2D Functions#

Two-dimensional test functions can be visualized as 3D surfaces and contour plots. They are ideal for understanding optimizer behavior and creating publication-quality figures.

Available 2D Functions#

Function	Type	Characteristics
`AckleyFunction`	Multimodal	Many local minima, global at origin
`RastriginFunction`	Multimodal	Highly multimodal, cosine waves
`RosenbrockFunction`	Unimodal	Banana-shaped valley
`HimmelblausFunction`	Multimodal	Four identical local minima
`BealeFunction`	Unimodal	Sharp valley
`BoothFunction`	Unimodal	Simple, smooth
`GoldsteinPriceFunction`	Multimodal	Complex landscape
`EasomFunction`	Unimodal	Flat with narrow peak
`CrossInTrayFunction`	Multimodal	Four global minima
`EggholderFunction`	Multimodal	Difficult, many local minima
`DropWaveFunction`	Multimodal	Concentric waves
`SchafferFunctionN2`	Multimodal	Circular ridges
`LeviFunctionN13`	Multimodal	Complex interactions
`MatyasFunction`	Unimodal	Simple, elliptical
`McCormickFunction`	Unimodal	Asymmetric valley
`ThreeHumpCamelFunction`	Multimodal	Three local minima

Classic Examples#

Ackley Function#

The Ackley function is a widely used multimodal test function with many local minima surrounding a global minimum at the origin.

from surfaces.test_functions.algebraic import AckleyFunction

func = AckleyFunction()
result = func({"x0": 0.0, "x1": 0.0})  # Global minimum: 0.0

Properties:

Global minimum: f(0, 0) = 0
Many local minima
Tests ability to escape local optima

Rastrigin Function#

A highly multimodal function with a cosine wave pattern.

from surfaces.test_functions.algebraic import RastriginFunction

func = RastriginFunction(n_dim=2)
result = func({"x0": 0.0, "x1": 0.0})  # Global minimum: 0.0

Properties:

Global minimum: f(0, 0) = 0
Regular grid of local minima
Number of local minima grows exponentially with dimension

Rosenbrock Function#

The famous “banana function” with a narrow, curved valley.

from surfaces.test_functions.algebraic import RosenbrockFunction

func = RosenbrockFunction(n_dim=2)
result = func({"x0": 1.0, "x1": 1.0})  # Global minimum: 0.0

Properties:

Global minimum: f(1, 1) = 0
Non-separable (variables interact)
Tests ability to follow narrow valleys

Himmelblau’s Function#

A function with four identical local minima.

from surfaces.test_functions.algebraic import HimmelblausFunction

func = HimmelblausFunction()
# Four global minima, all with f(x) = 0

Properties:

Four global minima at: - (3.0, 2.0) - (-2.805, 3.131) - (-3.779, -3.283) - (3.584, -1.848)
Tests multi-basin exploration

Visualization#

2D functions can be visualized with Surfaces’ built-in tools:

from surfaces.test_functions.algebraic import AckleyFunction
from surfaces.visualization import plot_surface, plot_contour

func = AckleyFunction()

# 3D surface plot
fig_surface = plot_surface(func, title="Ackley Function")
fig_surface.show()

# 2D contour plot
fig_contour = plot_contour(func, title="Ackley Contour")
fig_contour.show()

See the function_gallery for visualizations of all 2D functions.

Next Steps#

N-D Functions - Scalable N-dimensional functions
Visualization - Visualization guide
Algebraic Functions - Complete API reference