12.1.1 Symbol Conventions and Definitions

Symbols and definitions used in this book are described as follows. By default, all vectors refer to column vectors which are depicted by lowercase letter and typeset in bold.

• [·] indicates the nearest integer value

• ⌊·⌋ indicates the largest integer less than or equal to

• x_i denotes ith element of vector x

• f(·), g(·) and G(·) denote multivariable functions

• f_opt denotes optimal (minimal) value of function f

• x^opt denotes optimal solution vector, such that f(x^opt) = f_opt

• R denotes normalized orthogonal matrix for rotation

• D denotes dimension

• 1 = (1,…, 1)^T denotes all one vector

12.1.2 General Setup

The general setup of the test suit is presented as follows:

• Dimensions: The test suit is scalable in terms of dimension. Within the hardware limit, any dimension D ≥ 2 works. However, to construct a real hybrid function, D should be at least 10.

• Search Space: All functions are defined and can be evaluated over ${\mathcal{R}}^D$, while the actual search domain is given as [−100, 100]^D.

• f_opt: All functions, by definition, have a minimal value of 0, a bias (f_opt) can be added to each function. The selection can be arbitrary, f_opt for each function in the test suit is listed in Table 12.1.

• x^opt: The optimum point of each function is located at original, which is randomly distributed in [−70, 70]^D, and is selected as the new optimum.

• Rotation Matrix: To derive nonseparable functions from separable ones, the search space is rotated by a normalized orthogonal matrix R. For a given function in one dimension, a different R is used. Variables are divided into three (almost) equal-sized subcomponents randomly. The rotation matrix for each subcomponent is generated from standard normally distributed entries by Gram–Schmidt orthonormalization. Then, these matrices consist of the R actually used.

12.1.3 CUDA Interface and Implementation

A simple description of the interface and implementation is given below. For detail, see the source code and its accompanied readme file.

12.1.4 Test Suite Summary

The whole of the test functions fall into four categories: unimodal functions, basic multimodal functions, hybrid functions, and composition functions. The summary of the suit is listed in Table 12.1. Detailed information of each function will be given in the following sections.

12.3 Unimodal Functions

12.3.1 Shifted and Rotated Sphere Function

$f_1\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^D{\mathbf{z}}_i^2+f_{\mathit{opt}}},$

where z = R(x − x^opt).

Properties: Unimodal; Nonseparable; Highly symmetric, in particular rotationally invariant.

12.3.2 Shifted and Rotated Ellipsoid Function

$f_2\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^{D}i·{\mathbf{z}}_i^2+f_{\mathit{opt}}},$

where z = R(x − x^opt).

Properties: Unimodal; Nonseparable.

12.3.3 Shifted and Rotated High-Conditioned Elliptic Function

$f_3\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^D{\left({10}^6\right)}^{\frac{i-1}{D-1}}{\mathbf{z}}_i^2+f_{\mathit{opt}}},$

where z = R(x − x^opt).

Properties: Unimodal; Nonseparable; Quadratic ill-conditioned; Smooth local irregularities.

12.3.4 Shifted and Rotated Discus Function

$f_4\left(\mathbf{x}\right)={10}^6·{\mathbf{z}}_1^2+{\displaystyle \sum _{i=2}^D{\mathbf{z}}_i^2+f_{\mathit{opt}}},$

where z = R(x − x^opt).

Properties: Unimodal; Nonseparable; Smooth local irregularities; With One sensitive direction.

12.3.5 Shifted and Rotated Bent Cigar Function

$f_5\left(\mathbf{x}\right)={\mathbf{z}}_1^2+{10}^6·{\displaystyle \sum _{i=2}^D{\mathbf{z}}_i^2+f_{\mathit{opt}}},$

where z = R(x − x^opt).

Properties: Unimodal; Nonseparable; Optimum located in a smooth but very narrow valley.

12.3.6 Shifted and Rotated Different Powers Function

$f_6\left(\mathbf{x}\right)=\sqrt{{\displaystyle \sum _{i=1}^D{\left|{\mathbf{z}}_i\right|}^{2+4\frac{i-1}{D-1}}}}+f_{\mathit{opt}},$

where z = R(0.01(x − x^opt)).

Properties: Unimodal; Nonseparable; Sensitivities of the z_i variables are different.

12.3.7 Shifted and Rotated Sharp Valley Function

$f_7\left(\mathbf{x}\right)={\mathbf{z}}_i^2+100·\sqrt{{\displaystyle \sum _{i=2}^D{\mathbf{z}}_i^2}}+f_{\mathit{opt}},$

where z = R(x − x^opt).

Properties: Unimodal; Nonseparable; Global optimum located in a sharp (non-differentiable) ridge.

12.4 Basic Multimodal Functions

12.4.1 Shifted and Rotated Step Function

$f_8\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^D{\lfloor {\mathbf{z}}_i+0.5\rfloor }^2+f_{\mathit{opt}}},$

where z = R(x − x^opt).

Properties: Many Plateaus of different sizes; Nonseparable.

12.4.2 Shifted and Rotated Weierstrass Function

$f_9\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^D\left({\displaystyle \sum _{k=0}^{k_{\mathit{max}}}{a}^k\text{cos}(2\pi b^k\left({\mathbf{z}}_i+0.5\right))}\right)}-D·{\displaystyle \sum _{k=0}^{k_{\mathit{max}}}{a}^k\text{cos}(2\pi b^k·0.5)+f_{\mathit{opt}}},$

where a = 0.5, b = 3, k_max = 20, z = R(0.005 · (x − x^opt)).

Properties: Multimodal; Nonseparable; Continuous everywhere but only differentiable on a set of points.

12.4.3 Shifted and Rotated Griewank Function

$f_{10}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^D\frac{{\mathbf{z}}_i^2}{4000}-{\displaystyle \prod _{i=1}^D\text{cos}(\frac{{\mathbf{z}}_i}{\sqrt{i}})+1+f_{\mathit{opt}}},}$

where z = R(6 · (x − x^opt)).

Properties: Multimodal; Nonseparable; With many regularly distributed local optima.

12.4.4 Shifted Rastrigin Function

$f_{11}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^D\left({\mathbf{z}}_i^2-10\text{cos}\left(2\pi {\mathbf{z}}_i\right)\right)+10·D+f_{\mathit{opt}}},$

where z = 0.0512 · (x − x^opt).

Properties: Multimodal; Separable; With many regularly distributed local optima.

12.4.5 Shifted and Rotated Rastrigin Function

$f_{12}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^D\left({\mathbf{z}}_i^2-10\text{cos}\left(2\pi {\mathbf{z}}_i\right)+10\right)+f_{\mathit{opt}}},$

where z = R(0.0512 · (x − x^opt)).

Properties: Multimodal; Nonseparable; With many regularly distributed local optima.

12.4.6 Shifted Rotated Schaffer’s F7 Function

$f_{13}\left(\mathbf{x}\right)={\left(\frac{1}{D-1}{\displaystyle \sum _{i=1}^{D-1}((1+{\text{sin}}^2(50·{\mathbf{w}}_i^{0.2}))·\sqrt{{\mathbf{w}}_i})}\right)}^2+f_{\mathit{opt}},$

where ${\mathbf{w}}_i=\sqrt{{\mathbf{z}}_i^2+{\mathbf{z}}_{i+1}^2}$, z = R(x − x^opt).

Properties: Multimodal; Nonseparable.

12.4.7 Expanded Griewank Plus Rosenbrock Function

$\begin{array}{c}\text{Rosenbrock}\text{Function}:g_2\left(x,y\right)=100{\left(x^2-y\right)}^2+{\left(x-1\right)}^2\\ \text{Griewank}\text{Function}:g_3\left(x\right)=x^2/4000-\text{cos}\left(x\right)+1\\ f_{14}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^{D-1}{g}_3\left(g_2\left({\mathbf{z}}_i,{\mathbf{z}}_{i+1}\right)\right)+g_3\left(g_2\left({\mathbf{z}}_D,{\mathbf{z}}_1\right)\right)+f_{\mathit{opt}},}\end{array}$

where z = R(0.05 · (x − x^opt)) + 1.

Properties: Multimodal; Nonseparable.

12.4.8 Shifted and Rotated Rosenbrock Function

$f_{15}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^{D-1}\left(100·{({\mathbf{z}}_i^2-{\mathbf{z}}_{i+1})}^2+{({\mathbf{z}}_i-1)}^2\right)+f_{\mathit{opt}}},$

where z = R(0.02048 · (x − x^opt)) + 1.

Properties: Multimodal; Nonseparable; With a long, narrow, parabolic shaped flat valley from local optima to global optima.

12.4.9 Shifted Modified Schwefel Function

$f_{16}\left(\mathbf{x}\right)=418.9829\times D-{\displaystyle \sum _{i=1}^D{g}_1\left({\mathbf{w}}_i\right)},{\mathbf{w}}_i={\mathbf{z}}_i+420.9687462275036,$

$g_1\left({\mathbf{w}}_i\right)=\left\{\begin{array}{ll}{\mathbf{w}}_i·\text{sin}\left(\sqrt{\left|{\mathbf{w}}_i\right|}\right)\hfill & \text{if}\left|{\mathbf{w}}_i\right|\le 500\hfill \\ \left(500-\text{mod}\left({\mathbf{w}}_i,500\right)\right)·\text{sin}\left(\sqrt{500-\text{mod}\left({\mathbf{w}}_i,500\right)}\right)-\frac{{\left({\mathbf{w}}_i-500\right)}^2}{10000D}\hfill & \text{if}{\mathbf{w}}_i>500\hfill \\ \left(\text{mod}\left(-{\mathbf{w}}_i,500\right)-500\right)·\text{sin}\left(\sqrt{500-\text{mod}\left(-{\mathbf{w}}_i,500\right)}\right)-\frac{{\left({\mathbf{w}}_i+500\right)}^2}{10000D}\hfill & \text{if}{\mathbf{w}}_i<-500\hfill \end{array}\right.,$

where z = 10 · (x − x^opt).

Properties: Multimodal; Separable; Having many local optima with the second better local optima far from the global optima.

12.4.10 Shifted Rotated Modified Schwefel Function

$f_{17}\left(\mathbf{x}\right)=418.9829\times D-{\displaystyle \sum _{i=1}^D{g}_1\left({\mathbf{w}}_i\right)},{\mathbf{w}}_i={\mathbf{z}}_i+420.9687462275036,$

where z = R(10 · (x − x^opt)) and g₁(·) is defined as Eq. (12.17).

Properties: Multimodal; Nonseparable; Having many local optima with the second better local optima far from the global optima.

12.4.11 Shifted Rotated Katsuura Function

$f_{18}\left(\mathbf{x}\right)=\frac{10}{D^2}{\displaystyle \prod _{i=1}^D{(1+i{\displaystyle \sum _{j=1}^{32}\frac{|2^j·{\mathbf{z}}_i-[2^j·{\mathbf{z}}_i]|}{2^j}})}^{\frac{10}{D^{1.2}}}-\frac{10}{D^2}}+f_{\mathit{opt}},$

where z = R(0.05 · (x − x^opt)).

Properties: Multimodal; Nonseparable; Continuous everywhere but differentiable nowhere.

12.4.12 Shifted and Rotated Lunacek Bi-Rastrigin Function

$f_{19}\left(\mathbf{x}\right)=\text{min}\left({\displaystyle \sum _{i=1}^D{({\mathbf{z}}_i-{\mu }_1)}^2,\mathit{dD}+s{\displaystyle \sum _{i=1}^D{({\mathbf{z}}_i-{\mu }_2)}^2)}}\right)+10·(D-{\displaystyle \sum _{i=1}^D\text{cos}(2\pi ({\mathbf{z}}_i-{\mu }_1))})+f_{\mathit{opt}},$

where z = R(0.1 · (x − x^opt) + 2.5 * 1), μ₁ = 2.5, μ₂ = −2.5, d = 1, s = 0.9.

Properties: Multimodal; Nonseparable; With two funnel around μ₁1 and μ₂1.

12.4.13 Shifted and Rotated Ackley Function

$f_{20}\left(\mathbf{x}\right)=-20·\text{exp}\left(-0.2\sqrt{\frac{1}{D}{\displaystyle \sum _{i=1}^D{\mathbf{x}}_i^2}}\right)-\text{exp}\left(\frac{1}{D}{\displaystyle \sum _{i=1}^D\text{cos}\left(2\pi {\mathbf{x}}_i\right)}\right)+20+e+f_{\mathit{opt}},$

where z = R(x − x^opt).

Properties: Multimodal; Nonseparable; Having many local optima with the global optima located in a very small basin.

12.4.14 Shifted Rotated HappyCat Function

$f_{21}\left(\mathbf{x}\right)={|{\displaystyle \sum _{i=1}^D{\mathbf{z}}_i^2-D}|}^{0.25}+(\frac{1}{2}{\displaystyle \sum _{j=1}^D{\mathbf{z}}_j^2}+{\displaystyle \sum _{j=1}^D{\mathbf{z}}_j})/D+0.5+f_{\mathit{opt}},$

where z = R(0.05 · (x − x^opt)) − 1.

Properties: Multimodal; Nonseparable; Global optima located in curved narrow valley.

12.4.15 Shifted Rotated HGBat Function

$f_{22}\left(\mathbf{x}\right)={|{({\displaystyle \sum _{i=1}^D{\mathbf{z}}_i^2})}^2-{({\displaystyle \sum _{j=1}^D{\mathbf{z}}_j})}^2|}^{0.5}+(\frac{1}{2}{\displaystyle \sum _{j=1}^D{\mathbf{z}}_j^2}+{\displaystyle \sum _{j=1}^D{\mathbf{z}}_j})/D+0.5+f_{\mathit{opt}},$

where z = R(0.05 · (x − x^opt)) − 1.

Properties: Multimodal; Nonseparable; Global optima located in curved narrow valley.

12.4.16 Expanded Schaffer’s F6 Function

$\begin{array}{c}\text{Schaffer’s}\text{F}6\text{Function}:g_4\left(x,y\right)=\frac{{\text{sin}}^2\left(\sqrt{x^2+y^2}\right)-0.5}{{(1+0.001·\left(x^2+y^2\right))}^2}+0.5,\\ f_{23}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^{D-1}{g}_4\left({\mathbf{z}}_i,{\mathbf{z}}_{i+1}\right)+g_4\left({\mathbf{z}}_D,{\mathbf{z}}_1\right)+f_{\mathit{opt}}},\end{array}$

where z = R(x − x^opt).

Properties: Multimodal; Nonseparable.

12.5.1 Hybrid Function 1

• N = 3

• p =[0.3, 0.3, 0.4]

• G₁: Modified Schwefel’s Function

• G₂: Rastrigin Function

• G₃: High Conditioned Elliptic Function

12.5.2 Hybrid Function 2

• N = 5

• p =[0.3, 0.3, 0.4]

• G₁: Bent Cigar Function

• G₂: HGBat Function

• G₃: Rastrigin Function

12.5.3 Hybrid Function 3

• N = 4

• p =[0.2, 0.2, 0.3, 0.3]

• G₁: Griewank Function

• G₂: Weierstrass Function

• G₃: Rosenbrock Function

• G₄: Expanded Scaffer’s F6 Function

12.5.4 Hybrid Function 4

• N = 4

• p =[0.2, 0.2, 0.3, 0.3]

• G₁: HGBat Function

• G₂: Discus Function

• G₃: Expanded Griewank plus Rosenbrock Function

• G₄: Rastrigin Function

12.5.5 Hybrid Function 5

• N = 5

• p =[0.1, 0.2, 0.2, 0.2, 0.3]

• G₁: Expanded Scaffer’s F6 Function

• G₂: HGBat Function

• G₃: Rosenbrock Function

• G₄: Modified Schwefel’s Function

• G₅: High Conditioned Elliptic Function

12.5.6 Hybrid Function 6

• N = 5

• p =[0.1, 0.2, 0.2, 0.2, 0.3]

• G₁: Katsuura Function

• G₂: HappyCat Function

• G₃: Expanded Griewank plus Rosenbrock Function

• G₄: Modified Schwefel’s Function

• G₅: Ackley Function

12.6.1 Composition Function 1

• N = 5

• σ =[10, 20, 30, 40, 50]

• λ = [1e − 10, 1e − 6, 1e − 26, 1e − 6, 1e − 6]

• bias = [0, 100, 200, 300, 400]

• G₁: Rotated Rosenbrock Function

• G₂: High Conditioned Elliptic Function

• G₃: Rotated Bent Cigar Function

• G₄: Rotated Discus Function

• G₅: High Conditioned Elliptic Function

12.6.2 Composition Function 2

• N = 3

• σ = [15, 15, 15]

• λ = [1, 1, 1]

• bias = [0, 100, 200]

• G₁: Expanded Schwefel Function

• G₂: Rotated Rstrigin Function

• G₃: Rotated HGBat Function

12.6.3 Composition Function 3

• N = 3

• σ = [20, 50, 40]

• λ = [0.25, 1, 1e − 7]

• bias = [0, 100, 200]

• G₁: Rotated Schwefel Function

• G₂: Rotated Rastrigin Function

• G₃: Rotated High Conditioned Elliptic Function

12.6.4 Composition Function 4

• N = 5

• σ = [20, 15, 10, 10, 40]

• λ = [2.5e − 2, 0.1, 1e − 8, 0.25, 1]

• bias = [0, 100, 200, 300, 400]

• G₁: Rotated Schwefel Function

• G₂: Rotated HappyCat Function

• G₃: Rotated High Conditioned Elliptic Function

• G₄: Rotated Weierstrass Function

• G₅: Rotated Griewank Function

12.6.5 Composition Function 5

• N = 5

• σ = [15, 15, 15, 15, 15]

• λ = [10, 10, 2.5, 2.5, 1e − 6]

• bias = [0, 100, 200, 300, 400]

• G₁: Rotated HGBat Function

• G₂: Rotated Rastrigin Function

• G₅: Rotated Schwefel Function

• G₄: Rotated Weierstrass Function

• G₃: Rotated High Conditioned Elliptic Function

12.6.6 Composition Function 6

• N = 5

• σ = [10, 20, 30, 40, 50]

• λ = [2.5, 10, 2.5, 5e − 4, 1e − 6]

• bias = [0, 100, 200, 300, 400]

• G₁: Rotated Expanded Griewank plus Rosenbrock Function

• G₂: Rotated HappyCat Function

• G₃: Rotated Schwefel Function

• G₄: Rotated Expanded Scaffer’s F6 Function

• G₅: High Conditioned Elliptic Function

12.6.7 Composition Function 7

• N = 3

• σ = [10, 30, 50]

• λ = [1, 1, 1]

• bias = [0, 100, 200]

• G₁: Hybrid Function 1

• G₂: Hybrid Function 2

• G₃: Hybrid Function 3

12.6.8 Composition Function 8

• N = 3

• σ = [10, 30, 50]

• λ = [1, 1, 1]

• bias = [0, 100, 200]

• G₁: Hybrid Function 4

• G₂: Hybrid Function 5

• G₃: Hybrid Function 6