 3 discrete variables, 2 dimensional variables
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4 sub-problems
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32 equivalent continuous problems
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2 constraints
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Initial data set size: 30 samples (e.g. 1 sample per dimension of each sub-problem)
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Number of infilled samples: 65
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10 repetitions
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Compared methods:
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Independent mixed-variable BO of each sub-problem (IO) with both CS and LV kernels
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SOMVSP (BA) with CS and LV kernels and values of a of 2 and 3
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Variable-size design space kernel BO with CS and LV kernels with both SPW and DVW approaches
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Acquisition function: EI under EV constraints
Multi-stage launch vehicle design
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18 continuous variables, 14 discrete variables, 3 dimensional variables
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6 sub-problems
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29136 equivalent continuous problems
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19 constraints
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Initial data set size: 122 samples (e.g. 1.5 sample per dimension of each sub-problem)
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Number of infilled samples: 58
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10 repetitions
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Compared methods:
-
Independent mixed-variable BO of each sub-problem (IO) with both CS and LV kernels
-
SOMVSP (BA) with CS and LV kernels and values of a of 3
-
Variable-size design space kernel BO with CS and LV kernels with both SPW and DVW approaches
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Acquisition function: EI under EV constraints
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Large number of continuous and discrete design variables. Large number of constraints.
Implementation
The results presented in the paper are obtained with the following implementation. The optimization routine overhead is written in Python 3.6. The GP models are created with the help of GPflow (Alexander et al. 2017), a Python-based toolbox for GP-based modeling relying on the Tensorflow framework (Abadi et al. 2015) (version 1.13). The surrogate model training is performed with the help of a Bounded Limited memory Broyden–Fletcher–Goldfarb Shanno (L-BFGS-B) algorithm (Byrd et al. 1995), whereas the acquisition functions are optimized the help of a constraint domination based mixed continuous/discrete genetic algorithm (Stelmack et al. 1998) implemented by relying on the Python-based toolbox DEAP (Fortin et al. 2012).
Appendix B: Variable-size design space Goldstein function
The variable-size design space variant of the Goldstein function which is considered for the testing discussed in Sect. 5 is characterized by a global design space with 5 continuous design variables, 4 discrete design variables and 2 dimensional design variables. Depending on the dimensional variable values, 8 different sub-problems can be identified, with total dimensions of 6 or 7, ranging from 2 continuous variables and 4 discrete variables to 5 continuous variables and 2 discrete variables. All of the sub-problem are subject to a variable-size design space constraint.
The resulting optimization problem can be defined as follows:
$$\begin{aligned} \min&\qquad f(\mathbf{x} ,\mathbf{z} ,\mathbf{w} ) \nonumber \\ \text {w.r.t.}&\qquad \mathbf{x} = \{x_1,\dots ,x_5\} \ \text{ with } \ x_i \in [0,100] \ \text{ for } i = 1,5 \nonumber \\&\qquad \mathbf{z} = \{z_1,\dots ,z_4\} \ \text{ with } \ z_i \in \{ 0,1,2\} \ \text{ for } i = 1,4 \nonumber \\&\qquad \mathbf{w} = \{w_1,w_2\} \ \text{ with } \ w_1 \in \{ 0,1,2,3\} \ \text{ and } w_2 \in \{ 0,1\} \nonumber \\ \text {s.t.:}&\qquad g(\mathbf{x} ,\mathbf{z} ,\mathbf{w} ) \le 0 \end{aligned}$$
(55)
where
$$\begin{aligned} f(\mathbf{x} ,\mathbf{z} ,\mathbf{w} ) = {\left\{ \begin{array}{ll} f_1(x_1,x_2,z_1,z_2,z_3,z_4) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 0\\ f_2(x_1,x_2,x_3,z_2,z_3,z_4) \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 0\\ f_3(x_1,x_2,x_4,z_1,z_3,z_4) \qquad \text{ if } w_1 = 2 \text{ and } w_2 = 0\\ f_4(x_1,x_2,x_3,x_4,z_3,z_4) \qquad \text{ if } w_1 = 3 \text{ and } w_2 = 0\\ f_5(x_1,x_2,x_5,z_1,z_2,z_3,z_4) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 1\\ f_6(x_1,x_2,x_3,x_5,z_2,z_3,z_4) \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 1\\ f_7(x_1,x_2,x_4,x_5,z_1,z_3,z_4) \qquad \text{ if } w_1 = 2 \text{ and } w_2 = 1\\ f_8(x_1,x_2,x_3,x_5,x_4,z_3,z_4) \qquad \text{ if } w_1 = 3 \text{ and } w_2 = 1 \end{array}\right. } \end{aligned}$$
(56)
and
$$\begin{aligned} g(\mathbf{x} ,\mathbf{z} ,\mathbf{w} ) = {\left\{ \begin{array}{ll} g_1(x_1,x_2,z_1,z_2) \qquad \text{ if } w_1 = 0 \\ g_2(x_1,x_2,z_2) \qquad \text{ if } w_1 = 1 \\ g_3(x_1,x_2,z_1) \qquad \text{ if } w_1 = 2 \\ g_4(x_1,x_2,z_3,z_4) \qquad \text{ if } w_1 = 3 \end{array}\right. } \end{aligned}$$
(57)
The objective functions \(f_1(\cdot ),\dots ,f_8(\cdot )\) are defined as follows:
$$\begin{aligned} \begin{aligned} f_1(x_1,x_2,z_1,z_2,z_3,z_4) = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} \end{aligned} \end{aligned}$$
(58)
where \(x_3\) and \(x_4\) are defined as a function of \(z_1\) and \(z_2\) according to the relations defined in Table 6.
Table 6 Characterization of the variable-dimension search space Goldstein function sub-problem N\({}^\circ \)1 discrete categories
$$\begin{aligned} \begin{aligned} f_2(x_1,x_2,x_3,z_2,z_3,z_4) = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} \end{aligned} \end{aligned}$$
(59)
where \(x_4\) is defined as a function of \(z_2\) according to the relations defined in Table 7.
Table 7 Characterization of the variable-dimension search space Goldstein function sub-problem N\({}^\circ \)2 discrete categories
$$\begin{aligned} \begin{aligned} f_3(x_1,x_2,x_4,z_1,z_3,z_4) = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} \end{aligned} \end{aligned}$$
(60)
where \(x_3\) is defined as a function of \(z_1\) according to the relations defined in Table 8.
Table 8 Characterization of the variable-dimension search space Goldstein function sub-problem N\({}^\circ \)2 discrete categories
$$\begin{aligned}&\begin{aligned} f_4(x_1,x_2,x_3,x_4,z_3,z_4) = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} \end{aligned} \end{aligned}$$
(61)
$$\begin{aligned}&\begin{aligned} f_5(x_1,x_2,z_1,z_2,z_3,z_4) = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} + 5 \cos (2 \pi \frac{x_5}{100})-2 \end{aligned} \end{aligned}$$
(62)
where \(x_3\) and \(x_4\) are defined as a function of \(z_1\) and \(z_2\) according to the relations defined in Table 9.
Table 9 Characterization of the variable-dimension search space Goldstein function sub-problem N\({}^\circ \)5 discrete categories
$$\begin{aligned} \begin{aligned} f_6(x_1,x_2,x_3,z_2,z_3,z_4) = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} + 5 \cos (2 \pi \frac{x_5}{100})-2 \end{aligned} \end{aligned}$$
(63)
where \(x_4\) is defined as a function of \(z_2\) according to the relations defined in Table 10.
Table 10 Characterization of the variable-dimension search space Goldstein function sub-problem N\({}^\circ \)6 discrete categories
$$\begin{aligned} \begin{aligned} f_7(x_1,x_2,x_4,z_1,z_3,z_4) = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} + 5 \cos (2 \pi \frac{x_5}{100})-2 \end{aligned} \end{aligned}$$
(64)
where \(x_3\) is defined as a function of \(z_1\) according to the relations defined in Table 11.
Table 11 Characterization of the variable-dimension search space Goldstein function sub-problem N\({}^\circ \)7 discrete categories
$$\begin{aligned} \begin{aligned} f_8(x_1,x_2,x_3,x_4,z_3,z_4 = \quad&53.3108+ 0.184901x_1-5.02914 x_1^3 \cdot 10^{-6} + 7.72522x_1^{z_3} \cdot 10^{-8}- \\&0.0870775x_2 - 0.106959 x_3 + 7.98772x_3^{z_4} \cdot 10^{-6} + \\&0.00242482 x_4 + 1.32851 x_4^3 \cdot 10^{-6} - 0.00146393 x_1 x_2 - \\&0.00301588 x_1 x_3 - 0.00272291 x_1 x_4+ 0.0017004 x_2 x_3 + \\&0.0038428 x_2 x_4 - 0.000198969 x_3x_4 + 1.86025 x_1x_2x_3 \cdot 10^{-5} - \\&1.88719 x_1 x_2 x_4 \cdot 10^{-6}+ 2.50923 x_1 x_3 x_4 \cdot 10^{-5} - \\&5.62199 x_2 x_3 x_4 \cdot 10^{-5} + 5 \cos (2 \pi \frac{x_5}{100})-2 \end{aligned} \end{aligned}$$
(65)
The constraints \(g_1(\cdot ),\dots ,g_4(\cdot )\) are defined are defined as follows:
$$\begin{aligned} g_1(x_1,x_2,z_1,z_2)= - (x_1-50)^2 - (x_2-50)^2 + (20+c_1*c_2)^2 \end{aligned}$$
(66)
where \(c_1\) and \(c_2\) are defined as a function of \(z_1\) and \(z_2\) according to the relations defined in Table 12.
Table 12 Characterization of the variable-dimension search space Goldstein function constraint
$$\begin{aligned} g_2(x_1,x_2,z_2) = - (x_1-50)^2 - (x_2-50)^2 + (20+c_1*c_2)^2 \end{aligned}$$
(67)
where \(c_1 = 0.5\) and \(c_2\) is defined as a function of \(z_2\) according to the relations defined in Table 13.
Table 13 Characterization of the variable-dimension search space Goldstein function constraint
$$\begin{aligned} g_3(x_1,x_2,z_1) = - (x_1-50)^2 - (x_2-50)^2 + (20+c_1*c_2)^2 \end{aligned}$$
(68)
where \(c_2 = 0.7\) and \(c_1\) is defined as a function of \(z_1\) according to the relations defined in Table 14.
Table 14 Characterization of the variable-dimension search space Goldstein function constraint
$$\begin{aligned} g_4(x_1,x_2,z_3,z_4) = - (x_1-50)^2 - (x_2-50)^2 + (20+c_1*c_2)^2 \end{aligned}$$
(69)
where \(c_1\) and \(c_2\) are defined as a function of \(z_3\) and \(z_4\) according to the relations defined in Table 15.
Table 15 Characterization of the variable-dimension search space Goldstein function constraint Appendix C: Variable-size design space Rosenbrock function
The variable-size design space variant of the Rosenbrock function which is considered for the testing discussed in Sect. 5 is characterized by a global design space with 8 continuous design variables, 3 discrete design variables and 2 dimensional design variables. Depending on the dimensional variable values, 4 different sub-problems can be identified, with total dimensions of 6 to 9, ranging from 4 continuous variables and 2 discrete variables to 6 continuous variables and 3 discrete variables. One of the 2 constraints is only active for half of the sub-problems
The resulting optimization problem can be defined as follows:
$$\begin{aligned} f(\mathbf{x} ,\mathbf{z} ,\mathbf{w} ) = {\left\{ \begin{array}{ll} f_1(x_1,x_2,x_3,x_4,z_1,z_2) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 0\\ f_2(x_1,x_2,x_5,x_6,z_1,z_2,z_3) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 1\\ f_3(x_1,x_2,x_3, x_4,x_7, x_8,z_1,z_2) \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 0\\ f_4(x_1,x_2,x_5,x_6,x_7, x_8,z_1,z_2,z_3) \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 1 \end{array}\right. } \end{aligned}$$
(70)
and:
$$\begin{aligned} g_1(\mathbf{x} ,\mathbf{z} ,\mathbf{w} ) = {\left\{ \begin{array}{ll} g_{1_1}(x_1,x_2,x_3,x_4) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 0\\ g_{1_2}(x_1,x_2,x_5,x_6) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 1\\ g_{1_3}(x_1,x_2,x_3, x_4,x_7, x_8) \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 0\\ g_{1_4}(x_1,x_2,x_5,x_6,x_7, x_8) \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 1 \end{array}\right. } \end{aligned}$$
(71)
and:
$$\begin{aligned} g_2(\mathbf{x} ,\mathbf{z} ,\mathbf{w} ) = {\left\{ \begin{array}{ll} g_{2_1}(x_1,x_2,x_3,x_4) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 0\\ g_{2_2}(x_1,x_2,x_3, x_4,x_7, x_8) \qquad \text{ if } w_1 = 0 \text{ and } w_2 = 1\\ \text{ Not } \text{ active } \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 0\\ \text{ Not } \text{ active } \qquad \text{ if } w_1 = 1 \text{ and } w_2 = 1 \end{array}\right. } \end{aligned}$$
(72)
The objective functions \(f_1(\cdot ),\dots ,f_4(\cdot )\) as well as the constraints \(g_{1_1}(\cdot ),\dots ,g_{1_4}(\cdot )\) and \(g_{2_1}(\cdot ),\dots ,g_{2_4}(\cdot )\) are defined as follows:
$$\begin{aligned} f_1(x_1,x_2,x_3,x_4,z_1,z_2) = {\left\{ \begin{array}{ll} 100z_0 + \sum _i a_1*a_2(x_{i+1} - x_i)^2+(a_1+a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 0 \\ 100z_0 + \sum _i 0.7a_1*a_2(x_{i+1} - x_i)^2+(a_1-a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 1 \end{array}\right. } \end{aligned}$$
(73)
with \(a_1 = 7\) and \(a_2 = 9\)
$$\begin{aligned} f_2(x_1,x_2,x_5,x_6,z_1,z_2,z_3) = {\left\{ \begin{array}{ll} 100z_0 - 35z_3 + \sum _i a_1*a_2(x_{i+1} - x_i)^2+(a_1+a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 0 \\ 100z_0 - 35z_3 + \sum _i 0.7a_1*a_2(x_{i+1} - x_i)^2+(a_1-a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 1 \end{array}\right. } \end{aligned}$$
(74)
with \(a_1 = 7\) and \(a_2 = 6\)
$$\begin{aligned} f_3(x_1,x_2,x_3,x_4,x_7,x_8,z_1,z_2) = {\left\{ \begin{array}{ll} 100z_0 + \sum _i a_1*a_2(x_{i+1} - x_i)^2+(a_1+a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 0 \\ 100z_0 + \sum _i 0.7a_1*a_2(x_{i+1} - x_i)^2+(a_1-a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 1 \end{array}\right. } \end{aligned}$$
(75)
with \(a_1 = 10\) and \(a_2 = 9\)
$$\begin{aligned} f_2(x_1,x_2,x_5,x_6,x_7,x_8,z_1,z_2,z_3) = {\left\{ \begin{array}{ll} 100z_0 - 35z_3 + \sum _i a_1*a_2(x_{i+1} - x_i)^2+(a_1+a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 0 \\ 100z_0 - 35z_3 + \sum _i 0.7a_1*a_2(x_{i+1} - x_i)^2+(a_1-a_2)/10(1-x_i)^2 \quad \text{ if } z_2 = 1 \end{array}\right. } \end{aligned}$$
(76)
with \(a_1 = 10\) and \(a_2 = 6\)
$$\begin{aligned} g_{1_1}(x_1,x_2,x_3,x_4)&= \sum _i -(x_i-1)^3 + x_{i+1} - 2.6 \end{aligned}$$
(77)
$$\begin{aligned} g_{1_2}(x_1,x_2,x_5,x_6)&= \sum _i -(x_i-1)^3 + x_{i+1} - 2.6 \end{aligned}$$
(78)
$$\begin{aligned} g_{1_3}(x_1,x_2,x_3,x_4,x_7,x_8)&= \sum _i -(x_i-1)^3 + x_{i+1} - 2.6 \end{aligned}$$
(79)
$$\begin{aligned} g_{1_4}(x_1,x_2,x_5,x_6,x_7,x_8)&= \sum _i -(x_i-1)^3 + x_{i+1} - 2.6 \end{aligned}$$
(80)
$$\begin{aligned} g_{2_1}(x_1,x_2,x_3,x_4)&= \sum _i -x_i - x_{i+1} + 0.4 \end{aligned}$$
(81)
$$\begin{aligned} g_{2_2}(x_1,x_2,x_5,x_6)&= \sum _i -x_i - x_{i+1} + 0.4 \end{aligned}$$
(82)