N-D Test Functions S¶

class Salomon(dimensions=2)¶

Salomon objective function.

This class defines the Salomon global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Salomon}}(x) = 1 - \cos \left (2 \pi \sqrt{\sum_{i=1}^{n} x_i^2} \right) + 0.1 \sqrt{\sum_{i=1}^n x_i^2}$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Salomon function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Sargan(dimensions=2)¶

Sargan objective function.

This class defines the Sargan global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Sargan}}(x) = \sum_{i=1}^{n} n \left (x_i^2 + 0.4 \sum_{i \neq j}^{n} x_ix_j \right)$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Sargan function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class SawtoothXY(dimensions=2)¶: SawtoothXY objective function.

Two-dimensional SawtoothXY function

class Schaffer01(dimensions=2)¶

Schaffer 1 objective function.

This class defines the Schaffer 1 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schaffer01}}(x) = 0.5 + \frac{\sin^2 (x_1^2 + x_2^2)^2 - 0.5} {1 + 0.001(x_1^2 + x_2^2)^2}$

with $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional Schaffer01 function

Global optimum: $f(x) = 0$ for $x = [0, 0]$ for $i = 1, 2$

Mishra, S. Some new test functions for global optimization and performance of repulsive particle swarm method. Munich Personal RePEc Archive, 2006, 2718

class Schaffer02(dimensions=2)¶

Schaffer 2 objective function.

This class defines the Schaffer 2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schaffer02}}(x) = 0.5 + \frac{\sin^2 (x_1^2 - x_2^2)^2 - 0.5} {1 + 0.001(x_1^2 + x_2^2)^2}$

with $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional Schaffer02 function

Global optimum: $f(x) = 0$ for $x = [0, 0]$ for $i = 1, 2$

Mishra, S. Some new test functions for global optimization and performance of repulsive particle swarm method. Munich Personal RePEc Archive, 2006, 2718

class Schaffer03(dimensions=2)¶

Schaffer 3 objective function.

This class defines the Schaffer 3 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schaffer03}}(x) = 0.5 + \frac{\sin^2 \left( \cos \lvert x_1^2 - x_2^2 \rvert \right ) - 0.5}{1 + 0.001(x_1^2 + x_2^2)^2}$

with $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional Schaffer03 function

Global optimum: $f(x) = 0.00156685$ for $x = [0, 1.253115]$

Mishra, S. Some new test functions for global optimization and performance of repulsive particle swarm method. Munich Personal RePEc Archive, 2006, 2718

class Schaffer04(dimensions=2)¶

Schaffer 4 objective function.

This class defines the Schaffer 4 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schaffer04}}(x) = 0.5 + \frac{\cos^2 \left( \sin(x_1^2 - x_2^2) \right ) - 0.5}{1 + 0.001(x_1^2 + x_2^2)^2}^2$

with $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional Schaffer04 function

Global optimum: $f(x) = 0.292579$ for $x = [0, 1.253115]$

Mishra, S. Some new test functions for global optimization and performance of repulsive particle swarm method. Munich Personal RePEc Archive, 2006, 2718

class SchmidtVetters(dimensions=3)¶

Schmidt-Vetters objective function.

This class defines the Schmidt-Vetters global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{SchmidtVetters}}(x) = \frac{1}{1 + (x_1 - x_2)^2} + \sin \left(\frac{\pi x_2 + x_3}{2} \right) + e^{\left(\frac{x_1+x_2}{x_2} - 2\right)^2}$

with $x_i \in [0, 10]$ for $i = 1, 2, 3$ .

Global optimum: $f(x) = 2.99643266$ for $x = [0.79876108, 0.79962581, 0.79848824]$

Todo

equation seems right, but [7.07083412 , 10., 3.14159293] produces a lower minimum, 0.193973

class Schwefel01(dimensions=2)¶

Schwefel 1 objective function.

This class defines the Schwefel 1 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{Schwefel01}}(x) = \left(\sum_{i=1}^n x_i^2 \right)^{\alpha}$

Where, in this exercise, $\alpha = \sqrt{\pi}$ .

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Schwefel01 function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Schwefel02(dimensions=2)¶

Schwefel 2 objective function.

This class defines the Schwefel 2 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{Schwefel02}}(x) = \sum_{i=1}^n \left(\sum_{j=1}^i x_i \right)^2$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Schwefel02 function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Schwefel04(dimensions=2)¶

Schwefel 4 objective function.

This class defines the Schwefel 4 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schwefel04}}(x) = \sum_{i=1}^n \left[(x_i - 1)^2 + (x_1 - x_i^2)^2 \right]$

Here, $n$ represents the number of dimensions and $x_i \in [0, 10]$ for $i = 1, ..., n$ .

Two-dimensional Schwefel04 function

Global optimum: $f(x) = 0$ for:math:x_i = 1 for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Schwefel06(dimensions=2)¶

Schwefel 6 objective function.

This class defines the Schwefel 6 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{Schwefel06}}(x) = \max(\lvert x_1 + 2x_2 - 7 \rvert, \lvert 2x_1 + x_2 - 5 \rvert)$

with $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional Schwefel06 function

Global optimum: $f(x) = 0$ for $x = [1, 3]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Schwefel20(dimensions=2)¶

Schwefel 20 objective function.

This class defines the Schwefel 20 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{Schwefel20}}(x) = \sum_{i=1}^n \lvert x_i \rvert$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Schwefel20 function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

Jamil #122 is incorrect. There shouldn’t be a leading minus sign.

class Schwefel21(dimensions=2)¶

Schwefel 21 objective function.

This class defines the Schwefel 21 global optimization problem. This is a unimodal minimization problem defined as follows:

$f_{\text{Schwefel21}}(x) = \smash{\displaystyle\max_{1 \leq i \leq n}} \lvert x_i \rvert$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Schwefel21 function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Schwefel22(dimensions=2)¶

Schwefel 22 objective function.

This class defines the Schwefel 22 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schwefel22}}(x) = \sum_{i=1}^n \lvert x_i \rvert + \prod_{i=1}^n \lvert x_i \rvert$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Schwefel22 function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Schwefel26(dimensions=2)¶

Schwefel 26 objective function.

This class defines the Schwefel 26 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schwefel26}}(x) = 418.9829n - \sum_{i=1}^n x_i \sin(\sqrt{|x_i|})$

Here, $n$ represents the number of dimensions and $x_i \in [-500, 500]$ for $i = 1, ..., n$ .

Two-dimensional Schwefel26 function

Global optimum: $f(x) = 0$ for $x_i = 420.968746$ for $i = 1, ..., n$

Gavana, A. Global Optimization Benchmarks and AMPGO

class Schwefel36(dimensions=2)¶

Schwefel 36 objective function.

This class defines the Schwefel 36 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Schwefel36}}(x) = -x_1x_2(72 - 2x_1 - 2x_2)$

with $x_i \in [0, 500]$ for $i = 1, 2$ .

Two-dimensional Schwefel36 function

Global optimum: $f(x) = -3456$ for $x = [12, 12]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Shekel05(dimensions=4)¶

Shekel 5 objective function.

This class defines the Shekel 5 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Shekel05}}(x) = \sum_{i=1}^{m} \frac{1}{c_{i} + \sum_{j=1}^{n} (x_{j} - a_{ij})^2 }`$

Where, in this exercise:

$a = \begin{bmatrix} 4.0 & 4.0 & 4.0 & 4.0 \\ 1.0 & 1.0 & 1.0 & 1.0 \\ 8.0 & 8.0 & 8.0 & 8.0 \\ 6.0 & 6.0 & 6.0 & 6.0 \\ 3.0 & 7.0 & 3.0 & 7.0 \end{bmatrix}$

$c = \begin{bmatrix} 0.1 \ 0.2 \ 0.2 \ 0.4 \ 0.4 \end{bmatrix}$

Here, $n$ represents the number of dimensions and $x_i \in [0, 10]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = -10.15319585$ for $x_i = 4$ for $i = 1, ..., 4$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

this is a different global minimum compared to Jamil#130. The minimum is found by doing lots of optimisations. The solution is supposed to be at [4] * N, is there any numerical overflow?

class Shekel07(dimensions=4)¶

Shekel 7 objective function.

This class defines the Shekel 7 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Shekel07}}(x) = \sum_{i=1}^{m} \frac{1}{c_{i} + \sum_{j=1}^{n} (x_{j} - a_{ij})^2 }`$

Where, in this exercise:

$a = \begin{bmatrix} 4.0 & 4.0 & 4.0 & 4.0 \\ 1.0 & 1.0 & 1.0 & 1.0 \\ 8.0 & 8.0 & 8.0 & 8.0 \\ 6.0 & 6.0 & 6.0 & 6.0 \\ 3.0 & 7.0 & 3.0 & 7.0 \\ 2.0 & 9.0 & 2.0 & 9.0 \\ 5.0 & 5.0 & 3.0 & 3.0 \end{bmatrix}$

$c = \begin{bmatrix} 0.1 \ 0.2 \ 0.2 \ 0.4 \ 0.4 \ 0.6 \ 0.3 \end{bmatrix}$

with $x_i \in [0, 10]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = -10.4028188$ for $x_i = 4$ for $i = 1, ..., 4$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

this is a different global minimum compared to Jamil#131. This minimum is obtained after running lots of minimisations! Is there any numerical overflow that causes the minimum solution to not be [4] * N?

class Shekel10(dimensions=4)¶

Shekel 10 objective function.

This class defines the Shekel 10 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Shekel10}}(x) = \sum_{i=1}^{m} \frac{1}{c_{i} + \sum_{j=1}^{n} (x_{j} - a_{ij})^2 }`$

Where, in this exercise:

$a = \begin{bmatrix} 4.0 & 4.0 & 4.0 & 4.0 \\ 1.0 & 1.0 & 1.0 & 1.0 \\ 8.0 & 8.0 & 8.0 & 8.0 \\ 6.0 & 6.0 & 6.0 & 6.0 \\ 3.0 & 7.0 & 3.0 & 7.0 \\ 2.0 & 9.0 & 2.0 & 9.0 \\ 5.0 & 5.0 & 3.0 & 3.0 \\ 8.0 & 1.0 & 8.0 & 1.0 \\ 6.0 & 2.0 & 6.0 & 2.0 \\ 7.0 & 3.6 & 7.0 & 3.6 \end{bmatrix}$

$c = \begin{bmatrix} 0.1 \ 0.2 \ 0.2 \ 0.4 \ 0.4 \ 0.6 \ 0.3 \ 0.7 \ 0.5 \ 0.5 \end{bmatrix}$

with $x_i \in [0, 10]$ for $i = 1, ..., 4$ .

Global optimum: $f(x) = -10.5362837$ for $x_i = 4$ for $i = 1, ..., 4$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

Found a lower global minimum than Jamil#132... Is this numerical overflow?

class Shubert01(dimensions=2)¶

Shubert 1 objective function.

This class defines the Shubert 1 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Shubert01}}(x) = \prod_{i=1}^{n}\left(\sum_{j=1}^{5} cos(j+1)x_i+j \right )$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i = 1, ..., n$ .

Two-dimensional Shubert01 function

Global optimum: $f(x) = -186.7309$ for $x = [-7.0835, 4.8580]$ (and many others).

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Jamil#133 is missing a prefactor of j before the cos function.

class Shubert03(dimensions=2)¶

Shubert 3 objective function.

This class defines the Shubert 3 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Shubert03}}(x) = \sum_{i=1}^n \sum_{j=1}^5 -j \sin((j+1)x_i + j)$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i = 1, ..., n$ .

Two-dimensional Shubert03 function

Global optimum: $f(x) = -24.062499$ for $x = [5.791794, 5.791794]$ (and many others).

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Jamil#134 has wrong global minimum value, and is missing a minus sign before the whole thing.

class Shubert04(dimensions=2)¶

Shubert 4 objective function.

This class defines the Shubert 4 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Shubert04}}(x) = \left(\sum_{i=1}^n \sum_{j=1}^5 -j \cos ((j+1)x_i + j)\right)$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i = 1, ..., n$ .

Two-dimensional Shubert04 function

Global optimum: $f(x) = -29.016015$ for $x = [-0.80032121, -7.08350592]$ (and many others).

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Jamil#135 has wrong global minimum value, and is missing a minus sign before the whole thing.

class Simpleton(dimensions=10)¶: Simpleton objective function.

class SineEnvelope(dimensions=2)¶

SineEnvelope objective function.

This class defines the SineEnvelope global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{SineEnvelope}}(x) = -\sum_{i=1}^{n-1}\left[\frac{\sin^2( \sqrt{x_{i+1}^2+x_{i}^2}-0.5)} {(0.001(x_{i+1}^2+x_{i}^2)+1)^2} + 0.5\right]$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional SineEnvelope function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Gavana, A. Global Optimization Benchmarks and AMPGO

Todo

Jamil #136

class Sinusoidal(dimensions=2)¶: Sinusoidal objective function.

Two-dimensional Sinusoidal function

class SixHumpCamel(dimensions=2)¶

Six Hump Camel objective function.

This class defines the Six Hump Camel global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{SixHumpCamel}}(x) = 4x_1^2+x_1x_2-4x_2^2-2.1x_1^4+ 4x_2^4+\frac{1}{3}x_1^6$

with $x_i \in [-5, 5]$ for $i = 1, 2$ .

Two-dimensional SixHumpCamel function

Global optimum: $f(x) = -1.031628453489877$ for $x = [0.08984201368301331 , -0.7126564032704135]$ or $x = [-0.08984201368301331, 0.7126564032704135]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Sodp(dimensions=2)¶

Sodp objective function.

This class defines the Sum Of Different Powers global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Sodp}}(x) = \sum_{i=1}^{n} \lvert{x_{i}}\rvert^{i + 1}$

Here, $n$ represents the number of dimensions and $x_i \in [-1, 1]$ for $i = 1, ..., n$ .

Two-dimensional Sodp function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Gavana, A. Global Optimization Benchmarks and AMPGO

class Sphere(dimensions=2)¶

Sphere objective function.

This class defines the Sphere global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Sphere}}(x) = \sum_{i=1}^{n} x_i^2$

Here, $n$ represents the number of dimensions and $x_i \in [-1, 1]$ for $i = 1, ..., n$ .

Two-dimensional Sphere function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

Jamil has stupid limits

class SphericalSinc(dimensions=2)¶: SphericalSinc objective function.

Two-dimensional SphericalSinc function

class Spike(dimensions=2)¶: Spike objective function.

Two-dimensional Spike function

class Step01(dimensions=2)¶

Step objective function.

This class defines the Step global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Step}}(x) = \sum_{i=1}^{n} \left ( \lfloor x_i + 0.5 \rfloor \right )^2$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Step01 function

Global optimum: $f(x) = 0$ for $x_i = 0.5$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Step02(dimensions=2)¶

Step objective function.

This class defines the Step 2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Step}}(x) = \sum_{i=1}^{n} \left ( \lfloor x_i + 0.5 \rfloor \right )^2$

Here, $n$ represents the number of dimensions and $x_i \in [-100, 100]$ for $i = 1, ..., n$ .

Two-dimensional Step02 function

Global optimum: $f(x) = 0$ for $x_i = 0.5$ for $i = 1, ..., n$

Gavana, A. Global Optimization Benchmarks and AMPGO

class Step03(dimensions=2)¶: Step 3 objective function.

Two-dimensional Step03 function

class Stochastic(dimensions=2)¶

Stochastic objective function.

This class defines the Stochastic global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Stochastic}}(x) = \sum_{i=1}^{n} \epsilon_i \left | {x_i - \frac{1}{i}} \right |$

The variable $\epsilon_i, (i=1,...,n)$ is a random variable uniformly distributed in $[0, 1]$ .

Here, $n$ represents the number of dimensions and $x_i \in [-5, 5]$ for $i = 1, ..., n$ .

Two-dimensional Stochastic function

Global optimum: $f(x) = 0$ for $x_i = [1/n]$ for $i = 1, ..., n$

Gavana, A. Global Optimization Benchmarks and AMPGO

class StretchedV(dimensions=2)¶

StretchedV objective function.

This class defines the Stretched V global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{StretchedV}}(x) = \sum_{i=1}^{n-1} t^{1/4} [\sin (50t^{0.1}) + 1]^2$

Where, in this exercise:

$t = x_{i+1}^2 + x_i^2$

Here, $n$ represents the number of dimensions and $x_i \in [-10, 10]$ for $i = 1, ..., n$ .

Two-dimensional StretchedV function

Global optimum: $f(x) = 0$ for $x = [0., 0.]$ when $n = 2$ .

Adorio, E. MVF - “Multivariate Test Functions Library in C for Unconstrained Global Optimization”, 2005

Todo

All the sources disagree on the equation, in some the 1 is in the brackets, in others it is outside. In Jamil#142 it’s not even 1. Here we go with the Adorio option.

class StyblinskiTang(dimensions=2)¶

StyblinskiTang objective function.

This class defines the Styblinski-Tang global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{StyblinskiTang}}(x) = \sum_{i=1}^{n} \left(x_i^4 - 16x_i^2 + 5x_i \right)$

Here, $n$ represents the number of dimensions and $x_i \in [-5, 5]$ for $i = 1, ..., n$ .

Two-dimensional StyblinskiTang function

Global optimum: $f(x) = -39.16616570377142n$ for $x_i = -2.903534018185960$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

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