opytimark.markers.n_dimensional

class opytimark.markers.n_dimensional.Ackley1(name: Optional[str] = 'Ackley1', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Ackley1 class implements the Ackley’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = -20e^{-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_i^2}}-e^{\frac{1}{n}\sum_{i=1}^{n}cos(2 \pi x_i)}+ 20 + e\]

Domain:

The function is commonly evaluated using \(x_i \in [-32, -32] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Ackley4(name: Optional[str] = 'Ackley4', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Ackley4 class implements the Ackley’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-1}(e^{-0.2}\sqrt{x_i^2+x_{i+1}^2}+3(cos(2x_i)+sin(2x_{i+1})))\]

Domain:

The function is commonly evaluated using \(x_i \in [-35, -35] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = −4.590101633799122 \mid \mathbf{x^*} = (-1.51, -0.755)\).

class opytimark.markers.n_dimensional.Alpine1(name: Optional[str] = 'Alpine1', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Alpine1 class implements the Alpine’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}|x_i sin(x_i)+0.1x_i|\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Alpine2(name: Optional[str] = 'Alpine2', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Alpine2 class implements the Alpine’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \prod_{i=1}^{n}\sqrt{x_i}sin(x_i)\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 2.808^n \mid \mathbf{x^*} = (7.917, 7.917, \ldots, 7.917)\).

class opytimark.markers.n_dimensional.Brown(name: Optional[str] = 'Brown', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Brown class implements the Brown’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-1}(x_i^2)^{(x_{i+1}^{2}+1)}+(x_{i+1}^2)^{(x_{i}^{2}+1)}\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 4] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.ChungReynolds(name: Optional[str] = 'ChungReynolds', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

ChungReynolds class implements the Chung Reynolds’ benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = (\sum_{i=1}^{n} x_i^2)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.CosineMixture(name: Optional[str] = 'CosineMixture', dims: Optional[int] = -1, continuous: Optional[bool] = False, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

CosineMixture class implements the Cosine Mixture’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = -0.1\sum_{i=1}^{n}cos(5 \pi x_i) - \sum_{i=1}^{n}x_i^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 1] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -0.1n \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Csendes(name: Optional[str] = 'Csendes', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Csendes class implements the Csendes’ benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}x_i^6(2 + sin(\frac{1}{x_i}))\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 1] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Deb1(name: Optional[str] = 'Deb1', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Deb1 class implements the Deb’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = -\frac{1}{n}\sum_{i=1}^{n}sin^6(5 \pi x_i)\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 1] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -1 \mid \mathbf{x^*} = (-0.9, -0.7, \ldots, 0.9)\).

class opytimark.markers.n_dimensional.Deb3(name: Optional[str] = 'Deb3', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Deb3 class implements the Deb’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = -\frac{1}{n}\sum_{i=1}^{n}sin^6(5 \pi (x_i^{\frac{3}{4}}-0.05))\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 1] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = ? \mid \mathbf{x^*} = (?, ?, \ldots, ?)\).

class opytimark.markers.n_dimensional.DixonPrice(name: Optional[str] = 'DixonPrice', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

DixonPrice class implements the Dixon & Price’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = (x_1 - 1)^2 + \sum_{i=2}^{n}i(2x_i^2 - x_{i-1})^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid x_i^* = 2^{-\frac{2^i-2}{2^i}}\).

class opytimark.markers.n_dimensional.Exponential(name: Optional[str] = 'Exponential', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Exponential class implements the Exponential’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = e^{-0.5\sum_{i=1}^n{x_i^2}}\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 1] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.F8F2(name: Optional[str] = 'F8F2', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

F8F2 class implements the Shifted Expanded Griewank’s plus Rosenbrock’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = f(x_1, x_2) + f(x_2, x_3) + \ldots + f(x_n, f_1)\]

Domain:

The function is commonly evaluated using \(x_i \in [-5, 5] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 1, \ldots, 1)\).

class opytimark.markers.n_dimensional.Griewank(name: Optional[str] = 'Griewank', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Griewank class implements the Griewank’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = 1 + \sum_{i=1}^{n}\frac{x_i^2}{4000} - \prod cos(\frac{x_i}{\sqrt{i}})\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.HappyCat(name: Optional[str] = 'HappyCat', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

HappyCat class implements the HappyCat’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = [(||\mathbf{x}||_2 - n)^2]^{\alpha} + \frac{1}{n}(\frac{1}{2}||\mathbf{x}||_2 + \sum_{i=1}^{n}x_i) + \frac{1}{2}\]

Domain:

The function is commonly evaluated using \(x_i \in [-2, 2] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (-1, -1, \ldots, -1)\).

class opytimark.markers.n_dimensional.HighConditionedElliptic(name: Optional[str] = 'HighConditionedElliptic', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

HighConditionedElliptic class implements the High Conditioned Elliptic’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n} (10^6)^\frac{i-1}{n-1} x_i^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Levy(name: Optional[str] = 'Levy', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Levy class implements the Levy’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = sin^2(\pi w_1) + \sum_{i=1}^{n-1}(w_i-1)^2 [1+10sin^2(\pi w_i + 1)]\]

\[+ (w_n - 1)^2 [1 + sin^2(2 \pi w_n)] \mid w_i = 1 + \frac{x_i - 1}{4}\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 1, \ldots, 1)\).

class opytimark.markers.n_dimensional.Michalewicz(name: Optional[str] = 'Michalewicz', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Michalewicz class implements the Michalewicz’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = - \sum_{i=1}^{n}sin(x_i)sin^{20}(\frac{ix_i^2}{\pi})\]

Domain:

The function is commonly evaluated using \(x_i \in [0, \pi] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = ? \mid \mathbf{x^*} = (?, ?, \ldots, ?)\).

class opytimark.markers.n_dimensional.NonContinuousExpandedScafferF6(name: Optional[str] = 'NonContinuousExpandedScafferF6', dims: Optional[int] = -1, continuous: Optional[bool] = False, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

NonContinuousExpandedScafferF6 class implements the Non-Continuous Expanded Scaffer’s F6 benchmarking function.

\[f(\mathbf{y}) = f(y_1, y_2, \ldots, y_n) = f(y_1, y_2) + f(y_2, y_3) + \ldots + f(y_n, y_1) \mid y_i = round(2x_i)/2, |x_i| >= 0.5\]

Domain:

The function is commonly evaluated using \(y_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{y^*}) = 0 \mid \mathbf{y^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.NonContinuousRastrigin(name: Optional[str] = 'NonContinuousRastrigin', dims: Optional[int] = -1, continuous: Optional[bool] = False, convex: Optional[bool] = True, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

NonContinuousRastrigin class implements the Non-Continuous Rastrigin’s benchmarking function.

\[f(\mathbf{x}) = f(y_1, y_2, \ldots, y_n) = 10n + \sum_{i=1}^{n}(y_i^2 - 10cos(2 \pi y_i)) \mid y_i = round(2x_i)/2, |x_i| >= 0.5\]

Domain:

The function is commonly evaluated using \(y_i \in [-5.12, 5.12] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{y^*}) = 0 \mid \mathbf{y^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Pathological(name: Optional[str] = 'Pathological', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Pathological class implements the Pathological’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-1}0.5 + \frac{sin^2(\sqrt{100x_i^2+x_{i+1}^2})-0.5}{1 + 0.001(x_i^2 - 2x_i x_{i+1} + x_{i+1}^2)^2}\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Periodic(name: Optional[str] = 'Periodic', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Periodic class implements the Periodic’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = 1 + \sum_{i=1}^{n}sin^2(x_i) - 0.1e^{\sum_{i=1}^{n}x_i^2}\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0.9 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Perm0DBeta(name: Optional[str] = 'Perm0DBeta', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Perm0DBeta class implements the Perm 0, D, Beta’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}(\sum_{j=1}^{n} (j + 10)(x_j^i - \frac{1}{j^i}))^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-n, n] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, \frac{1}{2}, \ldots, \frac{1}{n})\).

class opytimark.markers.n_dimensional.PermDBeta(name: Optional[str] = 'PermDBeta', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

PermDBeta class implements the Perm D, Beta’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}(\sum_{j=1}^{n} (j^i + 10)((\frac{x_j}{j})^i - 1))^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-n, n] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 2, \ldots, n)\).

class opytimark.markers.n_dimensional.PowellSingular2(name: Optional[str] = 'PowellSingular2', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

PowellSingular2 class implements the Powell’s Singular 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-2}(x_{i-1}+10x_i)^2 + 5(x_{i+1} - x_{i+2})^2 + (x_i - 2x_{i+1})^4 + 10(x_{i-1} - x_{i+2})^4\]

Domain:

The function is commonly evaluated using \(x_i \in [-4, 5] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.PowellSum(name: Optional[str] = 'PowellSum', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

PowellSum class implements the Powell’s Sum benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}|x_i|^{i+1}\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 1] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Qing(name: Optional[str] = 'Qing', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Qing class implements the Qing’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}(x_i^2 - i)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-500, 500] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid x_i^* = (\pm \sqrt{i}, \pm \sqrt{i}, \ldots, \pm \sqrt{i})\).

class opytimark.markers.n_dimensional.Quartic(name: Optional[str] = 'Quartic', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Quartic class implements the Quartic’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}ix_i^4 + rand()\]

Domain:

The function is commonly evaluated using \(x_i \in [-1.28, 1.28] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 + rand() \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Quintic(name: Optional[str] = 'Quintic', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Quintic class implements the Quintic’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}|x_i^5 - 3x_i^4 + 4x_i^3 + 2x_i^2 - 10x_i - 4|\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (-1 or 2, -1 or 2, \ldots, -1 or 2)\).

class opytimark.markers.n_dimensional.Rana(name: Optional[str] = 'Rana', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Rana class implements the Rana’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-2}(x_{i+1} + 1)cos(t_2)sin(t_1) + x_i cos(t_1)sin(t_2)\]

Domain:

The function is commonly evaluated using \(x_i \in [-500, 500] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Rastrigin(name: Optional[str] = 'Rastrigin', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Rastrigin class implements the Rastrigin’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = 10n + \sum_{i=1}^{n}(x_i^2 - 10cos(2 \pi x_i))\]

Domain:

The function is commonly evaluated using \(x_i \in [-5.12, 5.12] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Ridge(name: Optional[str] = 'Ridge', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Ridge class implements the Ridge’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = x_1 + (\sum_{i=2}^{n}x_i^2)^{0.5}\]

Domain:

The function is commonly evaluated using \(x_i \in [-\lambda, \lambda]^n \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -\lambda \mid \mathbf{x^*} = (-\lambda, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Rosenbrock(name: Optional[str] = 'Rosenbrock', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Rosenbrock class implements the Rosenbrock’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-1}[100(x_{i+1}-x_i^2)^2 + (x_i - 1)^2]\]

Domain:

The function is commonly evaluated using \(x_i \in [-30, 30] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 1, \ldots, 1)\).

class opytimark.markers.n_dimensional.RotatedExpandedScafferF6(name: Optional[str] = 'RotatedExpandedScafferF6', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

RotatedExpandedScafferF6 class implements the Rotated Expanded Scaffer’s F6 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = f(x_1, x_2) + f(x_2, x_3) + \ldots + f(x_n, x_1)\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.RotatedHyperEllipsoid(name: Optional[str] = 'RotatedHyperEllipsoid', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

RotatedHyperEllipsoid class implements the Rotated Hyper-Ellipsoid’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}\sum_{j=1}^{i}x_j^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-65.536, 65.536] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Salomon(name: Optional[str] = 'Salomon', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Salomon class implements the Salomon’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = 1 - cos(2 \pi \sqrt{\sum_{i=1}^{n}x_i^2}) + 0.1\sqrt{\sum_{i=1}^{n}x_i^2}\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.SchumerSteiglitz(name: Optional[str] = 'SchumerSteiglitz', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

SchumerSteiglitz class implements the Schumer Steiglitz’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}x_i^4\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Schwefel(name: Optional[str] = 'Schwefel', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schwefel class implements the Schwefel’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = 418.9829n -\sum_{i=1}^{n} x_i sin(\sqrt{|x_i|})\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (420.9687, 420.9687, \ldots, 420.9687)\).

class opytimark.markers.n_dimensional.Schwefel220(name: Optional[str] = 'Schwefel220', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Schwefel220 class implements the Schwefel’s 2.20 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}|x_i|\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Schwefel221(name: Optional[str] = 'Schwefel221', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Schwefel221 class implements the Schwefel’s 2.21 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \max_{i=1, \ldots, n}|x_i|\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Schwefel222(name: Optional[str] = 'Schwefel222', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schwefel222 class implements the Schwefel’s 2.22 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}|x_i| + \prod_{i=1}^{n}|x_i|\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Schwefel223(name: Optional[str] = 'Schwefel223', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schwefel223 class implements the Schwefel’s 2.23 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}x_i^{10}\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Schwefel225(name: Optional[str] = 'Schwefel225', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Schwefel225 class implements the Schwefel’s 2.25 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=2}^{n}(x_i - 1)^2 + (x_1 - x_i^2)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 1, \ldots, 1)\).

class opytimark.markers.n_dimensional.Schwefel226(name: Optional[str] = 'Schwefel226', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Schwefel226 class implements the Schwefel’s 2.26 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = -\frac{1}{n} \sum_{i=1}^{n}x_i sin(\sqrt{|x_i|})\]

Domain:

The function is commonly evaluated using \(x_i \in [-500, 500] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -418.983 \mid \mathbf{x^*} = (\pm[\pi (0.5+k)]^2, \pm[\pi (0.5+k)]^2, \ldots, \pm[\pi (0.5+k)]^2)\).

class opytimark.markers.n_dimensional.Shubert(name: Optional[str] = 'Shubert', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Shubert class implements the Shubert’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \prod_{i=1}^n \sum_{j=1}^{5}cos((j+1)x_i+j)\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -186.7309 \mid \mathbf{x^*} = \text{multiple solutions}\).

class opytimark.markers.n_dimensional.Shubert3(name: Optional[str] = 'Shubert3', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Shubert3 class implements the Shubert’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n \sum_{j=1}^{5}j sin((j+1)x_i+j)\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -29.6733337 \mid \mathbf{x^*} = (?, ?, \ldots, ?)\).

class opytimark.markers.n_dimensional.Shubert4(name: Optional[str] = 'Shubert4', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Shubert4 class implements the Shubert’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n \sum_{j=1}^{5}j cos((j+1)x_i+j)\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -25.740858 \mid \mathbf{x^*} = (?, ?, \ldots, ?)\).

class opytimark.markers.n_dimensional.SchafferF6(name: Optional[str] = 'SchafferF6', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

SchafferF6 class implements the Schaffer’s F6 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-1}0.5 + \frac{sin^2(\sqrt{x_i^2+x_{i+1}^2})-0.5}{[1 + 0.001(x_i^2 + x_{i+1}^2)]^2}\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Sphere(name: Optional[str] = 'Sphere', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Sphere class implements the Sphere’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n} x_i^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-5.12, 5.12] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.SphereWithNoise(name: Optional[str] = 'SphereWithNoise', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

SphereWithNoise class implements the Sphere with Noise’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = (\sum_{i=1}^{n} x_i^2)(1 + 0.1|N(0,1)|)\]

Domain:

The function is commonly evaluated using \(x_i \in [-5.12, 5.12] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Step(name: Optional[str] = 'Step', dims: Optional[int] = -1, continuous: Optional[bool] = False, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Step class implements the Step’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n} ⌊x_i⌋\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Step2(name: Optional[str] = 'Step2', dims: Optional[int] = -1, continuous: Optional[bool] = False, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Step2 class implements the Step’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n} ⌊x_i + 0.5⌋^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (-0.5, -0.5, \ldots, -0.5)\).

class opytimark.markers.n_dimensional.Step3(name: Optional[str] = 'Step3', dims: Optional[int] = -1, continuous: Optional[bool] = False, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Step3 class implements the Step’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n} ⌊x_i^2⌋\]

Domain:

The function is commonly evaluated using \(x_i \in [-100, 100] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.StrechedVSineWave(name: Optional[str] = 'StrechedVSineWave', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

StrechedVSineWave class implements the Streched V Sine Wave’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n-1}(x_{i+1}^2 + x_i^2)^{0.25}[sin^2(50(x_{i+1}^2 + x_i^2)^{0.1})+0.1]\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.StyblinskiTang(name: Optional[str] = 'StyblinskiTang', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

StyblinskiTang class implements the Styblinski-Tang’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \frac{1}{2}\sum_{i=1}^{n}(x_i^4 - 16x_i^2 + 5x_i)\]

Domain:

The function is commonly evaluated using \(x_i \in [-5, 5] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = −39.16599n \mid \mathbf{x^*} = (-2.903534, -2.903534, \ldots, -2.903534)\).

class opytimark.markers.n_dimensional.SumDifferentPowers(name: Optional[str] = 'SumDifferentPowers', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

SumDifferentPowers class implements the Sum of Different Powers’ benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}|x_i|^{i+1}\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 1] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.SumSquares(name: Optional[str] = 'SumSquares', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

SumSquares class implements the Sum of Squares’ benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}ix_i^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Trid(name: Optional[str] = 'Trid', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Trid class implements the Trid’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}(x_i - 1)^2 - \sum_{i=2}^{n}x_i x_{i-1}\]

Domain:

The function is commonly evaluated using \(x_i \in [-n^2, n^2] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -\frac{n(n+4)(n-1)}{6} \mid x_i = i(n+1-i)\).

class opytimark.markers.n_dimensional.Trigonometric1(name: Optional[str] = 'Trigonometric1', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Trigonometric1 class implements the Trigonometric’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}[n - \sum_{j=1}^{n} cos(x_j) + i(1 - cos(x_i) - sin(x_i))]^2\]

Domain:

The function is commonly evaluated using \(x_i \in [0, \pi] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Trigonometric2(name: Optional[str] = 'Trigonometric2', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Trigonometric2 class implements the Trigonometric’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = 1 + \sum_{i=1}^{n}8sin^2[7(x_i - 0.9)^2] + 6sin^2[14(x_1-0.9)^2] + (x_i-0.9)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-500, 500] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 1 \mid \mathbf{x^*} = (0.9, 0.9, \ldots, 0.9)\).

class opytimark.markers.n_dimensional.Wavy(name: Optional[str] = 'Wavy', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Wavy class implements the Wavy’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = 1 - \frac{1}{n} \sum_{i=1}^{n}cos(10x_i)e^{\frac{-x_i^2}{2}}\]

Domain:

The function is commonly evaluated using \(x_i \in [-\pi, \pi] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Weierstrass(name: Optional[str] = 'Weierstrass', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Weierstrass class implements the Weierstrass’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n} (\sum_{k=0}^{20} [0.5^k cos(2\pi 3^k(x_i+0.5))]) - n \sum_{k=0}^{20}[0.5^k cos(2\pi 3^k 0.5)]\]

Domain:

The function is commonly evaluated using \(x_i \in [-0.5, 0.5] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.XinSheYang(name: Optional[str] = 'XinSheYang', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

XinSheYang class implements the Xin-She Yang’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{n}\epsilon_i|x_i|^i\]

Domain:

The function is commonly evaluated using \(x_i \in [-5, 5] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.XinSheYang2(name: Optional[str] = 'XinSheYang2', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

XinSheYang2 class implements the Xin-She Yang’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = (\sum_{i=1}^{n}|x_i|)e^{-\sum_{i=1}^{n}sin(x_i^2)}\]

Domain:

The function is commonly evaluated using \(x_i \in [-2 \pi, 2 \pi] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.XinSheYang3(name: Optional[str] = 'XinSheYang3', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

XinSheYang3 class implements the Xin-She Yang’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = e^{-\sum_{i=1}^{n}(\frac{x_i}{\beta})^{2m}} - 2e^{-\sum_{i=1}^{n}x_i^2} \prod_{i=1}^{n} cos^2(x_i)\]

Domain:

The function is commonly evaluated using \(x_i \in [-2 \pi, 2 \pi] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -1 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.XinSheYang4(name: Optional[str] = 'XinSheYang4', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

XinSheYang4 class implements the Xin-She Yang’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = (\sum_{i=1}^{n} sin^2(x_i) - e^{-\sum_{i=1}^{n}x_i^2})e^{-\sum_{i=1}^{n}sin^2(\sqrt{|x_i|})}\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = -1 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).

class opytimark.markers.n_dimensional.Zakharov(name: Optional[str] = 'Zakharov', dims: Optional[int] = -1, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Zakharov class implements the Zakharov’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^n x_i^{2}+(\sum_{i=1}^n 0.5ix_i)^2 + (\sum_{i=1}^n 0.5ix_i)^4\]

Domain:

The function is commonly evaluated using \(x_i \in [-5, 10] \mid i = \{1, 2, \ldots, n\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, \ldots, 0)\).