Alphabetical List of Benchmark Functions

Generated automatically on 2025-12-19 from package metadata.

Scalable functions are shown with their default dimension (default_n). Use fixed(tf; n=...) to create fixed-dimension instances.

ackley

Description: Ackley function – one of the most famous deceptive multimodal benchmarks. Nearly flat outer region with a deep central hole and countless cosine-induced local minima. Systematically misleads gradient-based and local optimizers away from the global minimum at zero.
Formula: f(\mathbf{x}) = -20\exp!\left(-0.2\sqrt{\frac{1}{n}\sum xi^2}\right) - \exp!\left(\frac{1}{n}\sum\cos(2\pi xi)\right) + 20 + e
Dimension: scalable (default n = 10)
Bounds / Minimum: Bounds: [-32.768, ..., -32.768] to [32.768, ..., 32.768]; Global minimum: 0.0 at [0.0, ..., 0.0]
Properties: bounded, continuous, deceptive, differentiable, multimodal, non-convex, non-separable, scalable
Reference: Ackley (1987); Molga & Smutnicki (2005); Jamil & Yang (2013); Lehman & Stanley (2011, arXiv:1106.2128) – classic deceptive function
Note: Scalable function – shown with default_n = 10. Use fixed(tf; n=...) for custom dimensions.

ackley2

Description: Ackley2 function: Continuous, partially differentiable, non-separable, non-scalable, unimodal.
Formula: \f(x) = -200 e^{-0.02 \sqrt{x1^2 + x2^2}}
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-32.0, -32.0] to [32.0, 32.0]; Global minimum: -200.0 at [0.0, 0.0]
Properties: bounded, continuous, non-convex, non-separable, partially differentiable, unimodal
Reference: Jamil & Yang (2013): f2

ackley4

Description: Modified Ackley Function (Ackley 4). Properties adapted from Jamil & Yang (2013, p. 5) for variant with √(xi² + x{i+1}²) in exponential and no x_i² term; originally from Rónkkónen (2009). Highly multimodal with local minimum near origin; global minimum approaches -6 near boundaries. Implemented as fixed n=2 due to limited metadata for higher dimensions.
Formula: f(\mathbf{x}) = \sum{i=1}^{1} \left[ e^{-0.2 \sqrt{xi^2 + x{i+1}^2}} + 3 (\cos(2xi) + \sin(2x_{i+1})) \right].
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-35.0, -35.0] to [35.0, 35.0]; Global minimum: -5.297009385988958 at [-1.5812644, -0.79063191]
Properties: bounded, continuous, controversial, differentiable, highly multimodal, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 5)

adjiman

Description: Properties based on Jamil & Yang (2013, p. 3); Multimodal, non-convex, non-separable, differentiable, bounded test function with a single global minimum.
Formula: f(\mathbf{x}) = \cos x1 \sin x2 - \frac{x1}{x2^2 + 1}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [2.0, 1.0]; Global minimum: -2.021806783359787 at [2.0, 0.10578347]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 3)

alpinen1

Description: Properties based on Jamil & Yang (2013, p. 5); Contains absolute value terms leading to non-differentiability at certain points (gradient returns NaN there).
Formula: f(\mathbf{x}) = \sum{i=1}^n |xi \sin xi + 0.1 xi|.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, multimodal, non-convex, partially differentiable, scalable, separable
Reference: Jamil & Yang (2013, p. 5)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

alpinen2

Description: Properties based on Jamil & Yang (2013, p. 5); Fully differentiable on [0,10]^n.
Formula: f(\mathbf{x}) = -\prod{i=1}^n \sqrt{xi} \sin xi \quad (xi \geq 0).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [10.0, 10.0]; Global minimum: -7.885600724127536 at [7.9170527, 7.9170527]
Properties: bounded, continuous, differentiable, multimodal, non-convex, scalable, separable
Reference: Jamil & Yang (2013, p. 5)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

axisparallelhyperellipsoid

Description: Properties based on Jamil & Yang (2013, p. 4); Convex, quadratic function.
Formula: f(\mathbf{x}) = \sum{i=1}^n i xi^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-5.12, -5.12] to [5.12, 5.12]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, convex, differentiable, scalable, separable
Reference: Jamil & Yang (2013, p. 4)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

bartelsconn

Description: Properties based on Jamil & Yang (2013, p. 6); Contains absolute value terms leading to non-differentiability at certain points (gradient returns NaN there).
Formula: f(\mathbf{x}) = |x1^2 + x2^2 + x1 x2| + |\sin(x1)| + |\cos(x2)|.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 1.0 at [0.0, 0.0]
Properties: bounded, continuous, multimodal, non-convex, non-separable, partially differentiable
Reference: Jamil & Yang (2013, p. 6)

beale

Description: Beale function: Unimodal, non-convex, non-separable, differentiable, fixed, bounded, continuous.
Formula: (1.5 - x1 + x1 x2)^2 + (2.25 - x1 + x1 x2^2)^2 + (2.625 - x1 + x1 x_2^3)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-4.5, -4.5] to [4.5, 4.5]; Global minimum: 0.0 at [3.0, 0.5]
Properties: bounded, continuous, differentiable, non-convex, non-separable, unimodal
Reference: Jamil & Yang (2013, entry 10)

becker_lago

Description: Becker and Lago function from Price (1977). Properties: continuous, separable, multimodal, bounded. Contains absolute value terms (non-differentiable at x_i=0). Four minima at (5, 5), (5, -5), (-5, 5), (-5, -5). Possibly misidentified as Price 3 (Jamil & Yang, 2013, No. 96) in some contexts.
Formula: f(\mathbf{x}) = (|x1| - 5)^2 + (|x2| - 5)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [5.0, 5.0]
Properties: bounded, continuous, multimodal, separable
Reference: Price (1977); referenced in Jamil & Yang (2013), https://arxiv.org/abs/1308.4008

biggsexp2

Description: Properties based on Jamil & Yang (2013, p. 12); Sum-of-squares function with exact global minimum 0 at [1,10] for n=2.
Formula: f(\mathbf{x}) = \sum{i=1}^{10} \left( e^{-ti x1} - 5 e^{-ti x2} - yi \right)^2, \quad ti=0.1i, \ yi = e^{-ti} - 5 e^{-10 ti}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [20.0, 20.0]; Global minimum: 0.0 at [1.0, 10.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 12)

biggsexp3

Description: Properties based on Jamil & Yang (2013, p. 12); Multimodal sum-of-squares function with exact global minimum 0 at [1,10,5] for n=3.
Formula: f(\mathbf{x}) = \sum{i=1}^{10} \left( e^{-ti x1} - x3 e^{-ti x2} - yi \right)^2, \quad ti=0.1i, \ yi = e^{-ti} - 5 e^{-10 t_i}.
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0] to [20.0, 20.0, 20.0]; Global minimum: 0.0 at [1.0, 10.0, 5.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 12)

biggsexp4

Description: Properties based on Jamil & Yang (2013, p. 12); Multimodal sum-of-squares function with exact global minimum 0 at [1,10,1,5] for n=4.
Formula: f(\mathbf{x}) = \sum{i=1}^{10} \left( x3 e^{-ti x1} - x4 e^{-ti x2} - yi \right)^2, \quad ti=0.1i, \ yi = e^{-ti} - 5 e^{-10 ti}.
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0, 0.0] to [20.0, 20.0, 20.0, 20.0]; Global minimum: 0.0 at [1.0, 10.0, 1.0, 5.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 12)

biggsexp5

Description: Properties based on Jamil & Yang (2013, p. 12); Multimodal sum-of-squares function with exact global minimum 0 at [1,10,1,5,4] for n=5.
Formula: f(\mathbf{x}) = \sum{i=1}^{11} \left( x3 e^{-ti x1} - x4 e^{-ti x2} + 3 e^{-ti x5} - yi \right)^2, \quad ti = 0.1i, \ yi = e^{-ti} - 5 e^{-10 ti} + 3 e^{-4 t_i}.
Dimension: fixed (n = 5)
Bounds / Minimum: Bounds: [0.0, ..., 0.0] to [20.0, ..., 20.0]; Global minimum: 0.0 at [1.0, ..., 4.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 12)

biggsexp6

Description: Properties based on Jamil & Yang (2013, p. 12); Multimodal sum-of-squares function with exact global minimum 0 at [1,10,1,5,4,3] for n=6.
Formula: f(\mathbf{x}) = \sum{i=1}^{13} \left( x3 e^{-ti x1} - x4 e^{-ti x2} + x6 e^{-ti x5} - yi \right)^2, \quad ti=0.1i, \ yi = e^{-ti} - 5 e^{-10 ti} + 3 e^{-4 ti}.
Dimension: fixed (n = 6)
Bounds / Minimum: Bounds: [-20.0, ..., -20.0] to [20.0, ..., 20.0]; Global minimum: 0.0 at [1.0, ..., 3.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 12)

bird

Description: Properties based on Jamil & Yang (2013, p. 9); Highly multimodal with two symmetric global minima. The reported minima are high-precision numerical approximations obtained via global optimization; they are not critical points (∇f ≠ 0) and the exact analytical positions are unknown. Gradient norm at reported minima ≈ 4.95.
Formula: f(\mathbf{x}) = \sin x1 \exp\left((1 - \cos x2)^2\right) + \cos x2 \exp\left((1 - \sin x1)^2\right) + (x1 - x2)^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-6.28318531, -6.28318531] to [6.28318531, 6.28318531]; Global minimum: -106.76453674926468 at [4.70104313, 3.1529385]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 9)

bohachevsky1

Description: Bohachevsky 1 function. Scalable, separable, multimodal test function with global minimum 0 at the origin. Properties based on Jamil & Yang (2013, p. 10).
Formula: f(\mathbf{x}) = \sum{i=1}^{n-1} \left[ xi^2 + 2 x{i+1}^2 - 0.3 \cos(3 \pi xi) - 0.4 \cos(4 \pi x_{i+1}) + 0.7 \right].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, scalable, separable
Reference: Jamil & Yang (2013, p. 10)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

bohachevsky2

Description: Bohachevsky 2 function. Non-scalable (n=2), non-separable, multimodal test function with global minimum 0 at the origin. Properties based on Jamil & Yang (2013, p. 11).
Formula: f(\mathbf{x}) = x1^2 + 2 x2^2 - 0.3 \cos(3 \pi x1) \cos(4 \pi x2) + 0.3.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 11)

bohachevsky3

Description: Bohachevsky 3 function (Bohachevsky et al., 1986). Non-scalable (n=2), non-separable, multimodal test function with global minimum 0 at the origin. Properties based on Jamil & Yang (2013, p. 11).
Formula: f(\mathbf{x}) = x1^2 + 2 x2^2 - 0.3 \cos(3 \pi x_1 + 4 \pi x__2) + 0.3.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 11)

booth

Description: Booth function: Unimodal, convex, separable, differentiable, bounded quadratic test function with a single global minimum at (1, 3). Properties based on [Surjanovic & Bingham (2013)]; originally from [Booth (1976)].
Formula: (x1 + 2x2 - 7)^2 + (2x1 + x2 - 5)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 3.0]
Properties: bounded, continuous, convex, differentiable, separable, unimodal
Reference: Surjanovic & Bingham (2013), Virtual Library of Simulation Experiments: Test Functions and Datasets, retrieved from https://www.sfu.ca/~ssurjano/booth.html

boxbetts

Description: Box-Betts Quadratic Sum function: Continuous, differentiable, non-separable, nonscalable, multimodal.
Formula: \sum{i=1}^{10} \left( e^{-0.1i x1} - e^{-0.1i x2} - (e^{-0.1i} - e^{-i}) x3 \right)^2
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [0.9, 9.0, 0.9] to [1.2, 11.2, 1.2]; Global minimum: 0.0 at [1.0, 10.0, 1.0]
Properties: continuous, differentiable, multimodal
Reference: MVF - Multivariate Test Functions Library in C, Ernesto P. Adorio, Revised January 14, 2005

brad

Description: Brad function (Brad, 1970). Non-scalable (n=3), continuous, differentiable, non-separable, multimodal least-squares problem from exponential fitting.
Formula: f(\mathbf{x}) = \sum{i=1}^{15} \left( yi - x1 - \frac{i}{(16-i)x2 + \min(i,16-i)x_3} \right)^2
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [-0.25, 0.01, 0.01] to [0.25, 2.5, 2.5]; Global minimum: 0.008214877306578994 at [0.08241056, 1.13303609, 2.34369518]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Brad (1970); data from Moré et al. (1981)

branin

Description: Classic Branin function with three global minima. Properties based on Jamil & Yang (2013, p. 9).
Formula: f(\mathbf{x}) = a(x2 - b x1^2 + c x1 - r)^2 + s(1-t)\cos(x1) + s

\quad\text{with}\quad a=1,\; b=\frac{5.1}{4\pi^2},\; c=\frac{5}{\pi},\; r=6,\; s=10,\; t=\frac{1}{8\pi}

Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, 0.0] to [10.0, 15.0]; Global minimum: 0.39788735772973816 at [-3.14159265, 12.275]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 9)

braninrcos2

Description: Branin RCOS 2 function (Muntenau & Lazarescu, 1998). Multimodal, non-separable test function with a deep global minimum. The original paper reports an incorrect minimum of ≈5.56 – numerical verification yields ≈-39.196.
Formula: f(x1,x2) = \left(x2 - \frac{5.1 x1^2}{4\pi^2} + \frac{5 x1}{\pi} - 6\right)^2 + 10\left(1 - \frac{1}{8\pi}\right)\cos(x1)\cos(x2)\ln(x1^2 + x_2^2 + 1) + 10
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [15.0, 15.0]; Global minimum: -39.19565391797774 at [-3.17210415, 12.5856748]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Muntenau & Lazarescu (1998, p. 27)

brent

Description: The Brent function is a non-scalable, unimodal test function with a parabolic bowl centered at (-10, -10) and a Gaussian bump at the origin. Properties based on [Jamil & Yang (2013, p. 9)]; originally from [Brent (1960)].
Formula: f(\mathbf{x}) = (x1 + 10)^2 + (x2 + 10)^2 + \exp(-x1^2 - x2^2)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [-10.0, -10.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 9)

brown

Description: Brown function: Unimodal, non-separable, differentiable, scalable, continuous, bounded. Highly sensitive to changes in variables due to exponential terms. Properties based on [Jamil & Yang (2013, p. 5)]; originally from [Brown (1966)].
Formula: \sum{i=1}^{n-1} \left[ (xi^2)^{(x{i+1}^2 + 1)} + (x{i+1}^2)^{(x_i^2 + 1)} \right]
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [4.0, 4.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, non-separable, scalable, unimodal
Reference: Jamil & Yang (2013, p. 5)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

bukin2

Description: Bukin Function N.2. Highly multimodal with a narrow curving ridge. Contains absolute value → not differentiable at x₁ = -10. Global minimum = 0 at (-10, 1).
Formula: f(\mathbf{x}) = 100 (x2 - 0.01 x1^2)^2 + 0.01 |x_1 + 10|
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-15.0, -3.0] to [-5.0, 3.0]; Global minimum: 0.0 at [-10.0, 1.0]
Properties: bounded, continuous, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 34)

bukin4

Description: Properties based on Jamil & Yang (2013, p. 10); Contains absolute value terms leading to non-differentiability at x1=-10 (gradient returns NaN there).
Formula: f(\mathbf{x}) = 100 x2^2 + 0.01 |x1 + 10|.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-15.0, -3.0] to [-5.0, 3.0]; Global minimum: 0.0 at [-10.0, 0.0]
Properties: bounded, continuous, multimodal, non-convex, partially differentiable, separable
Reference: Jamil & Yang (2013, p. 10)

bukin6

Description: Properties based on Jamil & Yang (2013, p. 10); Contains absolute value and sqrt(abs) terms leading to non-differentiability at certain points (gradient returns NaN there).
Formula: f(\mathbf{x}) = 100 \sqrt{|x2 - 0.01 x1^2|} + 0.01 |x_1 + 10|.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-15.0, -3.0] to [-5.0, 3.0]; Global minimum: 0.0 at [-10.0, 1.0]
Properties: bounded, continuous, multimodal, non-convex, partially differentiable
Reference: Jamil & Yang (2013, p. 10)

carromtable

Description: Four global minima due to sign choices; Properties based on Jamil & Yang (2013, p. 34); originally from Mishra (2004).
Formula: f(\mathbf{x}) = -\frac{\left[ \left( \cos(x1) \cos(x2) \exp \left| 1 - \frac{\sqrt{x1^2 + x2^2}}{\pi} \right| \right)^2 \right]}{30}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -24.156815547391208 at [9.64616767, 9.64616767]
Properties: continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 34)

chen

Description: Chen V function: Multimodal, non-convex, non-separable, differentiable, bounded, continuous. Minimum: -2000.0 at (0.388888888888889, 0.722222222222222). Bounds: [-500, 500]^2. Dimensions: n=2. Formulated as a minimum problem by negating the maximum problem from [Naser et al. (2024)] and [al-roomi.org], where f(0.388888888888889, 0.722222222222222) = 2000. [Jamil & Yang (2013): f32] reports a different function with minimum f(-0.3888889, 0.7222222) = -2000.
Formula: -\left(\frac{0.001}{0.000001 + (x1 - 0.4 x2 - 0.1)^2} + \frac{0.001}{0.000001 + (2 x1 + x2 - 1.5)^2}\right)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: -2000.0 at [0.38888889, 0.72222222]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

chenbird

Description: Chen V function – Jamil & Yang (2013, f32). The version used in >95% of all papers. Three radial terms. Global minimum ≈ -2000.004. This is NOT the original Chen (2003) function!
Formula: f(\mathbf{x}) = -\left[ \frac{0.001}{(0.001)^2 + (x1^2 + x2^2 - 1)^2} + \frac{0.001}{(0.001)^2 + (x1^2 + x2^2 - 0.5)^2} + \frac{0.001}{(0.001)^2 + (x1^2 - x2^2)^2} \right]
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: -2000.0039999840003 at [0.5, 0.5]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 47)

chenv

Description: Original Chen V function (Chen, 2003). Three radial/hyperbolic terms. Eight symmetric near-global minima near (±0.5, ±0.5). Global minimum ≈ -2000.004. This is the REAL Chen V – not the incorrect linear versions from Jamil & Yang (2013) or others.
Formula: f(\mathbf{x}) = -\left[ \frac{0.001}{(0.001)^2 + (x1^2 + x2^2 - 1)^2} + \frac{0.001}{(0.001)^2 + (x1^2 + x2^2 - 0.5)^2} + \frac{0.001}{(0.001)^2 + (x1^2 - x2^2)^2} \right]
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: -2000.0039999840005 at [0.5, 0.5]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Chen (2003) – original publication

chichinadze

Description: Chichinadze function (continuous, differentiable, separable, non-scalable, multimodal). Source: Jamil & Yang (2013). Global minimum at approximately [6.18987, 0.5].
Formula: f(\mathbf{x}) = x1^2 - 12 x1 + 11 + 10 \cos\left(\frac{\pi x1}{2}\right) + 8 \sin\left(\frac{5 \pi x1}{2}\right) - \sqrt{\frac{1}{5}} \exp\left(-0.5 (x_2 - 0.5)^2 \right)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-30.0, -30.0] to [30.0, 30.0]; Global minimum: -42.944387018991 at [6.18986659, 0.5]
Properties: bounded, continuous, differentiable, multimodal, separable
Reference: Unknown Source

chungreynolds

Description: Chung Reynolds function (continuous, differentiable, partially separable, scalable, unimodal). Source: Jamil & Yang (2013) and Chung & Reynolds (1998). Global minimum at the origin.
Formula: f(\mathbf{x}) = \left( \sum{i=1}^n xi^2 \right)^2
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, partially separable, scalable, unimodal
Reference: Jamil & Yang (2013, Entry 34)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

cola

Description: Cola function (continuous, differentiable, non-separable, non-scalable, multimodal). Source: Adorio & Diliman (2005). 17D positioning problem with fixed points (x1=y1=y2=0). Literature global min f*=11.7464 (MVF-Library); approximate position from Jamil & Yang yields f≈11.828.
Formula: f(\mathbf{u}) = \sum{1 \le j < i \le 10} (r{i,j} - d{i,j})^2, \quad r{i,j} = \sqrt{(xi - xj)^2 + (yi - yj)^2}
Dimension: fixed (n = 17)
Bounds / Minimum: Bounds: [0.0, ..., -4.0] to [4.0, ..., 4.0]; Global minimum: 10.533719093221547 at [0.65771795, ..., -1.67742926]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Unknown Source

colville

Description: Colville function: Unimodal, non-convex, non-separable, differentiable, bounded, continuous. Known for its narrow, curved valleys challenging optimization algorithms. Properties based on [Jamil & Yang (2013, p. 13)]; originally from [Colville (1968)].
Formula: f(\mathbf{x}) = 100(x1^2 - x2)^2 + (x1 - 1)^2 + (x3 - 1)^2 + 90(x3^2 - x4)^2 + 10.1((x2 - 1)^2 + (x4 - 1)^2) + 19.8(x2 - 1)(x4 - 1)
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [-10.0, -10.0, -10.0, -10.0] to [10.0, 10.0, 10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0, 1.0, 1.0]
Properties: bounded, continuous, differentiable, non-convex, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 13)

corana

Description: Corana function (discontinuous, non-differentiable, separable, scalable, multimodal). Source: Corana et al. (1990). Global minimum f(x)=0 at x=(0,...,0).
Formula: f(\mathbf{x}) = \sum{i=1}^n \begin{cases} 0.15 di (zi - 0.05 \sgn(zi))^2 & |xi - zi| < 0.05 \ di xi^2 & \text{otherwise} \end{cases}, \ zi = 0.2 \left\lfloor \frac{|xi|}{0.2} + 0.49999 \right\rfloor \sgn(xi), \ di \text{ cycles over } [1, 1000, 10, 100].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, multimodal, scalable, separable
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

cosinemixture

Description: Cosine Mixture Function: A multimodal, separable benchmark with global minimum -0.1*n at origin.
Formula: f(\mathbf{x}) = -0.1 \sum{i=1}^n \cos(5 \pi xi) - \sum{i=1}^n xi^2
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: -0.2 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, scalable, separable
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

crossintray

Description: Cross-in-Tray function: A multimodal, non-convex, non-separable test function with four global minima at [1.349406575769872, 1.349406575769872], [-1.349406575769872, 1.349406575769872], [1.349406575769872, -1.349406575769872], [-1.349406575769872, -1.349406575769872].
Formula: -0.0001 \left( \left| \sin(x1) \sin(x2) \exp\left( \left| 100 - \frac{\sqrt{x1^2 + x2^2}}{\pi} \right| \right) \right| + 1 \right)^{0.1}
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -2.062611870822739 at [1.34940658, 1.34940658]
Properties: bounded, continuous, multimodal, non-convex, non-separable, partially differentiable
Reference: Unknown Source

csendes

Description: Csendes function (continuous, differentiable, separable, scalable, multimodal). Source: Csendes and Ratz (1997). Global minimum f(x)=0 at x=(0,...,0).
Formula: f(\mathbf{x}) = \sum{i=1}^n xi^2 (2 + \sin x_i).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, scalable, separable
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

cube

Description: Cube function (continuous, differentiable, non-separable, unimodal). Source: Vogel (1966). Global minimum f(x)=0 at x=(1,1). Properties based on [Jamil & Yang (2013, Entry 41)].
Formula: f(\mathbf{x}) = 100 (x2 - x1^3)^2 + (1 - x_1)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, Entry 41)

damavandi

Description: Damavandi function (Damavandi & Safavi-Naeini, 2005): A multimodal, non-separable 2D function with a global minimum of 0 at (2,2). Features a deceptive landscape due to the sinc-like terms.
Formula: f(\mathbf{x}) = \left[1 - \left| \frac{\sin[\pi (x1 - 2)]\sin[\pi (x2 - 2)]}{\pi^2 (x1 - 2)(x2 - 2)} \right|^5 \right] \left[ 2 + (x1 - 7)^2 + 2(x2 - 7)^2 \right]
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [14.0, 14.0]; Global minimum: 0.0 at [2.0, 2.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Unknown Source

deb1

Description: Deb's Function No.1 (Rönkkönen, 2009): A scalable, separable, multimodal function with 10^D evenly spaced global minima at f=-1.
Formula: f(\mathbf{x}) = -\frac{1}{D} \sum{i=1}^{D} \sin^{6}(5 \pi xi)
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: -1.0 at [-0.9, -0.9]
Properties: bounded, continuous, differentiable, multimodal, scalable, separable
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

deb3

Description: Deb's Function No.3 (Rönkkönen, 2009): A scalable, separable, multimodal function with 5^D evenly spaced global minima at f=-1.
Formula: f(\mathbf{x}) = -\frac{1}{D} \sum{i=1}^{D} \sin^6 \left(5 \pi (xi^{3/4} - 0.05)\right)
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: -1.0 at [0.07969939, 0.07969939]
Properties: bounded, continuous, differentiable, multimodal, scalable, separable
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

dejongf4

Description: De Jong F4 function: Unimodal, convex, separable, partially differentiable due to Gaussian noise, scalable to any dimension n, with global minimum at x = [0, ..., 0], f* ≈ 0 (depending on noise). Bounds are [-1.28, 1.28] per dimension.
Formula: \sum{i=1}^n i xi^4 + \text{Gaussian noise}
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.28, -1.28] to [1.28, 1.28]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, convex, has_noise, partially differentiable, scalable, separable, unimodal
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

dejongf5modified

Description: Modified De Jong F5 (Shekel's Foxholes): Multimodal, non-convex, non-separable, differentiable, bounded, finite at infinity, continuous. Minimum: 0.9980038377944507 at x ≈ [-31.97833, -31.97833]. Bounds: [-65.536, 65.536]^2. Dimensions: n=2.
Formula: -\left( \frac{1}{500} + \sum{i=1}^{25} \frac{1}{i + (x1 - a{1i})^6 + (x2 - a_{2i})^6} \right)^{-1}
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-65.536, -65.536] to [65.536, 65.536]; Global minimum: 0.9980038377944507 at [-31.97833, -31.97833]
Properties: bounded, continuous, differentiable, finiteatinf, multimodal, non-convex, non-separable
Reference: Unknown Source

dejongf5original

Description: Original De Jong F5 (Shekel variant) as per Molga & Smutnicki (2005): Multimodal, non-convex, non-separable, differentiable, bounded, continuous, finite at infinity. Minimum: ≈0.998001998667 at x ≈ [-32, -32]. Bounds: [-65.536, 65.536]^2. Dimensions: n=2.
Formula: f(x) = \left( 0.002 + \sum{j=1}^{25} \frac{1}{j + (x1 - a{1j})^6 + (x2 - a_{2j})^6} \right)^{-1}
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-65.536, -65.536] to [65.536, 65.536]; Global minimum: 0.9980038378086058 at [-31.97987349, -31.97987349]
Properties: bounded, continuous, differentiable, finiteatinf, multimodal, non-convex, non-separable
Reference: Unknown Source

dekkersaarts

Description: Dekkers-Aarts function: Multimodal with two global minima at [0, ±14.94511215174121]. Literature (Ali et al., 2005) reports minima at (0, ±15), f* ≈ -24777, but exact minima are at (0, ±14.94511215174121), f* ≈ -24776.51834231769.
Formula: 10^5 x1^2 + x2^2 - (x1^2 + x2^2)^2 + 10^{-5} (x1^2 + x2^2)^4
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-20.0, -20.0] to [20.0, 20.0]; Global minimum: -24776.518342317686 at [0.0, 14.94511215]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

devilliersglasser1

Description: De Villiers-Glasser Function 1, from De Villiers and Glasser (1981). Returns NaN if x2 < 0 to avoid complex exponentiation; literature bounds -500 ≤ x ≤ 500. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{24} \left[ x1 x2^{ti} \sin(x3 ti + x4) - yi \right]^2, \quad ti = 0.1(i-1), \quad yi = 60.137 \times 1.371^{ti} \sin(3.112 ti + 1.761).
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0, 0.0] to [500.0, 500.0, 500.0, 500.0]; Global minimum: 0.0 at [60.137, 1.371, 3.112, 1.761]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Unknown Source

devilliersglasser2

Description: De Villiers-Glasser function no. 2 (de Villiers and Glasser, 1981). Search space restricted to xi ≥ 1.0 to ensure real-valued x₂^{ti} (standard in modern benchmark implementations, e.g. AMPGO/Gavana, SciPy, pymoo, NLopt test suites). Original paper allows negative x₂ (complex values possible). Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{24} \left[ x1 x2^{ti} \tanh(x3 ti + \sin(x4 ti)) \cos(ti e^{x5}) - yi \right]^2,\quad ti = 0.1(i-1),\quad yi = 53.81 \cdot 1.27^{ti} \tanh(3.012 ti + \sin(2.13 ti)) \cos(e^{0.507} t_i).
Dimension: fixed (n = 5)
Bounds / Minimum: Bounds: [1.0, ..., 1.0] to [60.0, ..., 60.0]; Global minimum: 0.0 at [53.81, ..., 0.507]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: de Villiers and Glasser (1981); Jamil & Yang (2013)

dixonprice

Description: Dixon-Price function: Unimodal, non-convex, scalable function with global minimum at zero. Properties based on [Jamil & Yang (2013, Entry 14)]; originally from [Dixon & Price (1971)].
Formula: (x1 - 1)^2 + \sum{i=2}^n i (2 xi^2 - x{i-1})^2
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 0.70710678]
Properties: bounded, continuous, differentiable, non-convex, scalable, unimodal
Reference: Jamil & Yang (2013, Entry 14)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

dolan

Description: Dolan function (continuous, differentiable, non-separable, non-scalable, multimodal). Properties based on Jamil & Yang (2013). Global minimum corrected from literature (f=0 at [0,...] incorrect) using Al-Roomi (2015). Ill-conditioned Hessian.
Formula: f(\mathbf{x}) = (x1 + 1.7 x2) \sin(x1) - 1.5 x3 - 0.1 x4 \cos(x4 + x5 - x1) + 0.2 x5^2 - x2 - 1
Dimension: fixed (n = 5)
Bounds / Minimum: Bounds: [-100.0, ..., -100.0] to [100.0, ..., 100.0]; Global minimum: -529.8714387324576 at [98.96425831, ..., -0.24998753]
Properties: bounded, continuous, controversial, differentiable, multimodal, non-separable
Reference: Unknown Source

dropwave

Description: Properties based on Jamil & Yang (2013, p. 24); ursprünglich aus [Mutmaßliche Ursprungsquelle, falls bekannt].
Formula: f(\mathbf{x}) = -\frac{1 + \cos(12 \sqrt{x1^2 + x2^2})}{0.5 (x1^2 + x2^2) + 2}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.12, -5.12] to [5.12, 5.12]; Global minimum: -1.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 24)

easom

Description: Easom function. Properties based on Jamil & Yang (2013, p. 19); originally from Easom (1990).
Formula: f(\mathbf{x}) = -\cos(x1) \cos(x2) \exp\left( -((x1 - \pi)^2 + (x2 - \pi)^2) \right).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: -1.0 at [3.14159265, 3.14159265]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 19)

eggcrate

Description: Egg Crate test function as standardized in Jamil & Yang (2013). Multimodal, separable function in 2 dimensions. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = x1^2 + x2^2 + 25 (\sin^2 x1 + \sin^2 x2).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, separable
Reference: Jamil & Yang (2013)

eggholder

Description: Eggholder function: Multimodal, non-convex, non-separable, differentiable, bounded, continuous test function with a global minimum of -959.6406627208506 at [512.0, 404.2318058008512]. Bounds: [-512, 512]^2. Dimensions: n=2. Used for benchmarking nonlinear optimization algorithms due to its many local minima. See [Jamil & Yang (2013)] for details.
Formula: - (x2 + 47) \sin\left(\sqrt{\left|x2 + \frac{x1}{2} + 47\right|}\right) - x1 \sin\left(\sqrt{\left|x1 - (x2 + 47)\right|}\right)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-512.0, -512.0] to [512.0, 512.0]; Global minimum: -959.6406627208506 at [512.0, 404.2318058]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

elattavidyasagardutta

Description: El-Attar-Vidyasagar-Dutta test function from El-Attar et al. (1979), as standardized in Jamil & Yang (2013). Unimodal, non-separable function in 4 dimensions. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = (x1^2 + x2 - 10)^2 + 5(x3 - x4)^2 + (x2 - x3)^2 + 10(x_4 - 1)^2.
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [-5.2, -5.2, -5.2, -5.2] to [5.2, 5.2, 5.2, 5.2]; Global minimum: 0.0 at [3.0, 1.0, 1.0, 1.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, Entry 16)

exponential

Description: The Exponential test function, a scalable, non-separable function with a unique global minimum at the origin. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = -\exp\left( -\frac{1}{2} \sum{i=1}^n xi^2 \right).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: -1.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, non-separable, scalable, unimodal
Reference: Ramanujan et al. (2007)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

freudensteinroth

Description: Freudenstein-Roth function: Multimodal, non-convex, non-separable, differentiable, fixed dimension (n=2).
Formula: f(x) = \left( x1 - 13 + \left( (5 - x2)x2 - 2 \right)x2 \right)^2 + \left( x1 - 29 + \left( (x2 + 1)x2 - 14 \right)x2 \right)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [5.0, 4.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

giunta

Description: Giunta function: Multimodal, non-convex, separable, differentiable, bounded, continuous. Minimum: 0.06447 at [0.46732, 0.46732]. Bounds: [-1, 1]^2. Dimensions: n=2.
Formula: 0.6 + \sum{i=1}^2 \left[ \sin\left(\frac{16}{15}xi - 1\right) + \sin^2\left(\frac{16}{15}xi - 1\right) + \frac{1}{50} \sin\left(4 \left(\frac{16}{15}xi - 1\right)\right) \right]
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: 0.06447042053690566 at [0.46732003, 0.46732003]
Properties: bounded, continuous, differentiable, multimodal, non-convex, separable
Reference: Unknown Source

goldsteinprice

Description: Goldstein-Price function: Multimodal, non-convex, non-separable, differentiable, bounded (n=2 only).
Formula: (1 + (x1 + x2 + 1)^2 (19 - 14x1 + 3x1^2 - 14x2 + 6x1x2 + 3x2^2)) \cdot (30 + (2x1 - 3x2)^2 (18 - 32x1 + 12x1^2 + 48x2 - 36x1x2 + 27x2^2))
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-2.0, -2.0] to [2.0, 2.0]; Global minimum: 3.0 at [0.0, -1.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

griewank

Description: Griewank function – one of the most deceptive multimodal benchmarks in global optimization. The oscillating product term creates countless local minima that become increasingly misleading with higher dimensions. Gradient-based methods are systematically trapped far from the global minimum.
Formula: f(\mathbf{x}) = \sum{i=1}^n \frac{xi^2}{4000} - \prod{i=1}^n \cos!\left(\frac{xi}{\sqrt{i}}\right) + 1
Dimension: scalable (default n = 10)
Bounds / Minimum: Bounds: [-600.0, ..., -600.0] to [600.0, ..., 600.0]; Global minimum: 0.0 at [0.0, ..., 0.0]
Properties: bounded, continuous, deceptive, differentiable, multimodal, non-convex, non-separable, scalable
Reference: Griewank (1981); Molga & Smutnicki (2005, p. 19); Jamil & Yang (2013, p. 57); Lehman & Stanley (2011, arXiv:1106.2128) – classic deceptive benchmark; Locatelli & Schoen (2013) – deception increases with dimension
Note: Scalable function – shown with default_n = 10. Use fixed(tf; n=...) for custom dimensions.

gulfresearch

Description: Gulf Research and Development problem (least-squares). Properties based on Jamil & Yang (2013, p. 60).
Formula: f(\mathbf{x}) = \sum{i=1}^{99} \left[ \exp\left( -\frac{(ui - x2)^{x3}}{x1} \right) - 0.01 i \right]^2, \quad ui = 25 + [-50 \ln(0.01 i)]^{2/3}.
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [0.1, 0.0, 0.0] to [100.0, 25.6, 5.0]; Global minimum: 0.0 at [50.0, 25.0, 1.5]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 60)

hansen

Description: The Hansen function, a 2D multimodal separable function with multiple global minima. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=0}^{4} (i+1)\cos(i x1 + i+1) \sum{j=0}^{4} (j+1)\cos((j+2) x2 + j+1).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -176.5417931367457 at [-7.58989301, -7.70831374]
Properties: bounded, continuous, differentiable, multimodal, separable
Reference: Jamil & Yang (2013)

hartman6

Description: The Hartman Function 6, a 6D multimodal non-separable function. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = -\sum{i=1}^{4} ci \exp\left( -\sum{j=1}^{6} a{ij} (xj - p{ij})^2 \right).
Dimension: fixed (n = 6)
Bounds / Minimum: Bounds: [0.0, ..., 0.0] to [1.0, ..., 1.0]; Global minimum: -3.3223680114155147 at [0.20168951, ..., 0.65730053]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013)

hartmanf3

Description: Hartmann function: Multimodal, non-convex, non-separable, differentiable, defined for n=3 only, with a global minimum at [0.114614339099637, 0.5556488499706311, 0.8525469535196916].
Formula: -\sum{i=1}^4 \alphai \exp\left(-\sum{j=1}^3 A{ij} (xj - P{ij})^2\right), \quad \alpha=[1,1.2,3,3.2], \quad A=\begin{bmatrix}3 & 10 & 30 \ 0.1 & 10 & 35 \ 3 & 10 & 30 \ 0.1 & 10 & 35\end{bmatrix}, \quad P=\begin{bmatrix}0.3689 & 0.1170 & 0.2673 \ 0.4699 & 0.4387 & 0.7470 \ 0.1091 & 0.8732 & 0.5547 \ 0.03815 & 0.5743 & 0.8828\end{bmatrix}
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0] to [1.0, 1.0, 1.0]; Global minimum: -3.862782147820755 at [0.11461434, 0.55564885, 0.85254695]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

helicalvalley

Description: The Helical Valley function, a 3D multimodal non-separable function. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = 100 \left[ (x2 - 10\theta)^2 + \left(\sqrt{x1^2 + x2^2} - 1\right)^2 \right] + x3^2, \quad \theta = \frac{1}{2\pi} \atantwo(x2, x1).
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [-10.0, -10.0, -10.0] to [10.0, 10.0, 10.0]; Global minimum: 0.0 at [1.0, 0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013)

himmelblau

Description: Himmelblau function: Multimodal, non-convex, non-separable, differentiable, bounded, continuous. Minimum: 0.0 at [3.0, 2.0], [-2.805118, 3.131312], [-3.779310, -3.283186], [3.584428, -1.848126]. Bounds: [-5, 5]^2. Dimensions: n=2.
Formula: (x1^2 + x2 - 11)^2 + (x1 + x2^2 - 7)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: 0.0 at [3.0, 2.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

holder_table1

Description: Properties based on Jamil & Yang (2013, p. 34); four global minima; adapted formula from Surjano for consistency (sin/cos and abs in exp); originally from Mishra (2006). Non-separable due to interactions despite source claim.
Formula: f(\mathbf{x}) = -\left| \sin(x1) \cos(x2) \exp\left( \left| 1 - \frac{\sqrt{x1^2 + x2^2}}{\pi} \right| \right) \right|.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -19.208502567886743 at [8.05502348, -9.66459002]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 34)

holder_table2

Description: Properties based on Jamil & Yang (2013, p. 34); four global minima; adapted formula from Surjano/Al-Roomi for consistency (sin/sin and Euclidean norm in exp); source minimum -19.2085 not matching formula (computed -11.6839); originally from Mishra (2006). Non-separable due to interactions.
Formula: f(\mathbf{x}) = -\left| \sin(x1) \sin(x2) \exp\left( \left| 1 - \frac{\sqrt{x1^2 + x2^2}}{\pi} \right| \right) \right|.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -11.683926748430029 at [8.05100872, -10.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 34)

holdertable

Description: Holder Table function: Multimodal, continuous, partially differentiable, separable, bounded, non-convex. Four global minima at (±8.055023, ±9.664590), f* = -19.2085025678845. Bounds: [-10, 10]^2. Dimensions: n=2.
Formula: -\left| \sin(x1) \cos(x2) \exp\left(\left|1 - \sqrt{x1^2 + x2^2}/\pi\right|\right) \right|
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -19.2085025678845 at [8.055023, 9.66459]
Properties: bounded, continuous, multimodal, non-convex, partially differentiable, separable
Reference: Unknown Source

hosaki

Description: The Hosaki function, a 2D multimodal non-separable function. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \left(1 - 8x1 + 7x1^2 - \frac{7}{3}x1^3 + \frac{1}{4}x1^4\right) x2^2 e^{-x2}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [5.0, 6.0]; Global minimum: -2.345811576101292 at [4.0, 2.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013)

jennrichsampson

Description: The Jennrich-Sampson function. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{10} \left(2 + 2i - (e^{i x1} + e^{i x_2})\right)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: 124.36218235561489 at [0.25782521, 0.25782521]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jennrich and Sampson (1968)

keane

Description: Keane function: Multimodal, continuous, differentiable, non-separable, bounded, non-convex. Two global minima at (0.0, 1.39325) and (1.39325, 0.0), f* = -0.673668. Bounds: [0, 10]^2. Dimensions: n=2.
Formula: -\frac{\sin^2(x1 - x2) \sin^2(x1 + x2)}{\sqrt{x1^2 + x2^2}}
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [10.0, 10.0]; Global minimum: -0.6736675211468548 at [0.0, 1.39324908]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

kearfott

Description: Kearfott function: Multimodal, continuous, differentiable, non-separable, bounded, non-convex. Global minima where x₁² + x₂² = 1.25, e.g., at (0.7905694150420949, -0.7905694150420949) and (-0.7905694150420949, 0.7905694150420949), f* = 1.125. Note: Jamil & Yang (2013) incorrectly lists minima at (0.70710678, -0.70710678) and (-0.70710678, 0.70710678) with f* = 0; those yield f = 1.25.
Formula: (x1^2 + x2^2 - 2)^2 + (x1^2 + x2^2 - 0.5)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-3.0, -3.0] to [4.0, 4.0]; Global minimum: 1.125 at [0.79056942, -0.79056942]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

langermann

Description: Langermann function: Multimodal, non-convex, non-separable test function with unevenly distributed local minima. Minimum at [2.002992119907532, 1.0060959403343601] with value -5.162126159963982, confirmed by high-precision calculations in Al-Roomi (2015) and verified via tf.f(tf.meta:min_position) with atol=1e-6. Gradient norm at minimum is ~1.748e-9 within atol=0.01. Warning: Some sources (e.g., GlomPo, GEATbx) report a local minimum at approximately [2.002992, 1.006096] with value ≈-1.4, likely due to confusion with a local minimum or documentation error. See Al-Roomi (2015) for details.
Formula: -\sum{i=1}^m ci \exp \left[-\frac{1}{\pi} \sum{j=1}^n (xj - a{ij})^2 \right] \cos \left[ \pi \sum{j=1}^n (xj - a{ij})^2 \right]
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [10.0, 10.0]; Global minimum: -5.162126159963982 at [2.00299212, 1.00609594]
Properties: bounded, continuous, controversial, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

leon

Description: The Leon function. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = 100(x2 - x1^2)^2 + (1 - x_1)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-1.2, -1.2] to [1.2, 1.2]; Global minimum: 0.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Lavi and Vogel (1966)

levy

Description: Levy function. Properties based on Jamil & Yang (2013, p. 164); originally from Levy & Montalvo (1977).
Formula: f(\mathbf{x}) = \sin^2(\pi w1) + \sum{i=1}^{n-1} (wi - 1)^2 [1 + 10 \sin^2(\pi wi + 1)] + (wn - 1)^2 [1 + \sin^2(2\pi wn)], \quad wi = 1 + \frac{xi - 1}{4}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, scalable, separable
Reference: Jamil & Yang (2013, p. 164)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

levyjamil

Description: Levy function (Jamil & Yang, 2013): Multimodal, differentiable, non-convex, scalable, bounded, continuous. Global minimum at x* = (1, ..., 1), f* = 0. Bounds: [-10, 10]^n. Note: Follows Jamil & Yang (2013), which may contain typos (e.g., missing coefficient, incorrect 3π scaling). The standard Levy function (see levy.jl) uses wi = 1 + (xi - 1)/4. The function is also non-separable, but this property is omitted in tests for consistency.
Formula: \sin^2(3\pi x1) + \sum{i=1}^{n-1} (xi - 1)^2 [1 + \sin^2(3\pi x{i+1})] + (xn - 1)^2 [1 + \sin^2(2\pi xn)]
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, scalable
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

matyas

Description: Matyas function; Properties based on Jamil & Yang (2013, p. 20).
Formula: f(\mathbf{x}) = 0.26 (x1^2 + x2^2) - 0.48 x1 x2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, convex, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 20)

mccormick

Description: McCormick function: Multimodal, non-convex function with global minimum at approximately -1.91322295, defined for 2 dimensions.
Formula: \sin(x1 + x2) + (x1 - x2)^2 - 1.5 x1 + 2.5 x2 + 1
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-1.5, -3.0] to [4.0, 4.0]; Global minimum: -1.9132229549810367 at [-0.54719755, -1.54719755]
Properties: bounded, continuous, differentiable, multimodal, non-convex
Reference: molga&smutnicki(2005)

michalewicz

Description: Michalewicz function: Multimodal, non-separable, with many local minima.
Formula: f(x) = -\sum{i=1}^n \sin(xi) \sin^{2m}(i x_i^2 / \pi), m=10
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [3.14159265, 3.14159265]; Global minimum: -1.8013034100985528 at [2.20290552, 1.57079633]
Properties: bounded, continuous, differentiable, multimodal, non-separable, scalable
Reference: Unknown Source
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

mielcantrell

Description: The Miele-Cantrell function. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = (e^{x1} - x2)^4 + 100 (x2 - x3)^6 + [\tan(x3 - x4)]^4 + x_1^8
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [-1.0, -1.0, -1.0, -1.0] to [1.0, 1.0, 1.0, 1.0]; Global minimum: 0.0 at [0.0, 1.0, 1.0, 1.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Cragg and Levy (1969)

mishra1

Description: The Mishra Function 1. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \left(1 + gn\right)^{gn}, \quad gn = n - \sum{i=1}^{n-1} x_i
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [1.0, 1.0]; Global minimum: 2.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable, scalable
Reference: Mishra (2006a)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

mishra10

Description: Mishra Function 10: 2D multimodal function involving floor operations. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \left[ \lfloor x1 x2 \rfloor - \lfloor x1 \rfloor - \lfloor x2 \rfloor \right]^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Mishra (2006f), via Jamil & Yang (2013): f83

mishra11

Description: Mishra Function 11 (AMGM): Scalable multimodal function based on arithmetic-geometric mean inequality. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \left[ \frac{1}{n} \sum{i=1}^n |xi| - \left( \prod{i=1}^n |xi| \right)^{1/n} \right]^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable, scalable
Reference: Mishra (2006f), via Jamil & Yang (2013): f84
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

mishra2

Description: The Mishra Function 2. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = (1 + gn)^{gn}, \quad gn = n - \sum{i=1}^{n-1} \frac{xi + x{i+1}}{2}
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [1.0, 1.0]; Global minimum: 2.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable, scalable
Reference: Mishra (2006a)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

mishra3

Description: The Mishra Function 3. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \left| \cos \left( \sqrt{ | x1^2 + x2 | } \right) \right|^{0.5} + 0.01 (x1 + x2)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -0.184651333342989 at [-8.46661378, -9.99852131]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Mishra (2006f)

mishra4

Description: The Mishra Function 4. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sqrt{ \left| \sin \sqrt{ |x1^2 + x2| } \right| } + 0.01(x1 + x2)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -0.19941146886776687 at [-9.94114881, -10.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Mishra (2006f)

mishra5

Description: Mishra Function 5: A 2D multimodal test function. Properties based on Jamil & Yang (2013). Note: PDF reports erroneous min position; validated via optimization.
Formula: f(\mathbf{x}) = \left[ \sin^2 \left( (\cos x1 + \cos x2)^2 \right) + \cos^2 \left( (\sin x1 + \sin x2)^2 \right) + x1 \right]^2 + 0.01(x1 + x_2)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -0.11982951993 at [-1.98682, -10.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Mishra (2006f)

mishra6

Description: Mishra Function 6: A 2D multimodal test function. Properties based on Jamil & Yang (2013). Note: f3 term outside log; coeff. 0.1 (Jamil lists 0.01, likely typo).
Formula: f(\mathbf{x}) = -\ln \left[ \left( \sin^2 \left( (\cos x1 + \cos x2)^2 \right) - \cos^2 \left( (\sin x1 + \sin x2)^2 \right) + x1 \right)^2 \right] + 0.1 \left( (x1 - 1)^2 + (x_2 - 1)^2 \right)
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -2.28394983847 at [2.88630722, 1.82326033]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Mishra (2006)

mishra7

Description: Mishra Function 7: A 2D multimodal test function with multiple global minima where prod(x) = 2. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \left[ x1 x2 - 2! \right]^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 2.0]
Properties: continuous, differentiable, multimodal, non-separable
Reference: Mishra (2006f)

mishra8

Description: Mishra Function 8 (Decanomial): A 2D multimodal test function. Properties based on Jamil & Yang (2013). Note: Corrected coeff. 13340 for x1^4; multiplication between polys.
Formula: f(\mathbf{x}) = 0.001 \left[ \left| x1^{10} - 20 x1^9 + 180 x1^8 - 960 x1^7 + 3360 x1^6 - 8064 x1^5 + 13340 x1^4 - 15360 x1^3 + 11520 x1^2 - 5120 x1 + 2624 \right| \cdot \left| x2^4 + 12 x2^3 + 54 x2^2 + 108 x2 + 81 \right| \right]^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [2.0, -3.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Mishra (2006f)

mishra9

Description: Mishra Function 9 (Dodecal Polynomial): A 3D multimodal test function. Properties based on Jamil & Yang (2013). Corrected terms in b and c from al-roomi.org for f(1,2,3)=0.
Formula: f(\mathbf{x}) = \left[ f1 f2^2 f3 + f1 f2 f3^2 + f2^2 + (x1 + x2 - x3)^2 \right]^2 \ where \ f1 = 2x1^3 + 5x1 x2 + 4x3 - 2x1^2 x3 - 18, \ f2 = x1 + x2^3 + x1 x2^2 + x1 x3^2 - 22, \ f3 = 8x1^2 + 2x2 x3 + 2x2^2 + 3x2^3 - 52.
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [-10.0, -10.0, -10.0] to [10.0, 10.0, 10.0]; Global minimum: 0.0 at [1.0, 2.0, 3.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Mishra (2006f)

mishrabird

Description: Mishra's Bird function is a multimodal, non-separable function with two known global minima. It is often cited with the constraint (x+5)^2+(y+5)^2 < 25, but this implementation is unconstrained.
Formula: f(x, y) = \sin(y) e^{(1 - \cos(x))^2} + \cos(x) e^{(1 - \sin(y))^2} + (x - y)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -6.5] to [0.0, 0.0]; Global minimum: -106.76453674926466 at [-3.1302468, -1.58214218]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

mvf_shubert

Description: mvfShubert (Adorio variant, 2D fixed, additive sine with index shift j=0..4); separable with ≈400 local minima; properties based on Adorio (2005, p. 12–13); similar to Shubert 3 but with offsets.
Formula: f(\mathbf{x}) = -\sum{i=0}^{1}\sum{j=0}^{4}(j+1)\sin((j+2)x_i+(j+1)).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -24.062498884334282 at [-0.49139084, 5.79179447]
Properties: continuous, differentiable, multimodal, separable
Reference: Adorio (2005, p. 12)

mvf_shubert2

Description: mvfShubert2 (Adorio variant, 2D fixed, additive cosine with index shift j=0..4); separable with ≈400 local minima; properties based on Adorio (2005, p. 13); similar to Shubert 2 but with offsets.
Formula: f(\mathbf{x}) = \sum{i=0}^{1}\sum{j=0}^{4}(j+1)\cos((j+2)x_i+(j+1)).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -25.74177099545137 at [-1.42512843, -1.42512843]
Properties: continuous, differentiable, multimodal, separable
Reference: Adorio (2005, p. 13)

mvf_shubert3

Description: mvfShubert3 (Adorio nD generalization, additive sine with index shift j=0..4); separable with ≈400 local minima in 2D; properties based on Adorio (2005, p. 13); generalization of mvfShubert.
Formula: f(\mathbf{x}) = -\sum{i=0}^{n-1}\sum{j=0}^{4}(j+1)\sin((j+2)x_i+(j+1)).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -24.062498884334275 at [-0.49139083, -0.49139083]
Properties: continuous, differentiable, multimodal, scalable, separable
Reference: Adorio (2005, p. 13)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

parsopoulos

Description: Parsopoulos Function: 2D multimodal periodic function with infinite minima. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \cos^2(x1) + \sin^2(x2).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: 0.0 at [1.57079633, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, separable
Reference: Parsopoulos et al., via Jamil & Yang (2013): f85

pathological

Description: Pathological Function: Scalable multimodal function with interdependent variables. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{n-1} \left( 0.5 + \frac{\sin^2 \sqrt{100 xi^2 + x{i+1}^2} - 0.5}{1 + 0.001 (xi^2 - 2 xi x{i+1} + x_{i+1}^2)^2} \right).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable, scalable
Reference: Rahnamayan et al. (2007a), via Jamil & Yang (2013): f87
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

paviani

Description: Paviani Function: 10D multimodal function involving logs and product. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{10} \left[ \left(\ln (xi - 2)\right)^2 + \left(\ln (10 - xi)\right)^2 \right] - \left( \prod{i=1}^{10} x_i \right)^{0.2}.
Dimension: fixed (n = 10)
Bounds / Minimum: Bounds: [2.001, ..., 2.001] to [9.999, ..., 9.999]; Global minimum: -45.77846970744629 at [9.35026583, ..., 9.35026583]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Himmelblau (1972), via Jamil & Yang (2013): f88

penholder

Description: Pen Holder Function: 2D multimodal function with four global minima. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = -\exp\left( -\frac{1}{\left| \cos(x1) \cos(x2) \exp\left( \left| 1 - \frac{\sqrt{x1^2 + x2^2}}{\pi} \right| \right) \right|} \right).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-11.0, -11.0] to [11.0, 11.0]; Global minimum: -0.9635348327265058 at [9.64616767, 9.64616767]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Mishra (2006f), via Jamil & Yang (2013): f86

periodic

Description: Periodic function (also known as Price's Function No. 02) with 49 local minima at f=1 and global minimum at origin f=0.9. Non-separable due to coupled exponential term. Properties based on Jamil & Yang (2013), adapted for separability analysis.
Formula: f(\mathbf{x}) = 1 + \sin^2(x1) + \sin^2(x2) - 0.1 e^{-(x1^2 + x2^2)}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.9 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Ali et al. (2005)

pinter

Description: Pintér Function: Scalable multimodal function with cyclic dependencies. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^n i xi^2 + \sum{i=1}^n 20 i \sin^2 A + \sum{i=1}^n i \log{10} (1 + i B^2), \ A = x{i-1} \sin xi + \sin x{i+1}, \ B = x{i-1}^2 - 2 xi + 3 x{i+1} - \cos xi + 1 \ (cyclic: x0 = xn, x{n+1} = x1).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable, scalable
Reference: Pintér (1996), via Jamil & Yang (2013): f89
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

powell

Description: Powell function; Properties based on Jamil & Yang (2013, p. 28); originally from Powell (1962).
Formula: f(\mathbf{x}) = (x1 + 10x2)^2 + 5(x3 - x4)^2 + (x2 - 2x3)^4 + 10(x1 - x4)^4.
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [-5.0, -5.0, -5.0, -5.0] to [5.0, 5.0, 5.0, 5.0]; Global minimum: 0.0 at [0.0, 0.0, 0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 28)

powellsingular

Description: Powell Singular Function, a quartic function with singular Hessian at the global minimum. Scalable in blocks of 4 dimensions. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{n/4} \left[ (x{4i-3} + 10 x{4i-2})^2 + 5 (x{4i-1} - x{4i})^2 + (x{4i-2} - 2 x{4i-1})^4 + 10 (x{4i-3} - x_{4i})^4 \right].
Dimension: scalable (default n = 4)
Bounds / Minimum: Bounds: [-4.0, -4.0, -4.0, -4.0] to [5.0, 5.0, 5.0, 5.0]; Global minimum: 0.0 at [0.0, 0.0, 0.0, 0.0]
Properties: bounded, continuous, differentiable, non-separable, scalable, unimodal
Reference: Powell (1962)
Note: Scalable function – shown with default_n = 4. Use fixed(tf; n=...) for custom dimensions.

powellsingular2

Description: Powell Singular Function 2, a quartic function with singular Hessian at the global minimum. Scalable in blocks of 4 dimensions. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{n/4} \left[ (x{4i-3} + 10 x{4i-2})^2 + 5 (x{4i-1} - x{4i})^2 + (x{4i-2} - 2 x{4i-1})^4 + 10 (x{4i-3} - x_{4i})^4 \right].
Dimension: scalable (default n = 4)
Bounds / Minimum: Bounds: [-4.0, -4.0, -4.0, -4.0] to [5.0, 5.0, 5.0, 5.0]; Global minimum: 0.0 at [0.0, 0.0, 0.0, 0.0]
Properties: bounded, continuous, differentiable, non-separable, scalable, unimodal
Reference: Fu et al. (2006)
Note: Scalable function – shown with default_n = 4. Use fixed(tf; n=...) for custom dimensions.

powellsum

Description: Powell Sum Function with separable terms. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^n |xi|^{i+1}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, scalable, separable, unimodal
Reference: Rahnamyan et al. (2007a)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

price1

Description: Price Function 1 (Price, 1977). Properties based on Jamil & Yang (2013). Non-differentiable at x=0 due to absolute value terms. Has four global minima at the corners.
Formula: f(\mathbf{x}) = (|x1| - 5)^2 + (|x2| - 5)^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [5.0, 5.0]
Properties: continuous, multimodal, separable
Reference: Price (1977)

price2

Description: Price Function 2 (Price, 1977). Properties based on Jamil & Yang (2013). Combines trigonometric and exponential terms.
Formula: f(\mathbf{x}) = 1 + \sin^2(x1) + \sin^2(x2) - 0.1e^{-x1^2 - x2^2}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.9 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal
Reference: Price (1977)

price4

Description: Price 4 function from Jamil & Yang (2013, No. 97). Properties: continuous, differentiable, non-separable, multimodal, bounded. Multiple minima at (0, 0) and (2, 4).
Formula: f(\mathbf{x}) = (2x1^3 x2 - x2^3)^2 + (6x1 - x2^2 + x2)^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013), https://arxiv.org/abs/1308.4008

qing

Description: Qing Function. Properties based on Jamil & Yang (2013). Multiple global minima due to sign choices.
Formula: f(\mathbf{x}) = \sum{i=1}^D (xi^2 - i)^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [1.0, 1.41421356]
Properties: continuous, differentiable, multimodal, scalable, separable
Reference: Qing (2006)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

quadratic

Description: Quadratic function. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = -3803.84 - 138.08 x1 - 232.92 x2 + 128.08 x1^2 + 203.64 x2^2 + 182.25 x1 x2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -3873.72418218627 at [0.19388017, 0.48513391]
Properties: bounded, continuous, convex, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013)

quartic

Description: Quartic Function with additive uniform noise from [0,1). The deterministic part is strictly convex, but the overall function is non-differentiable due to random noise. Gradient implementation returns derivative of deterministic component only. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{n} i xi^4 + \mathcal{U}[0, 1).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.28, -1.28] to [1.28, 1.28]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, has_noise, scalable, separable, unimodal
Reference: Jamil & Yang (2013)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

quintic

Description: The Quintic function is a separable multimodal benchmark function with absolute value terms on a quintic polynomial per dimension. Note: Due to the absolute value, it is not strictly differentiable at the roots of the inner polynomial (x_i = -1 or 2), but a subgradient is provided. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^D |xi^5 - 3xi^4 + 4xi^3 + 2xi^2 - 10xi - 4|.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [-1.0, -1.0]
Properties: continuous, differentiable, multimodal, separable
Reference: Jamil & Yang (2013)

rana

Description: Rana function incorporates a complex iterative trigonometric mechanism on square roots of absolute differences and sums of pairs. Due to trigonometric sensitivity, minor input changes lead to significant output alterations. Operates in broad range [-500,500]; numerous local minima/maxima characterize its landscape. Properties based on Jamil & Yang (2013) and Naser et al. (2024). Note: Naser et al. reports f(x*) = -928.5478 at x=[-500,-500], but computed value is ≈ -464.274; possible discrepancy in formula interpretation or typo in source.
Formula: f(\mathbf{x}) = \sum{i=1}^{D-1} (x{i+1} + 1) \cos(t2) \sin(t1) + xi \cos(t1) \sin(t_2), \

t1 = \sqrt{|x{i+1} + xi + 1|}, \quad t2 = \sqrt{|x{i+1} - xi + 1|}.

Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: -464.27392770239135 at [-500.0, -500.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable, scalable
Reference: Price et al. (2005). Differential Evolution: A Practical Approach to Global Optimization. Springer; Naser et al. (2024). A Review of 315 Benchmark and Test Functions.
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

rastrigin

Description: Rastrigin function – highly multimodal and strongly deceptive. The regular arrangement of deep local minima systematically misleads gradient-based and local search methods away from the global minimum at zero. Classic example of a deceptive function in global optimization literature.
Formula: f(\mathbf{x}) = 10n + \sum{i=1}^n \left[ xi^2 - 10 \cos(2\pi x_i) \right]
Dimension: scalable (default n = 10)
Bounds / Minimum: Bounds: [-5.12, ..., -5.12] to [5.12, ..., 5.12]; Global minimum: 0.0 at [0.0, ..., 0.0]
Properties: bounded, continuous, deceptive, differentiable, multimodal, non-convex, scalable, separable
Reference: Molga & Smutnicki (2005), p. 24; Goldberg (1989) – classic deceptive benchmark; Jamil & Yang (2013), p. 88
Note: Scalable function – shown with default_n = 10. Use fixed(tf; n=...) for custom dimensions.

ripple1

Description: Ripple 1 function. Volatile due to high-frequency cosine and strong sin^6 power; complex range from oscillations, depending on x distribution/values. Represented in [0,1] with global min at x=[0.1,0.1], f=-2.2. Non-separable. Multimodal with 252004 local minima. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{2} -e^{-2 \ln 2 \left(\frac{xi - 0.1}{0.8}\right)^2} \left(\sin^6(5 \pi xi) + 0.1 \cos^2(500 \pi xi)\right).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [1.0, 1.0]; Global minimum: -2.2 at [0.1, 0.1]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

ripple25

Description: Ripple 25 function. Volatile due to high-frequency sine^6 term and Gaussian envelope; complex range from oscillations, depending on x distribution/values. Represented in [0,1] with global min at x=[0.1,0.1], f=-2.0. Non-separable. Multimodal with numerous local minima. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{2} -e^{-2 \ln 2 \left(\frac{xi - 0.1}{0.8}\right)^2} \sin^6(5 \pi x_i).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [1.0, 1.0]; Global minimum: -2.0 at [0.1, 0.1]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

rosenbrock

Description: Rosenbrock function: sum{i=1}^{n-1} [100 (x{i+1} - xi^2)^2 + (xi - 1)^2]. Features a narrow parabolic valley, making it ill-conditioned and challenging for gradient-based methods. Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = \sum{i=1}^{n-1} [100(x{i+1} - xi^2)^2 + (xi - 1)^2].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-30.0, -30.0] to [30.0, 30.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: bounded, continuous, differentiable, ill-conditioned, non-separable, scalable, unimodal
Reference: Jamil & Yang (2013), Benchmark Function #105
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

rosenbrock_modified

Description: Rosenbrock Modified function: 74 + 100(x2 - x1^2)^2 + (1 - x1)^2 - 400 exp(-[(x1 + 1)^2 + (x2 + 1)^2]/0.1). Multimodal variant with Gaussian perturbation creating a deceptive local minimum near (1,1). Computed minvalue via implementation for precision; literature values approximate (e.g., 34.04). Properties based on Jamil & Yang (2013).
Formula: f(\mathbf{x}) = 74 + 100(x2 - x1^2)^2 + (1 - x1)^2 - 400 e^{-\frac{(x1 + 1)^2 + (x_2 + 1)^2}{0.1}}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-2.0, -2.0] to [2.0, 2.0]; Global minimum: 34.04024310664062 at [-0.90955374, -0.95057171]
Properties: bounded, continuous, controversial, deceptive, differentiable, ill-conditioned, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013), Benchmark Function #106

rotatedellipse

Description: Rotated Ellipse function. A convex quadratic function with elliptical contours rotated due to the cross-term. Properties based on [Jamil & Yang (2013, Entry 107)].
Formula: f(\mathbf{x}) = 7x1^2 - 6\sqrt{3}x1x2 + 13x2^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, convex, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, Entry 107)

rotatedellipse2

Description: Rotated Ellipse N.2 function: A variant of the Rotated Ellipse, convex quadratic with elliptical contours rotated by cross-term. Properties based on [Jamil & Yang (2013, Entry 107)].
Formula: f(\mathbf{x}) = 7x1^2 - 6\sqrt{3}x1x2 + 13x2^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, convex, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, Entry 107)

rump

Description: The Rump function modified for optimization benchmarks: absolute value ensures non-negativity (global min 0), and denominator 2 + x_2 avoids division by zero. Adapted from original Moore (1988) via Jamil & Yang (2013). Properties based on Al-Roomi (2015).
Formula: f(\mathbf{x}) = \left| (333.75 - x1^2) x2^6 + x1^2 (11 x1^2 x2^2 - 121 x2^4 - 2) + 5.5 x2^8 + \frac{x1}{2 + x_2} \right|.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, non-separable, partially differentiable, unimodal
Reference: https://al-roomi.org/benchmarks/unconstrained/2-dimensions/128-rump-function

salomon

Description: The Salomon function. Properties based on Jamil & Yang (2013, p. 27); originally from Salomon (1996).
Formula: f(\mathbf{x}) = 1 - \cos\left(2\pi \sqrt{\sum{i=1}^D xi^2}\right) + 0.1 \sqrt{\sum{i=1}^D xi^2}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 27)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

sargan

Description: The Sargan function. Properties based on Jamil & Yang (2013, p. 27); originally from Dixon & Szegö (1978).
Formula: f(\mathbf{x}) = \sum{i=1}^D (xi^2 + 0.4 \sum{j \neq i} xi x_j).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 27)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schaffer1

Description: Schaffer's Function No. 01. Highly multimodal with concentric rings of local minima caused by the squared sine term. The function is non-separable due to coupling through x₁² + x₂².
Formula: f(\mathbf{x}) = 0.5 + \frac{\sin^2(x1^2 + x2^2) - 0.5}{[1 + 0.001(x1^2 + x2^2)]^2}, \quad D=2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 136, Function 112)

schaffer2

Description: 2D unimodal test function with fixed dimension (non-scalable); Properties based on Jamil & Yang (2013, p. 28); Note: The denominator in Jamil & Yang appears to omit the outer square (likely a typographical error with forgotten parentheses and square, deviating from referenced Mishra (2007)), implemented as per Jamil for consistency; originally from Schaffer.
Formula: f(\mathbf{x}) = 0.5 + \frac{\sin^2(x1^2 - x2^2) - 0.5}{1 + 0.001 (x1^2 + x2^2)^2}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 28)

schaffer3

Description: Schaffer Function No. 3; Properties based on Jamil & Yang (2013, p. 28); ursprünglich aus Schaffer (1984). Note: Denominator interpreted as [1 + 0.001(x1^2 + x2^2)]^2 to match reported minimum.
Formula: f(\mathbf{x}) = 0.5 + \sin^2(\cos |x1^2 - x2^2|) - \frac{0.5}{[1 + 0.001(x1^2 + x2^2)]^2}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0015668553065719681 at [0.0, 1.25311559]
Properties: continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 28)

schaffer6

Description: Schaffer's Function No. 06. Continuous, single-objective benchmark function for unconstrained global optimization in 2D. Highly multimodal with concentric rings of local minima due to the oscillatory sine term applied to the radius. The function is non-separable due to coupling through x₁² + x₂².
Formula: f(\mathbf{x}) = 0.5 + \frac{\sin^2(\sqrt{x1^2 + x2^2}) - 0.5}{[1 + 0.001(x1^2 + x2^2)]^2}, \quad D=2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Al-Roomi (2015, Schaffer's Function No. 06)

schafferf6

Description: Schaffer F6 function; multimodal with cyclic dependency; properties based on Jamil & Yang (2013, p. 32, f136); originally from Schaffer et al. (1989).
Formula: f(\mathbf{x}) = \sum{i=1}^n \frac{\sin^2 \sqrt{xi^2 + x{i+1}^2}}{(1 + 0.001 (xi^2 + x{i+1}^2))^2}, \quad x{n+1} = x_1.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 32)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schaffern2

Description: Schaffer N.2 function: A multimodal function with a global minimum at the origin and many local minima. Variant without sqrt in sine argument; Properties based on Mishra (2007, p. 4); originally from Schaffer.
Formula: f(\mathbf{x}) = 0.5 + \frac{\sin^2(x1^2 - x2^2) - 0.5}{(1 + 0.001(x1^2 + x2^2))^2}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Mishra (2007, p. 4)

schaffern4

Description: Properties based on Jamil & Yang (2013, p. 27); Adapted for variant with |x₁² - x₂²| from Al-Roomi (2015); Contains abs terms leading to non-differentiability at x₁² = x₂² (gradient returns NaN there).
Formula: f(\mathbf{x}) = 0.5 + \frac{\cos^2(\sin(|x1^2 - x2^2|)) - 0.5}{(1 + 0.001(x1^2 + x2^2))^2}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.29257863203598033 at [0.0, 1.25313183]
Properties: bounded, continuous, multimodal, non-convex, non-separable, partially differentiable
Reference: Jamil & Yang (2013, p. 27)

schmidtvetters

Description: Schmidt Vetters Function; Properties based on Jamil & Yang (2013, p. 116); Local minimum reported in source (rounded to 3, computed 2.998); global lower at boundary in wide bounds; partially differentiable due to singularity at x2=0; ursprünglich aus Schmidt & Vetter (1980).
Formula: f(\mathbf{x}) = \frac{1}{1 + (x1 - x2)^2} + \sin\left( \frac{\pi x2 + x3}{2} \right) + e^{\left( \frac{x1 + x2}{x_2} - 2 \right)^2}.
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0] to [10.0, 10.0, 10.0]; Global minimum: 0.19397252244395102 at [7.07083412, 10.0, 3.14159293]
Properties: controversial, multimodal, non-separable, partially differentiable
Reference: Jamil & Yang (2013, p. 116)

schumersteiglitz

Description: Schumer-Steiglitz function: A simple separable unimodal function with global minimum at the origin.
Formula: f(\mathbf{x}) = \sum{i=1}^{n} xi^4
Dimension: scalable (default n = 10)
Bounds / Minimum: Bounds: [-10.0, ..., -10.0] to [10.0, ..., 10.0]; Global minimum: 0.0 at [0.0, ..., 0.0]
Properties: continuous, differentiable, scalable, separable, unimodal
Reference: Schumer, M. A. and Steiglitz, K. (1968). Adaptive Step Size Random Search. IEEE Transactions on Automatic Control, 13(3), 270–276.
Note: Scalable function – shown with default_n = 10. Use fixed(tf; n=...) for custom dimensions.

schwefel

Description: The Schwefel Function [7] with α=2.0 (chosen for concrete unimodal variant; general α≥0 per source). Properties based on Jamil & Yang (2013, p. 118); originally from Schwefel (various works).
Formula: f(\mathbf{x}) = \left( \sum{i=1}^{D} xi^2 \right)^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, differentiable, partially separable, scalable, unimodal
Reference: Jamil & Yang (2013, p. 118)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schwefel12

Description: Schwefel 1.2 function; also known as Rotated Hyper-Ellipsoid or Double-Sum Function. Properties based on Jamil & Yang (2013, p. 29); originally from Schwefel (1977).
Formula: f(\mathbf{x}) = \sum{i=1}^n \left( \sum{j=1}^i x_j \right)^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, convex, differentiable, non-separable, scalable, unimodal
Reference: Jamil & Yang (2013, p. 29)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schwefel220

Description: Schwefel's Problem 2.20 test function (positive sum variant as in benchmarks); Properties based on Jamil & Yang (2013, p. 77) [adapted to standard form without erroneous n-factor or minus]; originally from Schwefel (1977). Global minimum at the origin.
Formula: f(\mathbf{x}) = \sum{i=1}^{D} |xi|.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, partially differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, p. 77)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schwefel221

Description: Schwefel's Problem 2.21 test function; Properties based on Jamil & Yang (2013, p. 123); originally from Schwefel (1977). Global minimum at the origin.
Formula: f(\mathbf{x}) = \max{1 \leq i \leq D} |xi|.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, partially differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, p. 123)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schwefel222

Description: Schwefel's Problem 2.22 test function; Properties based on Jamil & Yang (2013, p. 124); originally from Schwefel (1977). Global minimum at the origin.
Formula: f(\mathbf{x}) = \sum{i=1}^{D} |xi| + \prod{i=1}^{D} |xi|.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, non-separable, partially differentiable, scalable, unimodal
Reference: Jamil & Yang (2013, p. 124)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schwefel223

Description: Schwefel's Problem 2.23 test function; Properties based on Jamil & Yang (2013, p. 125) [adapted to 'separable' as function is additive]; originally from Schwefel (1977). Global minimum at the origin.
Formula: f(\mathbf{x}) = \sum{i=1}^{D} xi^{10}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, p. 125)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schwefel225

Description: Schwefel's Problem 2.25 test function; Properties based on Jamil & Yang (2013, p. 127) [separable per source, though coupled terms]; originally from Schwefel (1977).
Formula: f(\mathbf{x}) = (x2 - 1)^2 + (x1 - x_2^2)^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: continuous, differentiable, multimodal, separable
Reference: Jamil & Yang (2013, p. 127)

schwefel226

Description: Schwefel's Problem 2.26 test function; Properties based on Jamil & Yang (2013, p. 30); originally from Schwefel (1977). Multiple global minima due to periodic sin term.
Formula: f(\mathbf{x}) = -\frac{1}{D} \sum{i=1}^{D} xi \sin \sqrt{|x_i|}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: -418.9828872724338 at [420.96874358, 420.96874358]
Properties: continuous, differentiable, multimodal, scalable, separable
Reference: Jamil & Yang (2013, p. 30)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

schwefel236

Description: Schwefel's Problem 2.36 test function; Properties based on Jamil & Yang (2013, p. 129) [adapted: non-separable due to coupling, unimodal as quadratic]; originally from Schwefel (1981).
Formula: f(\mathbf{x}) = -x1 x2 (72 - 2 x1 - 2 x2).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [500.0, 500.0]; Global minimum: -3456.0 at [12.0, 12.0]
Properties: continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 129)

schwefel24

Description: Schwefel's Problem 2.4 test function; Properties based on Jamil & Yang (2013, p. 30); originally from Schwefel (1977).
Formula: f(\mathbf{x}) = \sum{i=1}^{2} (xi - 1)^2 + (x1 - xi^2)^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: continuous, differentiable, multimodal, separable
Reference: Jamil & Yang (2013, p. 30)

schwefel26

Description: Schwefel's Problem 2.6 test function (piecewise linear); Properties based on Jamil & Yang (2013, p. 31); originally from Schwefel (1981). Global minimum where both arguments are zero.
Formula: f(\mathbf{x}) = \max(|x1 + 2x2 - 7|, |2x1 + x2 - 5|).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [1.0, 3.0]
Properties: continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 31)

shekel

Description: Shekel function (m=10): Multimodal, finite at infinity, non-convex, non-separable, differentiable function defined for n=4, with multiple local minima and a global minimum near [4,4,4,4]. Properties based on Jamil & Yang (2013, p. 30) [f_{132}]; originally from Molga & Smutnicki (2005).
Formula: f(\mathbf{x}) = -\sum{i=1}^{10} \frac{1}{\sum{j=1}^4 (xj - a{ij})^2 + c_i}
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0, 0.0] to [10.0, 10.0, 10.0, 10.0]; Global minimum: -10.536409816692043 at [4.00074653, 4.00059293, 3.9996634, 3.9995098]
Properties: bounded, continuous, differentiable, finiteatinf, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 30)

shekel5

Description: Shekel 5 test function; Properties based on Jamil & Yang (2013, p. 130) [minimum controversial: source -10.1532 (rounded) and pos [4,4,4,4], precise -10.15319967905822 at ≈[4.00003715, 4.00013327, 4.00003715, 4.00013327]]; originally from Box (1966).
Formula: f(\mathbf{x}) = -\sum{i=1}^{5} \frac{1}{\sum{j=1}^{4} (xj - a{ij})^2 + ci}, \quad \mathbf{a}i \in A = \begin{bmatrix} 4 & 4 & 4 & 4 \ 1 & 1 & 1 & 1 \ 8 & 8 & 8 & 8 \ 6 & 6 & 6 & 6 \ 3 & 7 & 3 & 7 \end{bmatrix}, \quad \mathbf{c} = [0.1, 0.2, 0.2, 0.4, 0.4].
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0, 0.0] to [10.0, 10.0, 10.0, 10.0]; Global minimum: -10.15319967905822 at [4.00003715, 4.00013327, 4.00003715, 4.00013327]
Properties: continuous, controversial, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 130)

shekel7

Description: Shekel 7 test function; Properties based on Jamil & Yang (2013, p. 131) [minimum controversial: source -10.3999 (rounded) and pos [4,4,4,4], precise -10.402940566818653 at ≈[4.00057291, 4.00068936, 3.99948971, 3.99960616]]; originally from Box (1966).
Formula: f(\mathbf{x}) = -\sum{i=1}^{7} \frac{1}{\sum{j=1}^{4} (xj - a{ij})^2 + ci}, \quad \mathbf{a}i \in A = \begin{bmatrix} 4 & 4 & 4 & 4 \ 1 & 1 & 1 & 1 \ 8 & 8 & 8 & 8 \ 6 & 6 & 6 & 6 \ 3 & 7 & 3 & 7 \ 2 & 9 & 2 & 9 \ 5 & 5 & 3 & 3 \end{bmatrix}, \quad \mathbf{c} = [0.1, 0.2, 0.2, 0.4, 0.4, 0.6, 0.3].
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0, 0.0] to [10.0, 10.0, 10.0, 10.0]; Global minimum: -10.402940566818653 at [4.00057291, 4.00068936, 3.99948971, 3.99960616]
Properties: continuous, controversial, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 131)

shubertadditivecosine

Description: Additive cosine Shubert function (Shubert 2); separable with ≈400 local minima in 2D; properties based on Jamil & Yang (2013, p. 56, f135); originally from Yao (1999).
Formula: f(\mathbf{x}) = \sum{i=1}^n \sum{j=1}^5 j \cos((j+1)x_i + j).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -25.74177099545136 at [-1.42512843, -1.42512843]
Properties: continuous, differentiable, multimodal, scalable, separable
Reference: Jamil & Yang (2013, p. 56)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

shubertadditivesine

Description: Additive sine Shubert function (Shubert 3); separable and asymmetric with ≈400 local minima in 2D; properties based on Jamil & Yang (2013, p. 55, f134); originally from Yao (1999).
Formula: f(\mathbf{x}) = \sum{i=1}^n \sum{j=1}^5 j \sin((j+1)x_i + j).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -29.675900051421173 at [-7.397285, -7.397285]
Properties: continuous, differentiable, multimodal, scalable, separable
Reference: Jamil & Yang (2013, p. 55)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

shubert_classic

Description: Classical Shubert function (product of cosine sums); highly multimodal with 760 local minima in 2D; properties based on Jamil & Yang (2013, p. 55, f133); originally from Shubert (1970).
Formula: f(\mathbf{x}) = \prod{i=1}^{2} \left( \sum{j=1}^{5} j \cos((j+1) x_i + j) \right).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -186.73090883102375 at [4.85805688, 5.48286421]
Properties: continuous, controversial, differentiable, highly multimodal, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 55)

shubert_coupled

Description: Coupled Shubert function (Shubert 4); non-separable with 760 local minima in 2D; for n=2 identical to classical; properties based on Jamil & Yang (2013, p. 56, f135-related); originally from Yao (1999).
Formula: f(\mathbf{x}) = \sum{i=1}^{n-1} \left[\sum{j=1}^5 j \cos((j+1) xi + j)\right] \left[\sum{j=1}^5 j \cos((j+1) x_{i+1} + j)\right].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -186.73090883102375 at [4.85805688, 5.48286421]
Properties: continuous, differentiable, highly multimodal, multimodal, non-convex, non-separable, scalable
Reference: Jamil & Yang (2013, p. 56)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

shubert_generalized

Description: Generalized Shubert function (parametric additive cosine); separable with ≈400 local minima in 2D; standard params (aj=j, bj=j+1, c_j=j) identical to Shubert 2; properties based on Jamil & Yang (2013, p. 56, f135-Generalized); CEC 2008–2013.
Formula: f(\mathbf{x}) = \sum{i=1}^n \sum{j=1}^5 aj \cos(bj xi + cj).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -25.741770995451372 at [-1.42512843, -1.42512843]
Properties: continuous, differentiable, multimodal, scalable, separable
Reference: Jamil & Yang (2013, p. 56)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

shuberthybridrastrigin

Description: Hybrid Shubert-Rastrigin composition (weights 0.5/0.5, n=2); non-separable multimodal hybrid; properties based on Jamil & Yang (2013, p. 56, Hybrid-Variante); CEC 2020 F10.
Formula: f(\mathbf{x}) = 0.5 \cdot \text{Shubert}(\mathbf{x}) + 0.5 \cdot \text{Rastrigin}(\mathbf{x}).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -79.36953021020285 at [-0.81305187, -1.41787744]
Properties: bounded, continuous, differentiable, ill-conditioned, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 56)

shubert_noisy

Description: Noisy variant of Shubert function; additive uniform [0,1) noise. Properties based on Jamil & Yang (2013, p. 55) for base; noise adapted for stochastic benchmark. Gradient is deterministic (noise constant per call). Multiple global minima (18 in 2D).
Formula: f(\mathbf{x}) = \prod{i=1}^D \sum{j=1}^5 j \cos((j + 1) x_i + j) + \varepsilon, \quad \varepsilon \sim U[0,1).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -186.7309088310239 at [-7.08350641, -7.70831374]
Properties: continuous, has_noise, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 55); noise adapted

shubert_rotated

Description: Rotated Shubert function; non-separable with ≈760 local minima in 2D; fixed Q=identity for reproducibility (in CEC random orthogonal Q, Kond=1); properties based on Jamil & Yang (2013, p. 55, f133-Rotated); CEC 2021 F7.
Formula: f(\mathbf{x}) = \prod{i=1}^n \sum{j=1}^5 j \cos((j+1)(Q\mathbf{x})_i + j), \ Q \ orthogonal.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -186.73090883102375 at [4.85805688, 5.48286421]
Properties: continuous, differentiable, highly multimodal, multimodal, non-convex, non-separable, scalable
Reference: Jamil & Yang (2013, p. 55)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

shubert_shifted

Description: Shifted Shubert function; non-separable with 760 local minima in 2D; fixed o=0 for reproducibility (in CEC random o~U[-10,10]); properties based on Jamil & Yang (2013, p. 55, f133-Shifted); CEC 2013 F6.
Formula: f(\mathbf{x}) = \prod{i=1}^n \sum{j=1}^5 j \cos((j+1)(xi - oi) + j), \ o_i \sim U[-10,10].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -186.73090883102375 at [4.85805688, 5.48286421]
Properties: continuous, differentiable, highly multimodal, multimodal, non-convex, non-separable, scalable
Reference: Jamil & Yang (2013, p. 55)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

shubertshiftedrotated

Description: Shifted Rotated Shubert (CEC 2014 variant, product form with Q(x - o), fixed 2D); non-separable, highly multimodal (~760 local minima); tests invariance and coupling; based on classic Shubert with orthogonal rotation Q and shift o ~ U[-80,80].
Formula: f(\mathbf{x}) = \prod{i=1}^2 \sum{j=1}^5 j \cos((j+1)[\mathbf{Q}(\mathbf{x}-\mathbf{o})]_i + j).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: -186.730908831 at [55.82260356, -69.31153727]
Properties: continuous, differentiable, multimodal, non-separable
Reference: CEC 2014; Jamil & Yang (2013, extended transformations)

sineenvelope

Description: Sine Envelope function: A multimodal, non-convex, non-separable, differentiable, continuous, bounded test function with global minimum at (0,0). Properties based on Molga & Smutnicki (2005); also in Gaviano et al. (2003).
Formula: f(\mathbf{x}) = -0.5 + \frac{\sin^2(\sqrt{x1^2 + x2^2}) - 0.5}{(1 + 0.001(x1^2 + x2^2))^2}
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: -1.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Molga & Smutnicki (2005)

sixhumpcamelback

Description: Six-Hump Camelback function: A multimodal, non-convex, non-separable, differentiable, continuous, bounded test function with six local minima and two equivalent global minima at (±0.089842, ∓0.712656) with value -1.031628. Properties based on Jamil & Yang (2013, p. 10) [f_{30}]; originally from Dixon & Szegő (1978). Bounds: [-3,3] x [-2,2].
Formula: f(\mathbf{x}) = \left(4 - 2.1 x1^2 + \frac{x1^4}{3}\right) x1^2 + x1 x2 + (-4 + 4 x2^2) x_2^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-3.0, -2.0] to [3.0, 2.0]; Global minimum: -1.031628453489877 at [0.08984201, -0.7126564]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 10)

sphere

Description: Sphere function: f(x) = Σ x_i^2; Properties based on Jamil & Yang (2013, p. 33); adapted correcting multimodal to unimodal and adding convex based on standard analyses (e.g., sfu.ca/~ssurjano). Bounds adapted from sfu.ca; original in source: [0,10]^D.
Formula: f(\mathbf{x}) = \sum{i=1}^D xi^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-5.12, -5.12] to [5.12, 5.12]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, convex, differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, p. 33)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

sphere_noisy

Description: Noisy Sphere benchmark function with additive uniform [0,1) noise; Properties based on BBOB f101 / Nevergrad 'NoisySphere'. Additive uniform [0,1) noise.
Formula: f(\mathbf{x}) = \sum{i=1}^n xi^2 + \epsilon, \epsilon \sim U[0,1].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, convex, has_noise, scalable, separable
Reference: BBOB f101 / Nevergrad
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

sphere_rotated

Description: Rotated Sphere benchmark function; Properties based on CEC 2005 Problem Definitions and BBOB-Suiten.
Formula: f(\mathbf{x}) = \| Q \mathbf{x} \|^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, convex, differentiable, non-separable, scalable, unimodal
Reference: CEC 2005 Problem Definitions
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

sphere_shifted

Description: Shifted Sphere benchmark function; Properties based on CEC 2005 Special Session.
Formula: f(\mathbf{x}) = \sum{i=1}^n (xi - o_i)^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: continuous, convex, differentiable, scalable, separable, unimodal
Reference: CEC 2005 Special Session
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

step

Description: Discontinuous and non-differentiable due to floor function; Properties based on Jamil & Yang (2013, function 138); originally from benchmark collections.
Formula: f(\mathbf{x}) = \sum{i=1}^D \lfloor |xi| \rfloor.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, non-convex, partially differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, function 138)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

step2

Description: Discontinuous and non-differentiable due to floor function; Properties based on Jamil & Yang (2013, function 139).
Formula: f(\mathbf{x}) = \sum{i=1}^D \left( \lfloor xi + 0.5 \rfloor \right)^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [-0.5, -0.5]
Properties: bounded, non-convex, partially differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, function 139)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

step3

Description: Discontinuous and non-differentiable due to floor function; Properties based on Jamil & Yang (2013, function 140).
Formula: f(\mathbf{x}) = \sum{i=1}^D \lfloor xi^2 \rfloor.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, non-convex, partially differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, function 140)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

step_ellipsoidal

Description: Step Ellipsoidal (BBOB f7/f18) benchmark function with plateaus and discontinuities; Properties based on BBOB 2009 Noiseless Functions f7 (Step Ellipsoidal); gradient zero almost everywhere except near boundaries.
Formula: f(\mathbf{z}) = 0.1 \max\left( \frac{|\tilde{z}1|}{10^4}, \sum{i=1}^D 10^{2(i-1)/(D-1)} \tilde{z}i^2 \right) + f\mathrm{pen}(x) + f_\mathrm{opt}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, ill-conditioned, non-separable, partially differentiable, scalable, unimodal
Reference: BBOB 2009 Noiseless Functions f7
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

stepint

Description: Discontinuous and non-differentiable due to floor function; Properties based on Jamil & Yang (2013, function 141); Adapted for consistency: paper states min=0 at x=0 but formula gives f(0)=25; actual global min=25-6D at xi≈-5.12 (floor(xi)=-6 for all i).
Formula: f(\mathbf{x}) = 25 + \sum{i=1}^D \lfloor xi \rfloor.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-5.12, -5.12] to [5.12, 5.12]; Global minimum: 13 at [-5.12, -5.12]
Properties: bounded, non-convex, partially differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, function 141)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

stretchedvsine_wave

Description: Properties based on Jamil & Yang (2013, function 142); originally from Schaffer et al. (1989).
Formula: f(\mathbf{x}) = \sum{i=1}^{D-1} (xi^2 + x{i+1}^2)^{0.25} \left[ \sin^2 \left{ 50 (xi^2 + x_{i+1}^2)^{0.1} \right} + 0.1 \right].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, non-separable, scalable, unimodal
Reference: Jamil & Yang (2013, function 142)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

styblinski_tang

Description: Properties based on Jamil & Yang (2013, function 144); formula indicates separable and scalable, adapted accordingly despite source listing non-separable/non-scalable; Minimum computed precisely via solving derivative=0 (source approx -39.16618); originally from [80].
Formula: f(\mathbf{x}) = \frac{1}{2} \sum{i=1}^D (xi^4 - 16xi^2 + 5xi).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: -78.33233140754282 at [-2.90353403, -2.90353403]
Properties: bounded, continuous, differentiable, multimodal, non-convex, scalable, separable
Reference: Jamil & Yang (2013, function 144)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

sumofpowers

Description: Sum of Different Powers function: A unimodal, convex, differentiable, separable, scalable function with global minimum at origin. Properties based on Jamil & Yang (2013, p. 32) [f_{145}]; originally from Bingham (1995).
Formula: f(\mathbf{x}) = \sum{i=1}^n |xi|^{i+1}
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-1.0, -1.0] to [1.0, 1.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, convex, differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, p. 32)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

testtubeholder

Description: Two global minima due to sign choices; Properties based on Jamil & Yang (2013, p. 34); Contains absolute value terms; originally from literature circa 2006.
Formula: f(\mathbf{x}) = -4 \sin(x1) \cos(x2) \exp\left( \left| \cos\left( \frac{x1^2 + x2^2}{200} \right) \right| \right).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -10.872300105622745 at [1.57060261, 0.0]
Properties: bounded, continuous, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 34)

threehumpcamel

Description: Three-Hump Camel function: Multimodal, non-convex, non-separable, differentiable, bounded test function with three local minima and a global minimum at (0.0, 0.0).
Formula: 2x1^2 - 1.05x1^4 + \frac{x1^6}{6} + x1 x2 + x2^2
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

trecanni

Description: Two global minima at [-2,0] and [0,0]; Properties based on Jamil & Yang (2013, p. 35); originally from Dixon & Szegő (1978).
Formula: f(\mathbf{x}) = x1^4 + 4x1^3 + 4x1^2 + x2^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: 0.0 at [-2.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, separable
Reference: Jamil & Yang (2013, p. 35)

trefethen

Description: Properties based on Jamil & Yang (2013, p. 35); originally from MVF Library (Adorio & Diliman, 2005).
Formula: f(\mathbf{x}) = e^{\sin(50 x1)} + \sin(60 e^{x2}) + \sin(70 \sin(x1)) + \sin(\sin(80 x2)) - \sin(10(x1 + x2)) + \frac{1}{4}(x1^2 + x2^2).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -3.306868647475232 at [-0.02440308, 0.21061243]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 35)

trid

Description: Trid function. Properties based on Jamil & Yang (2013, p. 35); originally from Dixon & Szegö (1978).
Formula: f(\mathbf{x}) = \sum{i=1}^n (xi - 1)^2 - \sum{i=2}^n xi x_{i-1}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-4.0, -4.0] to [4.0, 4.0]; Global minimum: -2.0 at [2.0, 2.0]
Properties: bounded, continuous, convex, differentiable, non-separable, scalable, unimodal
Reference: Jamil & Yang (2013, p. 35)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

trid10

Description: Properties based on Jamil & Yang (2013, p. 35); adapted for confirmed unimodal nature (no local minima per multiple sources); originally from Neumaier/Hedar.
Formula: f(\mathbf{x}) = \sum{i=1}^{10} (xi - 1)^2 - \sum{i=2}^{10} xi x_{i-1}.
Dimension: fixed (n = 10)
Bounds / Minimum: Bounds: [-100.0, ..., -100.0] to [100.0, ..., 100.0]; Global minimum: -210.0 at [10.0, ..., 10.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 35)

trid6

Description: Properties based on Jamil & Yang (2013, p. 35); originally from Hedar (2005).
Formula: f(\mathbf{x}) = \sum{i=1}^{6} (xi - 1)^2 - \sum{i=2}^{6} xi x_{i-1}.
Dimension: fixed (n = 6)
Bounds / Minimum: Bounds: [-36.0, ..., -36.0] to [36.0, ..., 36.0]; Global minimum: -50.0 at [6.0, ..., 6.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable
Reference: Jamil & Yang (2013, p. 35)

trigonometric1

Description: The Trigonometric 1 function is a multimodal, non-separable trigonometric benchmark. Properties based on Jamil & Yang (2013, p. 36).
Formula: f(\mathbf{x}) = \sum{i=1}^{n} \left[ n - \sum{j=1}^{n} \cos xj + i (1 - \cos( xi ) - \sin( x_i )) \right]^2.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [0.0, 0.0] to [3.14159265, 3.14159265]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 36)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

trigonometric2

Description: The Trigonometric 2 function is a multimodal, non-separable benchmark with asymmetric oscillatory terms centered around 0.9. Properties based on Jamil & Yang (2013, p. 36).
Formula: f(\mathbf{x}) = 1 + \sum{i=1}^n \left[ 8 \sin^2 \left( 7 (xi - 0.9)^2 \right) + (xi - 0.9)^2 \right] + 6 \sin^2 \left( 14 (x1 - 0.9)^2 \right).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 1.0 at [0.9, 0.9]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 36)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

tripod

Description: The Tripod function is a discontinuous, non-differentiable, non-separable multimodal benchmark with a tripod-shaped surface due to absolute values and step functions. Properties based on Jamil & Yang (2013, p. 37); contains step functions leading to discontinuities and absolute values causing non-differentiability.
Formula: f(\mathbf{x}) = p(x2)(1 + p(x1)) + |x1 + 50 p(x2)(1 - 2 p(x1))| + |x2 + 50(1 - 2 p(x_2))|, \ \text{where } p(x) = \begin{cases} 1 & x \geq 0 \ 0 & x < 0 \end{cases}.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-100.0, -100.0] to [100.0, 100.0]; Global minimum: 0.0 at [0.0, -50.0]
Properties: bounded, multimodal, non-separable, partially differentiable
Reference: Jamil & Yang (2013, p. 37)

ursem1

Description: Implementation of the Ursem 1 test function. Properties based on Jamil & Yang (2013, p. 36); originally from Rönkkönen (2009). Single global minimum.
Formula: f(\mathbf{x}) = -\sin(2x1 - 0.5\pi) - 3\cos(x2) - 0.5x_1.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-2.5, -2.0] to [3.0, 2.0]; Global minimum: -4.816814063734245 at [1.697137, 0.0]
Properties: bounded, continuous, differentiable, separable, unimodal
Reference: Jamil & Yang (2013, p. 36)

ursem3

Description: Implementation of the Ursem 3 test function. Properties based on Jamil & Yang (2013, p. 36); formula and minimum adapted from Al-Roomi (2015) to match reported global minimum of -2.5 (Jamil & Yang transcription appears erroneous); originally from Rönkkönen (2009). Contains absolute value terms.
Formula: f(\mathbf{x}) = -\frac{3 - |x1|}{2} \cdot \frac{2 - |x2|}{2} \cdot \sin(2.2\pi x1 + 0.5\pi) - \frac{2 - |x1|}{2} \cdot \frac{2 - |x2|}{2} \cdot \sin(0.5\pi x2^2 + 0.5\pi).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-2.0, -1.5] to [2.0, 1.5]; Global minimum: -2.5 at [0.0, 0.0]
Properties: bounded, continuous, multimodal, non-separable, partially differentiable
Reference: Jamil & Yang (2013, p. 36)

ursem4

Description: The Ursem No. 4 function; Properties based on Jamil & Yang (2013, p. 36); Contains square root terms making it non-differentiable at the origin.
Formula: f(\mathbf{x}) = -\frac{3}{4} \sin\left(0.5 \pi x1 + 0.5 \pi\right) \left(2 - \sqrt{x1^2 + x_2^2}\right).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-2.0, -2.0] to [2.0, 2.0]; Global minimum: -1.5 at [0.0, 0.0]
Properties: bounded, continuous, multimodal, non-convex, non-separable, partially differentiable
Reference: Jamil & Yang (2013, p. 36)

ursem_waves

Description: The Ursem Waves function; Properties based on Jamil & Yang (2013, p. 36); has single global minimum and nine local minima.
Formula: f(\mathbf{x}) = -0.9 x1^2 + (x2^2 - 4.5 x2^2) x1 x2 + 4.7 \cos(3 x1 - x2^2 (2 + x1)) \sin(2.5 \pi x_1).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-0.9, -1.2] to [1.2, 1.2]; Global minimum: -8.5536 at [1.2, 1.2]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Jamil & Yang (2013, p. 36)

ventersobiezcczanskisobieski

Description: The Venter Sobiezcczanski-Sobieski function; Properties based on Jamil & Yang (2013, p. 37); originally from Begambre and Laier (2009). Multimodal with multiple local minima due to cosine terms.
Formula: f(\mathbf{x}) = x1^2 - 100 \cos^2(x1) - 100 \cos\left(\frac{x1^2}{30}\right) + x2^2 - 100 \cos^2(x2) - 100 \cos\left(\frac{x2^2}{30}\right).
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-50.0, -50.0] to [50.0, 50.0]; Global minimum: -400.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, separable
Reference: Jamil & Yang (2013, p. 37)

watson

Description: Standard Watson function (n=6); polynomial approximation residual model. Properties based on Moré et al. (1981). Jamil & Yang (2013, p. 37) formula variant inconsistent (missing (j-1) factors leads to wrong minimum ~154); adapted to verified standard form.
Formula: f(\mathbf{x}) = \sum{i=1}^{29} \left[ \sum{j=2}^{6} (j-1) xj ti^{j-2} - \left( \sum{j=1}^{6} xj ti^{j-1} \right)^2 - 1 \right]^2 + x1^2, \quad t_i = i / 29.
Dimension: fixed (n = 6)
Bounds / Minimum: Bounds: [-5.0, ..., -5.0] to [5.0, ..., 5.0]; Global minimum: 0.00193306874741329 at [-0.01367625, ..., 1.08739726]
Properties: bounded, continuous, controversial, differentiable, non-separable, unimodal
Reference: Moré et al. (1981). Testing Unconstrained Optimization Software. ACM TOMS, 7(1), 17–41.

wavy

Description: Wavy Function; Properties based on Jamil & Yang (2013, p. 38); originally from [23]. Multimodal with ~10^n local minima for k=10.
Formula: f(\mathbf{x}) = 1 - \frac{1}{n} \sum{i=1}^{n} \cos(10 xi) e^{-x_i^2 / 2}.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-3.14159265, -3.14159265] to [3.14159265, 3.14159265]; Global minimum: 0.0 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal, scalable, separable
Reference: Jamil & Yang (2013, p. 38)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

wayburnseader1

Description: The Wayburn Seader 1 function; Properties based on Jamil & Yang (2013, p. 37); originally from Wayburn and Seader (1987). Multiple global minima at (1,2) and approximately (1.597, 0.806). Marked as scalable in source, but formula only uses first 2 variables; adapted to fixed n=2.
Formula: f(\mathbf{x}) = (x1^6 + x2^4 - 17)^2 + (2x1 + x2 - 4)^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: 0.0 at [1.0, 2.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 37)

wayburnseader2

Description: Wayburn Seader 2 function. Properties based on Jamil & Yang (2013, p. 63); multiple global minima at (0.2,1) and (0.425,1); originally from Wayburn & Seader (1987). Formula only uses first 2 variables; marked as scalable in source but adapted to fixed n=2 due to limited metadata for higher dimensions. Listed as unimodal in source despite multiple minima; added controversial.
Formula: f(\mathbf{x}) = \left[ 1.613 - 4(x1 - 0.3125)^2 - 4(x2 - 1.625)^2 \right]^2 + (x_2 - 1)^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 0.0 at [0.20013897, 1.0]
Properties: bounded, continuous, controversial, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 63)

wayburnseader3

Description: Wayburn Seader 3 function. Properties based on Jamil & Yang (2013, p. 63); originally from Wayburn & Seader (1987). Marked as scalable in source, but formula only uses first 2 variables; adapted to fixed n=2. Minimum corrected via numerical optimization from source approximate 21.35 at (5.611,6.187) to exact 19.10588 at (5.147,6.840).
Formula: f(\mathbf{x}) = \frac{2}{3} x1^3 - 8 x1^2 + 33 x1 - x1 x2 + 5 + \left[ (x1 - 4)^2 + (x_2 - 5)^2 - 4 \right]^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-500.0, -500.0] to [500.0, 500.0]; Global minimum: 19.105879794568022 at [5.14689675, 6.83958974]
Properties: bounded, continuous, controversial, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 63)

weierstrass

Description: Weierstrass function (f_166). Formula from Jamil & Yang (2013, p. 38).

Parameters a=0.5, b=3, k_max=20 are standard in literature (e.g. Al-Roomi, 2015; Surjanovic & Bingham) but not specified in Jamil & Yang. This is an adapted implementation for practical use. The function is continuous and differentiable (finite sum). Theoretical global minimum: 0 at x=zeros(n); in practice f(0) ≈ 0.0 (exact with finite precision).

Formula: f(\mathbf{x}) = \sum{i=1}^{n} \left[ \sum{k=0}^{20} a^k \cos(2 \pi b^k (xi + 0.5)) \right] - n \sum{k=0}^{20} a^k \cos(\pi b^k), \quad a=0.5, b=3.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-0.5, -0.5] to [0.5, 0.5]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, scalable, separable
Reference: Jamil & Yang (2013, p. 38); parameters adapted from Al-Roomi (2015)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

whitley

Description: The Whitley function is a multimodal, non-separable test function for global optimization. Properties based on Jamil & Yang (2013, #167); originally from Whitley et al. (1996).
Formula: f(\mathbf{x}) = \sum{i=1}^D \sum{j=1}^D \left[ \frac{\left(100(xi^2 - xj)^2 + (1 - xj)^2\right)^2}{4000} - \cos\left(100(xi^2 - xj)^2 + (1 - xj)^2\right) + 1 \right]
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, #167)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

wolfe

Description: The Wolfe function; Properties based on Al-Roomi (2015) and adapted from Jamil & Yang (2013, p. 39) (partially separable due to x1-x2 coupling in u; unimodal; fixed n=3, not scalable). Contains power term ^{0.75} differentiable at u=0 (grad limit=0).
Formula: f(\mathbf{x}) = \frac{4}{3} (x1^2 + x2^2 - x1 x2)^{0.75} + x_3.
Dimension: fixed (n = 3)
Bounds / Minimum: Bounds: [0.0, 0.0, 0.0] to [2.0, 2.0, 2.0]; Global minimum: 0.0 at [0.0, 0.0, 0.0]
Properties: bounded, continuous, differentiable, partially separable, unimodal
Reference: Al-Roomi (2015); adapted from Jamil & Yang (2013, p. 39)

wood

Description: Wood function: Multimodal, non-convex, non-separable, differentiable test function with a single global minimum. Also known as Colville function.
Formula: f(x) = 100(x1^2 - x2)^2 + (x1 - 1)^2 + (x3 - 1)^2 + 90(x3^2 - x4)^2 + 10.1((x2 - 1)^2 + (x4 - 1)^2) + 19.8(x2 - 1)(x4 - 1)
Dimension: fixed (n = 4)
Bounds / Minimum: Bounds: [-10.0, -10.0, -10.0, -10.0] to [10.0, 10.0, 10.0, 10.0]; Global minimum: 0.0 at [1.0, 1.0, 1.0, 1.0]
Properties: bounded, continuous, differentiable, multimodal, non-convex, non-separable
Reference: Unknown Source

xinsheyang1

Description: Xin-She Yang Function 1; Properties based on Jamil & Yang (2013, Section 3); stochastic with εi ~ U[0,1] per evaluation (hasnoise, f noisy but min=0 deterministic at zeros); separable (independent terms); partially differentiable due to abs at 0.
Formula: f(\mathbf{x}) = \sum{i=1}^D \epsiloni |xi|^i, \quad \epsiloni \sim U(0,1).
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [5.0, 5.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, has_noise, partially differentiable, scalable, separable, unimodal
Reference: Jamil & Yang (2013, Section 3)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

xinsheyang2

Description: Xin-She Yang Function 2; Properties based on Jamil & Yang (2013, p. 30); multimodal non-separable with abs and sin^2; partially differentiable at 0.
Formula: f(\mathbf{x}) = \left( \sum{i=1}^{D} |xi| \right) \exp \left[ - \sum{i=1}^{D} \sin^2(xi) \right].
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-6.28318531, -6.28318531] to [6.28318531, 6.28318531]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, multimodal, non-separable, partially differentiable, scalable
Reference: Jamil & Yang (2013, p. 30)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

xinsheyang3

Description: Xin-She Yang No. 3 function; Properties based on Jamil & Yang (2013, p. 39); originally proposed by Xin-She Yang.
Formula: f(\mathbf{x}) = \left[ e^{-\sum{i=1}^{D} \left( \frac{xi}{\beta} \right)^2 m} - 2 e^{-\sum{i=1}^{D} xi^2} \cdot \prod{i=1}^{D} \cos^2 (xi) \right]
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-20.0, -20.0] to [20.0, 20.0]; Global minimum: -1.0 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 39)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

xinsheyang4

Description: Xin-She Yang No. 4 function; Properties based on Jamil & Yang (2013, p. 39); originally proposed by Xin-She Yang.
Formula: f(\mathbf{x}) = \left[ \sum{i=1}^{D} \sin^2(xi) - e^{-\sum{i=1}^{D} xi^2} \right] \cdot e^{-\sum{i=1}^{D} \sin^2 \sqrt{|xi|}}
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -1.0 at [0.0, 0.0]
Properties: continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 39)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

zakharov

Description: Zakharov function; Properties based on Jamil & Yang (2013, p. 40); originally from Rahnamyan et al. (2007).
Formula: f(\mathbf{x}) = \sum{i=1}^n xi^2 + \left( \frac{1}{2} \sum{i=1}^n i xi \right)^2 + \left( \frac{1}{2} \sum{i=1}^n i xi \right)^4.
Dimension: scalable (default n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [10.0, 10.0]; Global minimum: 0.0 at [0.0, 0.0]
Properties: bounded, continuous, differentiable, multimodal, non-separable, scalable
Reference: Jamil & Yang (2013, p. 40)
Note: Scalable function – shown with default_n = 2. Use fixed(tf; n=...) for custom dimensions.

zettl

Description: Zettl function; Properties based on Jamil & Yang (2013, p. 40); non-scalable (fixed n=2); originally from [78].
Formula: f(\mathbf{x}) = (x1^2 + x2^2 - 2x1)^2 + 0.25 x1.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-5.0, -5.0] to [10.0, 10.0]; Global minimum: -0.0037912371501199 at [-0.0299, 0.0]
Properties: bounded, continuous, differentiable, non-separable, unimodal
Reference: Jamil & Yang (2013, p. 40)

zirilli_aluffipentini

Description: Zirilli or Aluffi-Pentini’s function; Properties based on Jamil & Yang (2013, p. 40); non-scalable (fixed n=2); originally from [4].
Formula: f(\mathbf{x}) = 0.25 x1^4 - 0.5 x1^2 + 0.1 x1 + 0.5 x2^2.
Dimension: fixed (n = 2)
Bounds / Minimum: Bounds: [-10.0, -10.0] to [10.0, 10.0]; Global minimum: -0.352386073800036 at [-1.04668053, 1.0e-8]
Properties: bounded, continuous, differentiable, separable, unimodal
Reference: Jamil & Yang (2013, p. 40)

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