Network characteristics for neighborhood field algorithms

Abstract

Evolutionary algorithms (EAs) have been successfully applied to solve numerous optimization problems. Neighborhood field optimization algorithm (NFO) has been proposed to integrate the neighborhood field in EAs, which utilizes local cooperation behaviors to explore new solutions. In this paper, certain new NFO variants are proposed based on the cooperation of descendent neighbors. The competitive and cooperative behaviors of NFO variants provide a remarkable ability to accelerate information exchanges and achieve global search. Experimental results show that NFO variants perform better than basic and other state-of-the-art EAs under different benchmark functions. For NFO and other EAs, it is difficult to quantify benefits of local cooperation in the optimization process. For this purpose, the cooperation behaviors are analyzed in a new network approach in this paper. In the proposed NFO variants, population graph shows a scale-free network with power-law distribution. Network characteristics, i.e., degree distribution, cluster coefficient and average degree, are used to quantify the cooperation behaviors. Experimental results show that network characteristics can effectively indicate the optimization performance of NFO variants in terms of convergence rate and population diversity. NFO variants with large cluster coefficients and significant heterogeneous characteristics can achieve a significant performance improvement on numerous problems.

Appendix

1.1 Introduction to the IEEE-CEC 2014 benchmark suite

IEEE-CEC 2014 includes 30 benchmark functions as follows: unimodal functions (${{F_1}-{F_3}}$), simple multimodal functions (${{F_4}-{F_{16}}}$), hybrid functions (${{F_{17}}-{F_{22}}}$) and composition functions (${{F_{23}}-{F_{30}}}$). In this “Appendix”, I will list the details of functions ( ${F_1}$, ${F_4}$, ${F_{17}}$ and ${F_{23}}$ ) which come from unimodal functions, simple multimodal functions, hybrid functions and composition functions, respectively. The details of other functions are shown in the literature [56].

Rotated High Conditioned Elliptic Function: ${F_1}$

$$\begin{aligned} {F_1(\mathbf{x} )} = {f_1(\mathbf{M} (\mathbf{x} - \mathbf{o}_1)) + F_1^*} \end{aligned}$$

note:

${{F_1^*} = 100}$

${{f_1({\mathbf{x }})} = {\sum \nolimits _{i = 1}^D {(10^6)^{\frac{i-1}{D-1}} x_i^2}}}$

Shifted and Rotated Rosenbrock’s Function: ${F_4}$

$$\begin{aligned} {F_4({\mathbf{x}} )} = {f_4\left({\mathbf{M }}(\frac{2.048({\mathbf{x}} -{\mathbf{o }}_4)}{100})+1\right) + F_4^*} \end{aligned}$$

note:

${{F_4^*} = 400}$

${{f_4({\mathbf{x}} )} = {\sum \nolimits _{i = 1}^{D-1} {(100({x_i^2} - {x_{i+1}})^2 + ({x_i} - 1)^2)}} }$

Hybrid Function 1${(N=3)}$: ${F_{17}}$

$$\begin{aligned} {F_{17}(\mathbf x )} = {g_1(\mathbf{M }_1 \mathbf{z }_1) + g_2(\mathbf{M }_2 \mathbf{z }_2) +g_3(\mathbf{M }_3 \mathbf{z }_3) + F_{17}^*} \end{aligned}$$

note:

• ${{F_{17}^*} = 1700}$

• ${p=[0.3, 0.3, 0.4]}$ is used to control the percentage of ${g_i({\mathbf{x }})}$

• ${g_1}$: Modified Schwefel’s Function

• ${g_2}$: Rastrigin’s Function

• ${g_3}$: High Conditioned Elliptic Function

• ${{\mathbf{z }_1} = {[\mathbf{y }_{S_1}, \mathbf{y }_{S_2}, \mathbf{y }_{S_{n_1}} ]}}$

• ${{\mathbf{z }_2} = {[\mathbf{y }_{S_{n_1+1}}, \mathbf{y }_{S_{n_1+2}}, \mathbf{y }_{S_{n_1+n_2}} ]}}$

• ${{\mathbf{z }_3} = {[\mathbf{y }_{S_{n_1+n_2+1}}, \mathbf{y }_{S_{n_1+n_2+2}}, \mathbf{y }_{S_{n_1+n_2+n_3}} ]}}$

• ${{n_N} ={D-\sum \nolimits _{i = 1}^{N-1}{n_i} } }$

• ${{\mathbf{y }} = {{\mathbf{x }}-\mathbf{o }_i} }$

• ${{S} = {randperm(1:D)} }$

Composition Function 1${(N=5)}$: ${F_{23}}$

$$\begin{aligned} {F_{23}({\mathbf{x }})} = {\sum \limits _{i = 1}^{N}{{\omega _i * [\lambda _i g_i({\mathbf{x }}) + bias_i]}} + F_{23}^*} \end{aligned}$$

note:

• ${{F_{23}^*} = 2300}$

• ${\sigma =[10, 20, 30, 40, 50]}$

• ${\lambda =[1, 1e-6, 1e-26, 1e-6, 1e-6]}$

• ${bias=[0, 100, 200, 300, 400]}$

• ${g_1}$: Rotated Rosenbrock’s Function

• ${g_2}$: High Conditioned Elliptic Function

• ${g_3}$: Rotated Bent Cigar Function

• ${g_4}$: Rotated Discus Function

• ${g_5}$: High Conditioned Elliptic Function

• ${{\omega _i} = {\frac{1}{\sqrt{\sum \nolimits _{j = 1}^{D}{(x_i-o_{ij})^2}}} exp(- \frac{\sum \nolimits _{j = 1}^{D}{(x_i-o_{ij})^2}}{2 D \sigma _i^2}) } }$

Please note: All test functions are minimization problems; the details of mutual parameters are shown as following:

search range: ${[-100, 100]^D}$

• D: dimensions

• ${{\mathbf{x }}=[x_1,x_2,,...,x_D]^T}$

• ${\mathbf{o }_{i1}=[o_{i1},o_{i2},,...,o_{iD}]^T}$: the shifted global optimum, which is randomly distributed in ${[-80, 80]^D}$.

• ${\mathbf{M }_i}$: rotation matrix, and it for each subcomponents is generated from standard normally distributed.