opytimark.markers.two_dimensional

class opytimark.markers.two_dimensional.Ackley2(name: Optional[str] = 'Ackley2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Ackley2 class implements the Ackley’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -200e^{-0.2\sqrt{x_1^2+x_2^2}}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-32, 32], x_2 \in [-32, 32]\).

Global Minima:

\(f(\mathbf{x^*}) = -200 \mid \mathbf{x^*} = (0, 0)\).

class opytimark.markers.two_dimensional.Ackley3(name: Optional[str] = 'Ackley3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Ackley3 class implements the Ackley’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -200e^{-0.02\sqrt{x_1^2+x_2^2}} + 5e^{cos(3x_1) + sin(3x_2)}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-32, 32], x_2 \in [-32, 32]\).

Global Minima:

\(f(\mathbf{x^*}) \approx -195.62902823841935 \mid \mathbf{x^*} \approx (\pm 0.682584587365898, -0.36075325513719)\).

class opytimark.markers.two_dimensional.Adjiman(name: Optional[str] = 'Adjiman', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Adjiman class implements the Adjiman’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = cos(x_1) sin(x_2) - \frac{x_1}{x_2^2+1}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-1, 2], x_2 \in [-1, 1]\).

Global Minima:

\(f(\mathbf{x^*}) = -2.02181 \mid \mathbf{x^*} = (2, 0.10578)\).

class opytimark.markers.two_dimensional.BartelsConn(name: Optional[str] = 'BartelsConn', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

BartelsConn class implements the Bartels Conn’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = |x_1^2+x_2^2+x_1x_2| + |sin(x_1)| + |cos(x_2)|\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Beale(name: Optional[str] = 'Beale', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Beale class implements the Beale’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (1.5-x_1+x_1x_2)^2 + (2.25-x_1+x_1x_2^2)^2 + (2.625-x_1+x_1x_2^3)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-4.5, 4.5], x_2 \in [-4.5, 4.5]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (3, 0.5)\).

class opytimark.markers.two_dimensional.BiggsExponential2(name: Optional[str] = 'BiggsExponential2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

BiggsExponential2 class implements the Biggs Exponential’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = \sum_{i=1}^{10}(e^{-t_ix_1} - 5e^{-t_ix_2} - y_i)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [0, 20], x_2 \in [0, 20]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 10)\).

class opytimark.markers.two_dimensional.Bird(name: Optional[str] = 'Bird', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Bird class implements the Bird’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = sin(x_1)e^{(1-cos(x_2))^2}+cos(x_2)e^{(1-sin(x_1))^2}+(x_1-x_2)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-2\pi, 2\pi], x_2 \in [-2\pi, 2\pi]\).

Global Minima:

\(f(\mathbf{x^*}) = -106.764537 \mid \mathbf{x^*} = (4.70104, 3.15294) ~ or ~(−1.58214, −3.13024)\).

class opytimark.markers.two_dimensional.Bohachevsky1(name: Optional[str] = 'Bohachevsky1', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Bohachevsky1 class implements the Bohachevsky’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = x_1^2 + 2x_2^2 - 0.3cos(3\pi x_1) - 0.4cos(4\pi x_2) + 0.7\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Bohachevsky2(name: Optional[str] = 'Bohachevsky2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Bohachevsky2 class implements the Bohachevsky’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = x_1^2 + 2x_2^2 - 0.3cos(3\pi x_1)cos(4\pi x_2) + 0.7\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Bohachevsky3(name: Optional[str] = 'Bohachevsky3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Bohachevsky3 class implements the Bohachevsky’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = x_1^2 + 2x_2^2 - 0.3cos(3\pi x_1 + 4\pi x_2) + 0.3\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Booth(name: Optional[str] = 'Booth', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Booth class implements the Booth’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_1 + 2x_2 - 7)^2 + (2x_1 + x_2 - 5)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 3)\).

class opytimark.markers.two_dimensional.BraninHoo(name: Optional[str] = 'BraninHoo', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

BraninHoo class implements the Branin Hoo’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_2 - \frac{5.1x_1^2}{4\pi^2} + \frac{5x_1}{\pi} - 6)^2 + 10(1 - \frac{1}{8\pi})cos(x_1) + 10\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 10], x_2 \in [0, 15]\).

Global Minima:

\(f(\mathbf{x^*}) = 0.39788735775266204 \mid \mathbf{x^*} = (-\pi, 12.275) ~ or ~(\pi, 2.275) ~ or ~(3\pi, 2.425)\).

class opytimark.markers.two_dimensional.Brent(name: Optional[str] = 'Brent', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Brent class implements the Brent’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_1 + 10)^2 + (x_2 + 10)^2 + e^{-x_1^2 - x_2^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = e^{-200} \mid \mathbf{x^*} = (-10, -10)\).

class opytimark.markers.two_dimensional.Bukin2(name: Optional[str] = 'Bukin2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Bukin2 class implements the Bukin’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 100(x_2 - 0.01x_1^2 + 1) + 0.01(x_1 + 10)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-15, -5], x_2 \in [-3, 3]\).

Global Minima:

\(f(\mathbf{x^*}) = # \mid \mathbf{x^*} = (-10, 0)\).

class opytimark.markers.two_dimensional.Bukin4(name: Optional[str] = 'Bukin4', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Bukin4 class implements the Bukin’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 100x_2^2 + 0.01|x_1 + 10|\]

Domain:

The function is commonly evaluated using \(x_1 \in [-15, -5], x_2 \in [-3, 3]\).

Global Minima:

\(f(\mathbf{x^*}) = # \mid \mathbf{x^*} = (-10, 0)\).

class opytimark.markers.two_dimensional.Bukin6(name: Optional[str] = 'Bukin6', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Bukin6 class implements the Bukin’s 6th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 100sqrt{|x_2-0.01x_1^2|} + 0.01|x_1+10|\]

Domain:

The function is commonly evaluated using \(x_1 \in [-15, -5], x_2 \in [-3, 3]\).

Global Minima:

\(f(\mathbf{x^*}) = # \mid \mathbf{x^*} = (-10, 1)\).

class opytimark.markers.two_dimensional.Camel3(name: Optional[str] = 'Camel3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Camel3 class implements the Camel’s Three Hump benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 2x_1^2 - 1.05x_1^4 + \frac{x_1^6}{6} + x_1x_2 + x_2^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 5], x_2 \in [-5, 5]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Camel6(name: Optional[str] = 'Camel6', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Camel6 class implements the Camel’s Six Hump benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (4 - 2.1x_1^2 + \frac{x_1^4}{3})x_1^2 + x_1x_2 + (4x_2^2 - 4)x_2^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 5], x_2 \in [-5, 5]\).

Global Minima:

\(f(\mathbf{x^*}) = -1.0316284229280819 \mid \mathbf{x^*} = (−0.0898, 0.7126) ~ or ~(0.0898,−0.7126)\).

class opytimark.markers.two_dimensional.ChenBird(name: Optional[str] = 'ChenBird', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

ChenBird class implements the Chen Bird’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -\frac{0.001}{0.001^2 + (x_1^2 + x_2^2 - 1)^2} - \frac{0.001}{0.001^2 + (x_1^2 + x_2^2 - 0.5)^2} - \frac{0.001}{0.001^2 + (x_1^2 - x_2^2)^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 2000.0039999840003 \mid \mathbf{x^*} = (0.5, 0.5)\).

class opytimark.markers.two_dimensional.ChenV(name: Optional[str] = 'ChenV', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

ChenV class implements the Chen V’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = \frac{0.001}{0.001^2 + (x_1 - 0.4x_2 - 0.1)^2} + \frac{0.001}{0.001^2 + (2x_1 + x_2 - 1.5)^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 2000.0000000000002 \mid \mathbf{x^*} = (0.5, 0.5)\).

class opytimark.markers.two_dimensional.Chichinadze(name: Optional[str] = 'Chichinadze', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Chichinadze class implements the Chichinadze’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = x_1^2 - 12x_1 + 11 + 10cos(\frac{\pi x_1}{2}) + 8sin(\frac{5\pi x_1}{2}) - \frac{1}{5}^{0.5} e^{-0.5(x_2-0.5)^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-30, 30], x_2 \in [-30, 30]\).

Global Minima:

\(f(\mathbf{x^*}) = −43.3159 \mid \mathbf{x^*} = (5.90133, 0.5)\).

class opytimark.markers.two_dimensional.CrossTray(name: Optional[str] = 'CrossTray', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

CrossTray class implements the CrossTray’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -0.0001(|sin(x_1)sin(x_2)e^{|100-\frac{sqrt{x_1^2 + x_2^2}}{\pi}|}| + 1)^0.1\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = −2.06261218 \mid \mathbf{x^*} = (\pm 1.349406685353340, \pm 1.349406608602084)\).

class opytimark.markers.two_dimensional.Cube(name: Optional[str] = 'Cube', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Cube class implements the Cube’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 100(x_2 - x_1^3)^2 + (1 - x_1)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Damavandi(name: Optional[str] = 'Damavandi', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Damavandi class implements the Damavandi’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (1 - |\frac{sin(\pi (x_1-2))sin(\pi (x_2-2))}{\pi^2 (x_1-2)(x2_2)}|^5)(2 + (x_1-7)^2 + 2(x_2-7)^2)\]

Domain:

The function is commonly evaluated using \(x_1 \in [0, 14], x_2 \in [0, 14]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (2.00000001, 2.00000001)\).

class opytimark.markers.two_dimensional.DeckkersAarts(name: Optional[str] = 'DeckkersAarts', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

DeckkersAarts class implements the Deckkers Aarts’ benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 10^5x_1^2 + x_2^2 - (x_1^2 + x_2^2)^2 + 10^{-5}(x_1^2 + x_2^2)^4\]

Domain:

The function is commonly evaluated using \(x_1 \in [-20, 20], x_2 \in [-20, 20]\).

Global Minima:

\(f(\mathbf{x^*}) = -24771.093749999996 \mid \mathbf{x^*} = (0, \pm 15)\).

class opytimark.markers.two_dimensional.DropWave(name: Optional[str] = 'DropWave', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

DropWave class implements the Drop Wave’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = - \frac{1 + cos(12\sqrt{x_1^2+x_2^2})}{0.5(x_1^2+x_2^2) + 2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5.2, 5.2], x_2 \in [-5.2, 5.2]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Easom(name: Optional[str] = 'Easom', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Easom class implements the Easom’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -cos(x_1)cos(x_2)e^{-(x_1-\pi)^2 -(x_2-\pi)^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

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

class opytimark.markers.two_dimensional.ElAttarVidyasagarDutta(name: Optional[str] = 'ElAttarVidyasagarDutta', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

ElAttarVidyasagarDutta class implements the El-Attar-Vidyasagar-Dutta’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_1^2 + x_2 - 10)^2 + (x_1 + x_2^2 - 7)^2 + (x_1^2 + x_2^3 - 1)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 1.7127803548622027 \mid \mathbf{x^*} = (3.4091868222, −2.1714330361)\).

class opytimark.markers.two_dimensional.EggCrate(name: Optional[str] = 'EggCrate', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

EggCrate class implements the Egg Crate’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = x_1^2 + x_2^2 + 25(sin^2(x_1) + sin^2(x_2))\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 5], x_2 \in [-5, 5]\).

Global Minima:

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

class opytimark.markers.two_dimensional.EggHolder(name: Optional[str] = 'EggHolder', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

EggHolder class implements the Egg Holder’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -(x_2 + 47)sin(\sqrt{|x_2+\frac{x_1}{2}+47|})-x_1 sin(\sqrt{|x_1-(x_2+47)|})\]

Domain:

The function is commonly evaluated using \(x_1 \in [-512, 512], x_2 \in [-512, 512]\).

Global Minima:

\(f(\mathbf{x^*}) = -959.6406627106155 \mid \mathbf{x^*} = (512, 404.2319)\).

class opytimark.markers.two_dimensional.FreudensteinRoth(name: Optional[str] = 'FreudensteinRoth', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

FreudensteinRoth class implements the Freudenstein Roth’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_1 - 13 + ((5 - x_2)x_2 - 2)x_2) ** 2 + (x_1 - 29 + ((x_2 + 1)x_2 - 14)x_2) ** 2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (5, 4)\).

class opytimark.markers.two_dimensional.Giunta(name: Optional[str] = 'Giunta', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Giunta class implements the Giunta’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.6 + \sum_{i=1}^{2}[sin(\frac{16}{15}x_i-1) + sin^2(\frac{16}{15}x_i-1) + \frac{1}{50}sin(4(\frac{16}{15}x_i-1))]\]

Domain:

The function is commonly evaluated using \(x_1 \in [-1, 1], x_2 \in [-1, 1]\).

Global Minima:

\(f(\mathbf{x^*}) = 0.06447042053690571 \mid \mathbf{x^*} = (0.4673200277395354, 0.4673200169591304)\).

class opytimark.markers.two_dimensional.GoldsteinPrice(name: Optional[str] = 'GoldsteinPrice', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

GoldsteinPrice class implements the Goldstein Price’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = [1 + (x_1 + x_2 + 1)^2 (19 - 14x_1 + 3x_1^2 - 14x_2 + 6x_1x_2 + 3x_2^2)] [30 + (2x_1 - 3x_2)^2 (18 - 32x_1 + 12x_1^2 + 48x_2 - 36x_1x_2 + 27x_2^2)]\]

Domain:

The function is commonly evaluated using \(x_1 \in [-2, 2], x_2 \in [-2, 2]\).

Global Minima:

\(f(\mathbf{x^*}) = 3 \mid \mathbf{x^*} = (0, -1)\).

class opytimark.markers.two_dimensional.Himmelblau(name: Optional[str] = 'Himmelblau', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Himmelblau class implements the Himmelblau’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_1^2 + x_2 - 11)^2 + (x_1 + x_2^2 - 7)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 5], x_2 \in [-5, 5]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (3, 2)\).

class opytimark.markers.two_dimensional.HolderTable(name: Optional[str] = 'HolderTable', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

HolderTable class implements the HolderTable’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -|sin(x_1)cos(x_2)e^{|1 - \frac{\sqrt{x_1^2 + x_2^2}}{\pi}|}|\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -19.208502567767606 \mid \mathbf{x^*} = (\pm 8.05502, \pm 9.66459)\).

class opytimark.markers.two_dimensional.Hosaki(name: Optional[str] = 'Hosaki', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Hosaki class implements the Hosaki’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (1 - 8x_1 + 7x_1^2 - \frac{7}{3x_1^3} + \frac{1}{4x_1^4})x_2^2e^{-x_2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [0, 5], x_2 \in [0, 6]\).

Global Minima:

\(f(\mathbf{x^*}) = -2.345811576101292 \mid \mathbf{x^*} = (4, 2)\).

class opytimark.markers.two_dimensional.JennrichSampson(name: Optional[str] = 'JennrichSampson', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

JennrichSampson class implements the Jennrich Sampson’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = \sum_{i=1}^{10}(2 + 2i - (e^{ix_1} + e^{ix_2}))^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-1, 1], x_2 \in [-1, 1]\).

Global Minima:

\(f(\mathbf{x^*}) = 124.36218236181409 \mid \mathbf{x^*} = (0.257825, 0.257825)\).

class opytimark.markers.two_dimensional.Keane(name: Optional[str] = 'Keane', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Keane class implements the Keane’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = \frac{sin^2(x_1-x_2)sin^2(x_1+x_2)}{sqrt{x_1^2+x_2^2}}\]

Domain:

The function is commonly evaluated using \(x_1 \in [0, 10], x_2 \in [0, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = 0.6736675211468548 \mid \mathbf{x^*} = (1.393249070031784, 0) ~ or ~(0, 1.393249070031784)\).

class opytimark.markers.two_dimensional.Leon(name: Optional[str] = 'Leon', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Leon class implements the Leon’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 100(x_2 - x_1^2)^2 + (1 - x_1)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-1.2, 1.2], x_2 \in [-1.2, 1.2]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Levy13(name: Optional[str] = 'Levy13', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Levy13 class implements the Levy’s 13th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = sin^2(3\pi x_1)+(x_1-1)^2(1+sin^2(3\pi x_2))+(x_2-1)^2(1+sin^2(2\pi x_2))\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Matyas(name: Optional[str] = 'Matyas', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Matyas class implements the Matyas’ benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.26(x_1^2 + x_2^2) - 0.48x_1x_2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

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

class opytimark.markers.two_dimensional.McCormick(name: Optional[str] = 'McCormick', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

McCormick class implements the McCormick’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = sin(x_1 + x_2) + (x_1 - x_2)^2 - \frac{3}{2}x_1 + \frac{5}{2}x_2 + 1\]

Domain:

The function is commonly evaluated using \(x_1 \in [-1.5, 4], x_2 \in [-3, 3]\).

Global Minima:

\(f(\mathbf{x^*}) = -1.9132228873800594 \mid \mathbf{x^*} = (−0.547, −1.547)\).

class opytimark.markers.two_dimensional.Mishra3(name: Optional[str] = 'Mishra3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Mishra3 class implements the Mishra’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = |cos(sqrt{|x_1^2+x_2|})|^{0.5} + 0.01(x_1+x_2)\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -0.18465133334298883 \mid \mathbf{x^*} = (−8.466613775046579, −9.998521308999999)\).

class opytimark.markers.two_dimensional.Mishra4(name: Optional[str] = 'Mishra4', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Mishra4 class implements the Mishra’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = |sin(sqrt{|x_1^2+x_2|})|^{0.5} + 0.01(x_1+x_2)\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -0.1994069700888328 \mid \mathbf{x^*} = (−9.941127263635860, −9.999571661999983)\).

class opytimark.markers.two_dimensional.Mishra5(name: Optional[str] = 'Mishra5', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Mishra5 class implements the Mishra’s 5th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = [sin^2(cos(x_1)+cos(x_2))^2 + cos^2(sin(x_1)+sin(x_2)) + x_1]^2 + 0.01x_1 + 0.1x_2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -1.019829519930943 \mid \mathbf{x^*} = (−1.986820662153768, −10)\).

class opytimark.markers.two_dimensional.Mishra6(name: Optional[str] = 'Mishra6', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Mishra6 class implements the Mishra’s 6th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -log[sin^2(cos(x_1)+cos(x_2))^2 - cos^2(sin(x_1)+sin(x_2)) + x_1]^2 + 0.1((x_1-1)^2 + (x_2-1)^2)\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -2.2839498384747587 \mid \mathbf{x^*} = (2.886307215440481, 1.823260331422321)\).

class opytimark.markers.two_dimensional.Mishra8(name: Optional[str] = 'Mishra8', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Mishra8 class implements the Mishra’s 8th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.001(|g(x_1)| + |h(x_2)|)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (2, -3)\).

class opytimark.markers.two_dimensional.Parsopoulos(name: Optional[str] = 'Parsopoulos', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Parsopoulos class implements the Parsopoulos’ benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = cos(x_1)^2 + sin(x_2)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 5], x_2 \in [-5, 5]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (k\frac{\pi}{2}, \lambda \pi)\).

class opytimark.markers.two_dimensional.PenHolder(name: Optional[str] = 'PenHolder', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

PenHolder class implements the Pen Holder’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -e^{[cos(x_1)cos(x_2)e^{|1-[(x_1^2+x_2^2)]^{0.5} / \pi|}|]^{-1}}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-11, 11], x_2 \in [-11, 11]\).

Global Minima:

\(f(\mathbf{x^*}) = -0.9635348327265058 \mid \mathbf{x^*} = (\pm 9.646167671043401, \pm 9.646167671043401)\).

class opytimark.markers.two_dimensional.Periodic(name: Optional[str] = 'Periodic', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Periodic class implements the Periodic’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 1 + sin^2(x_1) + sin^2(x_2) - 0.1e^{-(x_1^2+x_2^2)}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Price1(name: Optional[str] = 'Price1', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Price1 class implements the Price’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (|x_1| - 5)^2 + (|x_2| - 5)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (-5, -5) ~ or ~(-5, 5) ~ or ~(5, -5) ~ or ~(5, 5)\).

class opytimark.markers.two_dimensional.Price2(name: Optional[str] = 'Price2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Price2 class implements the Price’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 1 + sin^2(x_1) + sin^2(x_2) - 0.1e^{-x_1^2-x_2^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Price3(name: Optional[str] = 'Price3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Price3 class implements the Price’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 100(x_2 - x_1^2)^2 + [6.4(x_2 - 0.5)^2 - x_1 - 0.6]^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 5], x_2 \in [-5, 5]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0.341307503353524, 0.116490811845416) ~ or ~(1, 1)\).

class opytimark.markers.two_dimensional.Price4(name: Optional[str] = 'Price4', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Price4 class implements the Price’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (2x_1^3x_2 - x_2^3)^2 + (6x_1 - x_2^2 + x_2)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0) ~ or ~(2, 4) ~ or ~(1.464, -2.506)\).

class opytimark.markers.two_dimensional.Quadratic(name: Optional[str] = 'Quadratic', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Quadratic class implements the Quadratic’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -3803.84 - 138.08x_1 - 232.92x_2 + 128.08x_1^2 + 203.64x_2^2 + 182.25x_1x_2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -3873.7243 \mid \mathbf{x^*} = (0.19388, 0.48513)\).

class opytimark.markers.two_dimensional.RotatedEllipse1(name: Optional[str] = 'RotatedEllipse1', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

RotatedEllipse1 class implements the Rotated Ellipse’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 7x_1^2 - 6\sqrt{3}x_1x_2 + 13x_2^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

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

class opytimark.markers.two_dimensional.RotatedEllipse2(name: Optional[str] = 'RotatedEllipse2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

RotatedEllipse2 class implements the Rotated Ellipse’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = x_1^2 - x_1x_2 + x_2^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Rump(name: Optional[str] = 'Rump', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Rump class implements the Rump’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (333.75 - x_1^2)x_2^6 + x_1^2(11x_1^2x_2^2 - 121x_2^4 - 2) + 5.5x_2^8 + \frac{x_1}{2x_2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Schaffer1(name: Optional[str] = 'Schaffer1', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schaffer1 class implements the Schaffer’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.5 + \frac{sin^2(x_1^2 + x_2^2)^2 - 0.5}{1 + 0.001(x_1^2 + x_2^2)^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Schaffer2(name: Optional[str] = 'Schaffer2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schaffer2 class implements the Schaffer’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.5 + \frac{sin^2(x_1^2 - x_2^2)^2 - 0.5}{1 + 0.001(x_1^2 + x_2^2)^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

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

class opytimark.markers.two_dimensional.Schaffer3(name: Optional[str] = 'Schaffer3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schaffer3 class implements the Schaffer’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.5 + \frac{sin^2(cos|x_1^2 + x_2^2|) - 0.5}{(1 + 0.001(x_1^2 + x_2^2))^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

\(f(\mathbf{x^*}) = 0.0015668545260126288 \mid \mathbf{x^*} = (0, 1.253115)\).

class opytimark.markers.two_dimensional.Schaffer4(name: Optional[str] = 'Schaffer4', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schaffer4 class implements the Schaffer’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.5 + \frac{cos^2(sin(x_1^2 - x_2^2)) - 0.5}{1 + 0.001(x_1^2 + x_2^2)^2}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

\(f(\mathbf{x^*}) = 0.29243850703298857 \mid \mathbf{x^*} = (0, 1.253115)\).

class opytimark.markers.two_dimensional.Schwefel26(name: Optional[str] = 'Schwefel26', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = False, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Schwefel26 class implements the Schwefel’s 2.6 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = \max(|x_1 + 2x_2 - 7|, |2x_1 + x_2 - 5|)\]

Domain:

The function is commonly evaluated using \(x_1 \in [-100, 100], x_2 \in [-100, 100]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 3)\).

class opytimark.markers.two_dimensional.Schwefel236(name: Optional[str] = 'Schwefel236', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Schwefel236 class implements the Schwefel’s 2.36 benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -x_1x_2(72 - 2x_1 - 2x_2)\]

Domain:

The function is commonly evaluated using \(x_1 \in [0, 500], x_2 \in [0, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = -3456 \mid \mathbf{x^*} = (12, 12)\).

class opytimark.markers.two_dimensional.Table1(name: Optional[str] = 'Table1', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Table1 class implements the Table’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -|cos(x_1)cos(x_2)e^{|1-(x_1+x_2)^{0.5}/\pi|}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -26.920335555515848 \mid \mathbf{x^*} = (\pm 9.646168, \pm 9.646168)\).

class opytimark.markers.two_dimensional.Table2(name: Optional[str] = 'Table2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Table2 class implements the Table’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -|cos(x_1)cos(x_2)e^{|1-(x_1+x_2)^{0.5}/\pi|}\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -19.20850256788675 \mid \mathbf{x^*} = (\pm 8.055023472141116, \pm 9.664590028909654)\).

class opytimark.markers.two_dimensional.Table3(name: Optional[str] = 'Table3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Table3 class implements the Table’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -\frac{1}{30}e^{2|1 - \frac{sqrt{x_1^2+x_2^2}{\pi}}}cos^2(x_1)cos^2(x_2)\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -19.20850256788675 \mid \mathbf{x^*} = (\pm 9.646157266348881, \pm 9.646134286497169)\).

class opytimark.markers.two_dimensional.TesttubeHolder(name: Optional[str] = 'TesttubeHolder', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

TesttubeHolder class implements the Testtube Holder’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = -4[(sin(x_1)cos(x_2)e^{|cos[(x_1^2+x_2^2)/200]|})]\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -10.872299901558 \mid \mathbf{x^*} = (\pm \frac{\pi}{2}, 0)\).

class opytimark.markers.two_dimensional.Trefethen(name: Optional[str] = 'Trefethen', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Trefethen class implements the Trefethen’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = e^{sin(50x_1)} + sin(60e^{x^2}) + sin(70sin(x_1)) + sin(sin(80x_2)) - sin(10(x_1 + x_2)) + \frac{1}{4}(x_1^2 + x_2^2)\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -3.3068686465567008 \mid \mathbf{x^*} = (−0.024403, 0.210612)\).

class opytimark.markers.two_dimensional.VenterSobiezcczanskiSobieski(name: Optional[str] = 'VenterSobiezcczanskiSobieski', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

VenterSobiezcczanskiSobieski class implements the Venter Sobiezcczanski-Sobieski’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = x_1^2 - 100cos(x_1)^2 - 100cos(\frac{x_1^2}{30}) + x_2^2 - 100cos(x_2)^2 - 100cos(\frac{x_2^2}{30})\]

Domain:

The function is commonly evaluated using \(x_1 \in [-50, 50], x_2 \in [-50, 50]\).

Global Minima:

\(f(\mathbf{x^*}) = -400 \mid \mathbf{x^*} = (0, 0)\).

class opytimark.markers.two_dimensional.WayburnSeader1(name: Optional[str] = 'WayburnSeader1', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

WayburnSeader1 class implements the WayburnSeader’s 1st benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_1^6 + x_2^4 - 17)^2 + (2x_1 + x_2 - 4)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 2) ~ or ~(1.596804153876933, 0.806391692246134)\).

class opytimark.markers.two_dimensional.WayburnSeader2(name: Optional[str] = 'WayburnSeader2', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

WayburnSeader2 class implements the WayburnSeader’s 2nd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = [1.613 - 4(x_1 - 0.3125)^2 - 4(x_2 - 1.625)^2]^2 + (x_2 - 1)^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0.200138974728779, 1) ~ or ~(0.424861025271221, 1)\).

class opytimark.markers.two_dimensional.WayburnSeader3(name: Optional[str] = 'WayburnSeader3', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

WayburnSeader3 class implements the WayburnSeader’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = \frac{2}{3}x_1^3 - 8x_1^2 + 33x_1 - x_1x_2 + 5 + [(x_1 - 4)^2 + (x_2 - 5)^2 - 4]^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-500, 500], x_2 \in [-500, 500]\).

Global Minima:

\(f(\mathbf{x^*}) = 19.105879794568 \mid \mathbf{x^*} = (5.146896745324582, 6.839589743000071)\).

class opytimark.markers.two_dimensional.Zettl(name: Optional[str] = 'Zettl', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Zettl class implements the Zettl’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = (x_1^2 + x_2^2 - 2x_1)^2 + 0.25x_1\]

Domain:

The function is commonly evaluated using \(x_1 \in [-5, 10], x_2 \in [-5, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -0.0037912371501199 \mid \mathbf{x^*} = (−0.0299, 0)\).

class opytimark.markers.two_dimensional.Zirilli(name: Optional[str] = 'Zirilli', dims: Optional[int] = 2, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = False, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Zirilli class implements the Zirilli’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2) = 0.25x_1^4 - 0.5x_1^2 + 0.1x_1 + 0.5x_2^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [-10, 10], x_2 \in [-10, 10]\).

Global Minima:

\(f(\mathbf{x^*}) = -0.3523860365437344 \mid \mathbf{x^*} = (-1.0465, 0)\).