Multilevel framework for large-scale global optimization

Abstract

Large-scale global optimization (LSGO) algorithms are crucially important to handle real-world problems. Recently, cooperative co-evolution (CC) algorithms have successfully been applied for solving many large-scale practical problems. Many applications have imbalanced subcomponents where the size of subcomponents and their contribution to the objective function value are different. CC algorithms often lose their efficiency on LSGO problems with the imbalanced subcomponents; since they do not consider the imbalance aspect of variables. In this paper, we propose a multilevel optimization framework based on variables effect (called MOFBVE) which optimizes several subcomponents of the most important variables at earlier stages of optimization procedure before optimizing the problem with the original search space at its last stage. Sensitivity analysis (SA) method determines how the variation in the outputs of the model can be influenced by the variation of its input parameters. MOFBVE computes the main effect of variables using an SA method, Morris screening, and then it employs the k-means clustering method to construct groups including variables with the similar effects on the fitness value. The constructed groups are sorted in the descending order based on their contribution on the fitness value and the top groups are selected as the levels of the important variables. MOFBVE can reduce the complexity of search space to work with a simplified model to achieve an efficient exploration. The performance of MOFBVE is benchmarked on the imbalanced LSGO problems, i.e., two individually modified CEC-2010 and the CEC-2013 LSGO benchmark functions. The simulated experiments confirmed that MOFBVE obtains a promising performance on the majority of the imbalanced LSGO test functions. Also, MOFBVE is compared with state-of-the-art CC algorithms; and the results show that it is better than or at least comparable to CC algorithms.

Appendices

Appendix A

The the normal coefficient

Tables 22, 23, and 24 present the normal coefficient corresponding to nonseparable subcomponents in the modified normal CEC-2010 test functions.

Appendix B

The benchmark functions

• CEC-2010 benchmark functions

Dimension: $D = 1000$

Group size: $m = 50$

$x = (x_1, x_2, \dots , x_D)$: The candidate solution

$o = (o_1, o_2, \dots , o_D)$: The (shifted) global optimum

$z = x - o, z = (z_1, z_2, \dots , z_D)$: The shifted candidate solution

P: A random permutation of ${1, 2, \dots ,D}$

$$\begin{aligned} F_\mathrm{elliptic}(x)=\sum _{\begin{array}{c} i=1 \end{array} }^{D} (10^6)\frac{i-1}{D-1} {x_i}^2 \end{aligned}$$

$$\begin{aligned} F_{\mathrm{rosenbrock}}=\sum _{\begin{array}{c} i=1 \end{array}}^{D-1}[100({x_i}^2-x_{i+1})^2+(x_i-1)^2] \end{aligned}$$

$$\begin{aligned} F_{\mathrm{rastrigin}}=\sum _{\begin{array}{c} i=1 \end{array}}^{D}[{x_i}^2-10\cos (2\pi )x_i+10] \end{aligned}$$

$$\begin{aligned} F_{\mathrm{ackley}}= & {} -20\mathrm{exp}\left( -0.2\sqrt{\frac{1}{D}\sum _{\begin{array}{c} i=1 \end{array}}^D{x_i}^2}\right) \\&-\,\mathrm{exp}\left( \frac{1}{D}\right) \sum _{\begin{array}{c} i=1 \end{array}}^D\cos (2\pi x_i))+20+c \end{aligned}$$

$$\begin{aligned} F_{\mathrm{schwefel}}=\sum _{\begin{array}{c} i=1 \end{array}}^{D}\left( \sum _{\begin{array}{c} j=1 \end{array}}^{i}x_i \right) ^2 \end{aligned}$$

$$\begin{aligned} F_{9}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{2m}}F_{\mathrm{rot}\_\mathrm{elliptic}}[z(P_{(k-1)*m+1} : P_{k*m})]\\&+F_{\mathrm{rot}_\mathrm{elliptic}}[z(P_{\frac{D}{2}+1} : P_D)] \end{aligned}$$

$$\begin{aligned} F_{10}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{2m}}F_{\mathrm{rot}\_\mathrm{rastrigin}}[z(P_{(k-1)*m+1} : P_{k*m})]\\&+F_{\mathrm{rastrigin}}[z(P_{\frac{D}{2}+1} : P_D)] \end{aligned}$$

$$\begin{aligned} F_{11}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{2m}}F_{\mathrm{rot}\_\mathrm{ackley}}[z(P_{(k-1)*m+1} : P_{k*m})]\\&+F_{\mathrm{ackley}}[z(P_{\frac{D}{2}+1} : P_D)] \end{aligned}$$

$$\begin{aligned} F_{12}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{2m}}F_{\mathrm{schwefel}}[z(P_{(k-1)*m+1} : P_{k*m})]\\&+F_\mathrm{sphere}[z(P_{\frac{D}{2}+1} : P_D)] \end{aligned}$$

$$\begin{aligned} F_{13}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{2m}}F_{\mathrm{rot}\_\mathrm{rosenbrock}}[z(P_{(k-1)*m+1} : P_{k*m})]\\&+F_\mathrm{sphere}[z(P_{\frac{D}{2}+1} : P_D)] \end{aligned}$$

$$\begin{aligned} F_{14}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{m}}F_{\mathrm{rot}\_\mathrm{elliptic}}[z(P_{(k-1)*m+1} : P_{k*m})] \end{aligned}$$

$$\begin{aligned} F_{15}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{m}}F_{\mathrm{rot}\_\mathrm{rastrigin}}[z(P_{(k-1)*m+1} : P_{k*m})] \end{aligned}$$

$$\begin{aligned} F_{16}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{m}}F_{\mathrm{rot}\_\mathrm{ackley}}[z(P_{(k-1)*m+1} : P_{k*m})] \end{aligned}$$

$$\begin{aligned} F_{17}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{m}}F_{\mathrm{schwefel}}[z(P_{(k-1)*m+1} : P_{k*m})] \end{aligned}$$

$$\begin{aligned} F_{18}(x)= & {} \sum _{\begin{array}{c} k=1 \end{array}}^{\frac{D}{m}}F_{\mathrm{rosenbrock}}[z(P_{(k-1)*m+1} : P_{k*m})] \end{aligned}$$

• CEC-2013 benchmark functions

Dimension: $D = 1000$

Group size: $m = 50$

$$\begin{aligned} S = {50, 25, 25, 100, 50, 25, 25, 700} \end{aligned}$$

$$\begin{aligned} S1= & {} \{50, 50, 25, 25, 100, 100, 25, 25, 50, 25, 100,\\&25, 100, 50, 25, 25, 25, 100, 50, 25\} \end{aligned}$$

$x^\mathrm{opt}$ : The optimum decision vector

P: A random permutation of ${1, 2, \dots ,D}$

$T_\mathrm{osz}$: A transformation function to create smooth local irregularities.

$T_\mathrm{asy}$: A transformation function to break the symmetry of the symmetric functions.

$\lambda$: A D-dimensional diagonal matrix with the diagonal elements is used to create ill-conditioning.

R: An orthogonal rotation matrix which is used to rotate the fitness landscape randomly around various axes

m: The overlap size between subcomponents

$$\begin{aligned} y = x - x^\mathrm{opt} \end{aligned}$$

$$\begin{aligned} y_i = y(P_{[C_{i-1}+1]} : P_{[C_i]}) \quad i \in {1, \ldots , |S|}, \end{aligned}$$

$$\begin{aligned} y_{i1} = y(P_{[C_{i-1}-(i-1)m+1]} : P_{[C_i-(i-1)m]}) \quad i \in {1, \ldots , |S|}, \end{aligned}$$

$$\begin{aligned} y_{i2}= & {} y(P_{[C_{i-1}-(i-1)m+1]} : P_{[C_i-(i-1)m]})\\&-x_i^\mathrm{opt} \quad i \in {1, \ldots , |S|}, \end{aligned}$$

$$\begin{aligned} z_i = T_\mathrm{osz}(R_iy_i), \quad i \in {1, \ldots , |S| - 1} \end{aligned}$$

$$\begin{aligned} z_{i1} = T_\mathrm{asy}^{0.2}T_\mathrm{osz}(R_iy_{i1}),\quad i \in {1, \ldots , |S| - 1} \end{aligned}$$

$$\begin{aligned} z_{i2} = T_\mathrm{asy}^{0.2}T_\mathrm{osz}(R_iy_{i2}), \quad i \in {1, \ldots , |S| - 1} \end{aligned}$$

$$\begin{aligned} z_{|S|}= T_\mathrm{osz}(y_{|S|}) \end{aligned}$$

$$\begin{aligned} R_i: a |S_i| \times |S_i| \end{aligned}$$

$$\begin{aligned} F_{4}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s|-1}w_i f_\mathrm{elliptic}(z_i)+f_\mathrm{elliptic}(z_{|s|}), \end{aligned}$$

$$\begin{aligned} F_{5}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s|-1}w_i f_{\mathrm{rastrigin}}(z_i)+f_{\mathrm{rastrigin}}(z_{|s|}), \end{aligned}$$

$$\begin{aligned} F_{6}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s|-1}w_i f_{\mathrm{ackley}}(z_i)+f_{\mathrm{ackley}}(z_{|s|}), \end{aligned}$$

$$\begin{aligned} F_{7}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s|-1}w_i f_{\mathrm{schwefel}}(z_i)+f_{\mathrm{schwefel}}(z_{|s|}), \end{aligned}$$

$$\begin{aligned} F_{8}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s1|}w_i f_\mathrm{elliptic}(z_i), \end{aligned}$$

$$\begin{aligned} F_{9}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s1|}w_i f_{\mathrm{rastrigin}}(z_i), \end{aligned}$$

$$\begin{aligned} F_{10}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s1|}w_i f_{\mathrm{ackley}}(z_i), \end{aligned}$$

$$\begin{aligned} F_{11}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s1|}w_i f_{\mathrm{schwefel}}(z_i), \end{aligned}$$

$$\begin{aligned} F_{13}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s1|}w_i f_{\mathrm{schwefel}}(z_{i1}), \end{aligned}$$

$$\begin{aligned} F_{14}(x)=\sum _{\begin{array}{c} k=1 \end{array}}^{|s1|}w_i f_{\mathrm{schwefel}}(z_{i2}), \end{aligned}$$