Alphabetical List of Benchmark Functions

Generated from package metadata on 2025-12-11. Functions are listed alphabetically with their details.

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
• Bounds/Minimum: Bounds: [-32.768, -32.768, -32.768, -32.768, -32.768, -32.768, -32.768, -32.768, -32.768, -32.768]; Min: 0.0 at [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
• Properties: multimodal, deceptive, non-separable, bounded, differentiable, scalable, continuous, non-convex
• Reference: Ackley (1987); Molga & Smutnicki (2005); Jamil & Yang (2013); Lehman & Stanley (2011, arXiv:1106.2128) – classic deceptive function

ackley2

• Description: Ackley2 function: Continuous, partially differentiable, non-separable, non-scalable, unimodal.
• Formula: \f(x) = -200 e^{-0.02 \sqrt{x1^2 + x2^2}}
• Bounds/Minimum: Bounds: [-32.0, -32.0]; Min: -200.0 at [0.0, 0.0]
• Properties: non-separable, bounded, unimodal, partially differentiable, continuous, non-convex
• 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].
• Bounds/Minimum: Bounds: [-35.0, -35.0]; Min: -5.297009385988958 at [-1.5812643986108843, -0.7906319137820829]
• Properties: multimodal, controversial, non-separable, bounded, differentiable, highly multimodal, continuous, non-convex
• 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}.
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: -2.021806783359787 at [2.0, 0.10578347]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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|.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, separable, bounded, partially differentiable, scalable, non-convex
• Reference: Jamil & Yang (2013, p. 5)

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).
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -7.885600724127536 at [7.917052698245946, 7.917052698245946]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Jamil & Yang (2013, p. 5)

axisparallelhyperellipsoid

• Description: Properties based on Jamil & Yang (2013, p. 4); Convex, quadratic function.
• Formula: f(\mathbf{x}) = \sum{i=1}^n i xi^2.
• Bounds/Minimum: Bounds: [-5.12, -5.12]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, convex, differentiable, scalable, continuous
• Reference: Jamil & Yang (2013, p. 4)

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)|.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 1.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, partially differentiable, continuous, non-convex
• 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
• Bounds/Minimum: Bounds: [-4.5, -4.5]; Min: 0.0 at [3.0, 0.5]
• Properties: non-separable, bounded, unimodal, differentiable, continuous, non-convex
• 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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [5.0, 5.0]
• Properties: multimodal, separable, bounded, continuous
• 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}.
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 0.0 at [1.0, 10.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0]; Min: 0.0 at [1.0, 10.0, 5.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0, 0.0]; Min: 0.0 at [1.0, 10.0, 1.0, 5.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0, 0.0, 0.0]; Min: 0.0 at [1.0, 10.0, 1.0, 5.0, 4.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-20.0, -20.0, -20.0, -20.0, -20.0, -20.0]; Min: 0.0 at [1.0, 10.0, 1.0, 5.0, 4.0, 3.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-6.283185307179586, -6.283185307179586]; Min: -106.76453674926468 at [4.701043130240078, 3.152938503721135]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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].
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Jamil & Yang (2013, p. 10)

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [1.0, 3.0]
• Properties: separable, convex, bounded, unimodal, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [0.9, 9.0, 0.9]; Min: 0.0 at [1.0, 10.0, 1.0]
• Properties: multimodal, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-0.25, 0.01, 0.01]; Min: 0.008214877306578994 at [0.08241056, 1.13303609, 2.34369518]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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}

• Bounds/Minimum: Bounds: [-5.0, 0.0]; Min: 0.39788735772973816 at [-3.141592653589793, 12.275]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: -39.19565391797774 at [-3.1721041516027824, 12.58567479697034]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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)
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [-10.0, -10.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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]
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: 0.0 at [0.0, 0.0]
• Properties: non-separable, bounded, unimodal, differentiable, scalable, continuous
• Reference: Jamil & Yang (2013, p. 5)

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|
• Bounds/Minimum: Bounds: [-15.0, -3.0]; Min: 0.0 at [-10.0, 1.0]
• Properties: multimodal, non-separable, bounded, continuous
• 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|.
• Bounds/Minimum: Bounds: [-15.0, -3.0]; Min: 0.0 at [-10.0, 0.0]
• Properties: multimodal, separable, bounded, partially differentiable, continuous, non-convex
• 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|.
• Bounds/Minimum: Bounds: [-15.0, -3.0]; Min: 0.0 at [-10.0, 1.0]
• Properties: multimodal, bounded, partially differentiable, continuous, non-convex
• 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}.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -24.156815547391208 at [9.646167670438874, 9.646167670438874]
• Properties: multimodal, non-separable, differentiable, continuous
• 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)
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: -2000.0 at [0.388888888888889, 0.722222222222222]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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]
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: -2000.0039999840003 at [0.5, 0.5]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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]
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: -2000.0039999840005 at [0.5, 0.5]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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)
• Bounds/Minimum: Bounds: [-30.0, -30.0]; Min: -42.944387018991 at [6.18986658696568, 0.5]
• Properties: multimodal, separable, bounded, differentiable, continuous
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: bounded, unimodal, differentiable, continuous, partially separable, scalable
• Reference: Jamil & Yang (2013, Entry 34)

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}
• Bounds/Minimum: Bounds: [0.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0, -4.0]; Min: 10.533719093221547 at [0.6577179521834655, 1.3410086494718647, 0.0621925866606903, -0.9215843284021408, -0.8587539108194528, 0.0398894904746407, -3.3508073710903923, 0.6714854553331792, -3.3960325842653383, 2.381549919707253, -1.3565015163235619, 1.3510478875312162, -3.3405083834260405, 1.8923144784852317, -2.7015951415440593, -0.9050732332838868, -1.677429264374116]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• Reference: Unknown

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)
• Bounds/Minimum: Bounds: [-10.0, -10.0, -10.0, -10.0]; Min: 0.0 at [1.0, 1.0, 1.0, 1.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous, non-convex
• 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].
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, separable, bounded, scalable
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: -0.2 at [0.0, 0.0]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Unknown

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}
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -2.062611870822739 at [1.349406575769872, 1.349406575769872]
• Properties: multimodal, non-separable, bounded, partially differentiable, continuous, non-convex
• Reference: Unknown

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).
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [1.0, 1.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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]
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 0.0 at [2.0, 2.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• Reference: Unknown

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)
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: -1.0 at [-0.9, -0.9]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable
• Reference: Unknown

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)
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: -1.0 at [0.07969939268869583, 0.07969939268869583]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable
• Reference: Unknown

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}
• Bounds/Minimum: Bounds: [-1.28, -1.28]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, convex, bounded, unimodal, partially differentiable, scalable, continuous, has_noise
• Reference: Unknown

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}
• Bounds/Minimum: Bounds: [-65.536, -65.536]; Min: 0.9980038377944507 at [-31.97833, -31.97833]
• Properties: multimodal, non-separable, bounded, differentiable, finiteatinf, continuous, non-convex
• Reference: Unknown

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}
• Bounds/Minimum: Bounds: [-65.536, -65.536]; Min: 0.9980038378086058 at [-31.97987349299719, -31.979873489712844]
• Properties: multimodal, non-separable, bounded, differentiable, finiteatinf, continuous, non-convex
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-20.0, -20.0]; Min: -24776.518342317686 at [0.0, 14.945112151891959]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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).
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0, 0.0]; Min: 0.0 at [60.137, 1.371, 3.112, 1.761]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• Reference: Unknown

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).
• Bounds/Minimum: Bounds: [1.0, 1.0, 1.0, 1.0, 1.0]; Min: 0.0 at [53.81, 1.27, 3.012, 2.13, 0.507]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [1.0, 0.7071067811865476]
• Properties: bounded, unimodal, differentiable, scalable, continuous, non-convex
• Reference: Jamil & Yang (2013, Entry 14)

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
• Bounds/Minimum: Bounds: [-100.0, -100.0, -100.0, -100.0, -100.0]; Min: -529.8714387324576 at [98.9642583122371, 100.0, 100.0, 99.2243236725547, -0.249987527588471]
• Properties: multimodal, controversial, non-separable, bounded, differentiable, continuous
• Reference: Unknown

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}.
• Bounds/Minimum: Bounds: [-5.12, -5.12]; Min: -1.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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).
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: -1.0 at [π, π]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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).
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, separable, bounded, differentiable, continuous
• 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)
• Bounds/Minimum: Bounds: [-512.0, -512.0]; Min: -959.6406627208506 at [512.0, 404.2318058008512]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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.
• Bounds/Minimum: Bounds: [-5.2, -5.2, -5.2, -5.2]; Min: 0.0 at [3.0, 1.0, 1.0, 1.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: -1.0 at [0.0, 0.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous, scalable
• Reference: Ramanujan et al. (2007)

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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [5.0, 4.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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]
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: 0.06447042053690566 at [0.46732002530945826, 0.46732002530945826]
• Properties: multimodal, separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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))
• Bounds/Minimum: Bounds: [-2.0, -2.0]; Min: 3.0 at [0.0, -1.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-600.0, -600.0, -600.0, -600.0, -600.0, -600.0, -600.0, -600.0, -600.0, -600.0]; Min: 0.0 at [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
• Properties: multimodal, deceptive, non-separable, bounded, differentiable, scalable, continuous, non-convex
• 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

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}.
• Bounds/Minimum: Bounds: [0.1, 0.0, 0.0]; Min: 0.0 at [50.0, 25.0, 1.5]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -176.5417931367457 at [-7.589893010800888, -7.708313735499348]
• Properties: multimodal, separable, bounded, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]; Min: -3.3223680114155147 at [0.20168951265373836, 0.15001069271431358, 0.4768739727643224, 0.2753324306183083, 0.31165161653706114, 0.657300534163256]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0]; Min: -3.862782147820755 at [0.1146143386186895, 0.5556488499736022, 0.8525469535210816]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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).
• Bounds/Minimum: Bounds: [-10.0, -10.0, -10.0]; Min: 0.0 at [1.0, 0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [3.0, 2.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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|.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -19.208502567886743 at [8.05502347573655, -9.664590019241274]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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|.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -11.683926748430029 at [8.051008722128277, -9.999999999999]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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|
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -19.2085025678845 at [8.055023, 9.66459]
• Properties: multimodal, separable, bounded, partially differentiable, continuous, non-convex
• Reference: Unknown

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}.
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -2.345811576101292 at [4.0, 2.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: 124.36218235561489 at [0.2578252136705121, 0.2578252136701835]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}}
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -0.6736675211468548 at [0.0, 1.3932490753257145]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-3.0, -3.0]; Min: 1.125 at [0.7905694150420949, -0.7905694150420949]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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]
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -5.162126159963982 at [2.002992119907532, 1.0060959403343601]
• Properties: multimodal, controversial, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

leon

• Description: The Leon function. Properties based on Jamil & Yang (2013).
• Formula: f(\mathbf{x}) = 100(x2 - x1^2)^2 + (1 - x_1)^2
• Bounds/Minimum: Bounds: [-1.2, -1.2]; Min: 0.0 at [1.0, 1.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [1.0, 1.0]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Jamil & Yang (2013, p. 164)

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)]
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [1.0, 1.0]
• Properties: multimodal, bounded, differentiable, scalable, continuous, non-convex
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [0.0, 0.0]
• Properties: convex, non-separable, bounded, unimodal, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-1.5, -3.0]; Min: -1.9132229549810367 at [-0.54719755, -1.54719755]
• Properties: multimodal, bounded, differentiable, continuous, 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
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -1.8013034100985528 at [2.2029055201726, 1.5707963267949]
• Properties: multimodal, non-separable, bounded, differentiable, scalable, continuous
• Reference: Unknown

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
• Bounds/Minimum: Bounds: [-1.0, -1.0, -1.0, -1.0]; Min: 0.0 at [0.0, 1.0, 1.0, 1.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 2.0 at [1.0, 1.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, scalable
• Reference: Mishra (2006a)

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.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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.
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 0.0 at [1.0, 1.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Mishra (2006f), via Jamil & Yang (2013): f84

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}
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 2.0 at [1.0, 1.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, scalable
• Reference: Mishra (2006a)

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)
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -0.184651333342989 at [-8.466613775046579, -9.998521309]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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)
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -0.19941146886776687 at [-9.94114880716358, -9.999999996365672]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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)
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -0.11982951993 at [-1.98682, -10.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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)
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -2.28394983847 at [2.88630721544, 1.82326033142]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [1.0, 2.0]
• Properties: multimodal, non-separable, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [2.0, -3.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-10.0, -10.0, -10.0]; Min: 0.0 at [1.0, 2.0, 3.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-10.0, -6.5]; Min: -106.76453674926466 at [-3.1302468034308637, -1.5821421769356672]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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)).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -24.062498884334282 at [-0.4913908362396234, 5.791794471580817]
• Properties: multimodal, separable, differentiable, continuous
• 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)).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -25.74177099545137 at [-1.42512843, -1.42512843]
• Properties: multimodal, separable, differentiable, continuous
• 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)).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -24.062498884334275 at [-0.4913908340773322, -0.4913908340773322]
• Properties: multimodal, separable, differentiable, continuous, scalable
• Reference: Adorio (2005, p. 13)

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).
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [1.5707963267948966, 0.0]
• Properties: multimodal, separable, bounded, differentiable, continuous, non-convex
• 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).
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Rahnamayan et al. (2007a), via Jamil & Yang (2013): f87

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}.
• Bounds/Minimum: Bounds: [2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001]; Min: -45.77846970744629 at [9.350265833069052, 9.350265833069052, 9.350265833069052, 9.350265833069052, 9.350265833069052, 9.350265833069052, 9.350265833069052, 9.350265833069052, 9.350265833069052, 9.350265833069052]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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).
• Bounds/Minimum: Bounds: [-11.0, -11.0]; Min: -0.9635348327265058 at [9.6461676710434, 9.6461676710434]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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)}.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.9 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Pintér (1996), via Jamil & Yang (2013): f89

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.
• Bounds/Minimum: Bounds: [-5.0, -5.0, -5.0, -5.0]; Min: 0.0 at [0.0, 0.0, 0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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].
• Bounds/Minimum: Bounds: [-4.0, -4.0, -4.0, -4.0]; Min: 0.0 at [0.0, 0.0, 0.0, 0.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous, scalable
• Reference: Powell (1962)

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].
• Bounds/Minimum: Bounds: [-4.0, -4.0, -4.0, -4.0]; Min: 0.0 at [0.0, 0.0, 0.0, 0.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous, scalable
• Reference: Fu et al. (2006)

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}.
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, bounded, unimodal, differentiable, continuous, scalable
• Reference: Rahnamyan et al. (2007a)

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.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [5.0, 5.0]
• Properties: multimodal, separable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.9 at [0.0, 0.0]
• Properties: multimodal, differentiable, continuous
• 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
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [1.0, 1.4142135623730951]
• Properties: multimodal, separable, differentiable, continuous, scalable
• Reference: Qing (2006)

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.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -3873.72418218627 at [0.193880172788953, 0.485133909126923]
• Properties: convex, non-separable, bounded, unimodal, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [-1.28, -1.28]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, bounded, unimodal, continuous, scalable, has_noise
• Reference: Jamil & Yang (2013)

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|.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [-1.0, -1.0]
• Properties: multimodal, separable, differentiable, continuous
• 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|}.

• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: -464.27392770239135 at [-500.0, -500.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, 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.

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]
• Bounds/Minimum: Bounds: [-5.12, -5.12, -5.12, -5.12, -5.12, -5.12, -5.12, -5.12, -5.12, -5.12]; Min: 0.0 at [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
• Properties: multimodal, separable, deceptive, bounded, differentiable, scalable, continuous, non-convex
• Reference: Molga & Smutnicki (2005), p. 24; Goldberg (1989) – classic deceptive benchmark; Jamil & Yang (2013), p. 88

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).
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -2.2 at [0.1, 0.1]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -2.0 at [0.1, 0.1]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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].
• Bounds/Minimum: Bounds: [-30.0, -30.0]; Min: 0.0 at [1.0, 1.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous, scalable, ill-conditioned
• Reference: Jamil & Yang (2013), Benchmark Function #105

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}}.
• Bounds/Minimum: Bounds: [-2.0, -2.0]; Min: 34.04024310664062 at [-0.9095537365025769, -0.9505717126589607]
• Properties: multimodal, deceptive, controversial, non-separable, bounded, differentiable, continuous, ill-conditioned, non-convex
• 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.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [0.0, 0.0]
• Properties: convex, non-separable, unimodal, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [0.0, 0.0]
• Properties: convex, non-separable, unimodal, differentiable, continuous
• 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|.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [0.0, 0.0]
• Properties: non-separable, bounded, partially differentiable, unimodal, continuous
• 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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 27)

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).
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 27)

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0015668553065719681 at [0.0, 1.253115587]
• Properties: non-separable, unimodal, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 32)

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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.29257863203598033 at [0.0, 1.253131828792882]
• Properties: multimodal, non-separable, bounded, partially differentiable, continuous, non-convex
• 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}.
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0]; Min: 0.19397252244395102 at [7.07083412, 10.0, 3.14159293]
• Properties: multimodal, controversial, 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
• Bounds/Minimum: Bounds: [-10.0, -10.0, -10.0, -10.0, -10.0, -10.0, -10.0, -10.0, -10.0, -10.0]; Min: 0.0 at [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
• Properties: separable, unimodal, differentiable, continuous, scalable
• Reference: Schumer, M. A. and Steiglitz, K. (1968). Adaptive Step Size Random Search. IEEE Transactions on Automatic Control, 13(3), 270–276.

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: unimodal, differentiable, continuous, partially separable, scalable
• Reference: Jamil & Yang (2013, p. 118)

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: convex, non-separable, bounded, unimodal, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 29)

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|.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, partially differentiable, unimodal, continuous, scalable
• Reference: Jamil & Yang (2013, p. 77)

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|.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, partially differentiable, unimodal, continuous, scalable
• Reference: Jamil & Yang (2013, p. 123)

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|.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: non-separable, partially differentiable, unimodal, continuous, scalable
• Reference: Jamil & Yang (2013, p. 124)

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}.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, unimodal, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 125)

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.
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 0.0 at [1.0, 1.0]
• Properties: multimodal, separable, differentiable, continuous
• 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|}.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: -418.9828872724338 at [420.96874357691473, 420.96874357691473]
• Properties: multimodal, separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 30)

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).
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: -3456.0 at [12.0, 12.0]
• Properties: non-separable, unimodal, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 0.0 at [1.0, 1.0]
• Properties: multimodal, separable, differentiable, continuous
• 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|).
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [1.0, 3.0]
• Properties: non-separable, unimodal, differentiable, continuous
• 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}
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0, 0.0]; Min: -10.536409816692043 at [4.000746531592147, 4.000592934138629, 3.9996633980404135, 3.9995098005868956]
• Properties: multimodal, non-separable, bounded, differentiable, finiteatinf, continuous, non-convex
• 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].
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0, 0.0]; Min: -10.15319967905822 at [4.00003715, 4.00013327, 4.00003715, 4.00013327]
• Properties: multimodal, controversial, non-separable, differentiable, continuous
• 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].
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0, 0.0]; Min: -10.402940566818653 at [4.00057291, 4.00068936, 3.99948971, 3.99960616]
• Properties: multimodal, controversial, non-separable, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -25.74177099545136 at [-1.42512843, -1.42512843]
• Properties: multimodal, separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 56)

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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -29.675900051421173 at [-7.397285, -7.397285]
• Properties: multimodal, separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 55)

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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -186.73090883102375 at [4.858056878468046, 5.482864206944743]
• Properties: multimodal, controversial, non-separable, differentiable, highly multimodal, continuous, non-convex
• 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].
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -186.73090883102375 at [4.858056878468046, 5.482864206944743]
• Properties: multimodal, non-separable, differentiable, highly multimodal, continuous, scalable, non-convex
• Reference: Jamil & Yang (2013, p. 56)

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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -25.741770995451372 at [-1.42512843, -1.42512843]
• Properties: multimodal, separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 56)

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}).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -79.36953021020285 at [-0.8130518668176654, -1.4178774389880957]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, ill-conditioned
• 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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -186.7309088310239 at [-7.083506405745021, -7.708313737307907]
• Properties: multimodal, non-separable, continuous, has_noise
• 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.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -186.73090883102375 at [4.858056878468046, 5.482864206944743]
• Properties: multimodal, non-separable, differentiable, highly multimodal, continuous, scalable, non-convex
• Reference: Jamil & Yang (2013, p. 55)

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].
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -186.73090883102375 at [4.858056878468046, 5.482864206944743]
• Properties: multimodal, non-separable, differentiable, highly multimodal, continuous, scalable, non-convex
• Reference: Jamil & Yang (2013, p. 55)

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).
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: -186.730908831 at [55.82260356132168, -69.31153726530631]
• Properties: multimodal, non-separable, differentiable, continuous
• 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}
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: -1.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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
• Bounds/Minimum: Bounds: [-3.0, -2.0]; Min: -1.031628453489877 at [0.08984201368301331, -0.7126564032704135]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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.
• Bounds/Minimum: Bounds: [-5.12, -5.12]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, convex, bounded, unimodal, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 33)

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].
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, convex, continuous, scalable, has_noise
• Reference: BBOB f101 / Nevergrad

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: convex, non-separable, unimodal, differentiable, continuous, scalable
• Reference: CEC 2005 Problem Definitions

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [1.0, 1.0]
• Properties: separable, convex, unimodal, differentiable, continuous, scalable
• Reference: CEC 2005 Special Session

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, bounded, partially differentiable, unimodal, scalable, non-convex
• Reference: Jamil & Yang (2013, function 138)

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [-0.5, -0.5]
• Properties: separable, bounded, partially differentiable, unimodal, scalable, non-convex
• Reference: Jamil & Yang (2013, function 139)

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.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, bounded, partially differentiable, unimodal, scalable, non-convex
• Reference: Jamil & Yang (2013, function 140)

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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, 0.0]
• Properties: non-separable, partially differentiable, unimodal, continuous, scalable, ill-conditioned
• Reference: BBOB 2009 Noiseless Functions f7

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.
• Bounds/Minimum: Bounds: [-5.12, -5.12]; Min: 13 at [-5.12, -5.12]
• Properties: separable, bounded, partially differentiable, unimodal, scalable, non-convex
• Reference: Jamil & Yang (2013, function 141)

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].
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [0.0, 0.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, function 142)

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).
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: -78.33233140754282 at [-2.903534027771177, -2.903534027771177]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable, non-convex
• Reference: Jamil & Yang (2013, function 144)

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}
• Bounds/Minimum: Bounds: [-1.0, -1.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, convex, bounded, unimodal, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 32)

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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -10.872300105622745 at [1.5706026141696665, 4.534236385313213e-12]
• Properties: multimodal, non-separable, bounded, continuous
• 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
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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.
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [-2.0, 0.0]
• Properties: multimodal, separable, bounded, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -3.306868647475232 at [-0.0244030799684242, 0.210612427873691]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-4, -4]; Min: -2.0 at [2, 2]
• Properties: convex, non-separable, bounded, unimodal, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 35)

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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0, -100.0, -100.0, -100.0, -100.0, -100.0, -100.0, -100.0, -100.0]; Min: -210.0 at [10.0, 18.0, 24.0, 28.0, 30.0, 30.0, 28.0, 24.0, 18.0, 10.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-36.0, -36.0, -36.0, -36.0, -36.0, -36.0]; Min: -50.0 at [6.0, 10.0, 12.0, 12.0, 10.0, 6.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [0.0, 0.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 36)

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).
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 1.0 at [0.9, 0.9]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 36)

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}.
• Bounds/Minimum: Bounds: [-100.0, -100.0]; Min: 0.0 at [0.0, -50.0]
• Properties: multimodal, non-separable, bounded, 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.
• Bounds/Minimum: Bounds: [-2.5, -2.0]; Min: -4.816814063734245 at [1.697137, 0.0]
• Properties: separable, bounded, unimodal, differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [-2.0, -1.5]; Min: -2.5 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, partially differentiable, continuous
• 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).
• Bounds/Minimum: Bounds: [-2.0, -2.0]; Min: -1.5 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, partially differentiable, continuous, non-convex
• 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).
• Bounds/Minimum: Bounds: [-0.9, -1.2]; Min: -8.5536 at [1.2, 1.2]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• 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).
• Bounds/Minimum: Bounds: [-50.0, -50.0]; Min: -400.0 at [0.0, 0.0]
• Properties: separable, bounded, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-5.0, -5.0, -5.0, -5.0, -5.0, -5.0]; Min: 0.00193306874741329 at [-0.01367625, 1.02929024, -0.33159222, 1.50042489, -1.75935227, 1.08739726]
• Properties: controversial, non-separable, bounded, unimodal, differentiable, continuous
• 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}.
• Bounds/Minimum: Bounds: [-3.141592653589793, -3.141592653589793]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 38)

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.
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [1.0, 2.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 0.0 at [0.200138974079519, 1.000000000083411]
• Properties: controversial, non-separable, bounded, unimodal, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-500.0, -500.0]; Min: 19.105879794568022 at [5.14689674946688, 6.83958974367702]
• Properties: controversial, non-separable, bounded, unimodal, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-0.5, -0.5]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, separable, bounded, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 38); parameters adapted from Al-Roomi (2015)

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]
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: 0.0 at [1.0, 1.0]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, #167)

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.
• Bounds/Minimum: Bounds: [0.0, 0.0, 0.0]; Min: 0.0 at [0.0, 0.0, 0.0]
• Properties: bounded, unimodal, differentiable, continuous, partially separable
• 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)
• Bounds/Minimum: Bounds: [-10.0, -10.0, -10.0, -10.0]; Min: 0.0 at [1.0, 1.0, 1.0, 1.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, non-convex
• Reference: Unknown

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).
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [0.0, 0.0]
• Properties: separable, bounded, unimodal, partially differentiable, scalable, has_noise
• Reference: Jamil & Yang (2013, Section 3)

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].
• Bounds/Minimum: Bounds: [-6.283185307179586, -6.283185307179586]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, partially differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 30)

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]
• Bounds/Minimum: Bounds: [-20.0, -20.0]; Min: -1.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 39)

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|}}
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -1.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 39)

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.
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: 0.0 at [0.0, 0.0]
• Properties: multimodal, non-separable, bounded, differentiable, continuous, scalable
• Reference: Jamil & Yang (2013, p. 40)

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.
• Bounds/Minimum: Bounds: [-5.0, -5.0]; Min: -0.0037912371501199 at [-0.0299, 0.0]
• Properties: non-separable, bounded, unimodal, differentiable, continuous
• 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.
• Bounds/Minimum: Bounds: [-10.0, -10.0]; Min: -0.352386073800036 at [-1.046680529537701, 5.558876e-9]
• Properties: separable, bounded, unimodal, differentiable, continuous
• Reference: Jamil & Yang (2013, p. 40)

Generation

Run julia --project=. examples/generate_functions_md.jl to update from TEST_FUNCTIONS.