N-D Functions#

N-dimensional (scalable) test functions work in any number of dimensions. They are essential for testing how optimizers scale with problem size.

Available N-D Functions#

Function

Type

Characteristics

SphereFunction

Unimodal

Simplest test function, convex

RastriginFunction

Multimodal

Highly multimodal, separable

RosenbrockFunction

Unimodal

Narrow valley, non-separable

GriewankFunction

Multimodal

Product term creates local minima

StyblinskiTangFunction

Multimodal

Asymmetric landscape

Sphere Function#

The simplest and most fundamental test function. Convex, separable, unimodal. If your optimizer cannot solve Sphere, something is wrong.

from surfaces.test_functions.algebraic import SphereFunction

# Any dimension
func = SphereFunction(n_dim=10)

# Global minimum at origin
result = func({f"x{i}": 0.0 for i in range(10)})  # 0.0

Properties:

  • Formula: f(x) = sum(x_i^2)

  • Global minimum: f(0, …, 0) = 0

  • Convex, separable, unimodal

  • Difficulty increases with dimension

Rastrigin Function#

A highly multimodal function with a regular pattern of local minima.

from surfaces.test_functions.algebraic import RastriginFunction

func = RastriginFunction(n_dim=10)

# Global minimum at origin
result = func({f"x{i}": 0.0 for i in range(10)})  # 0.0

Properties:

  • Formula: f(x) = 10n + sum(x_i^2 - 10*cos(2*pi*x_i))

  • Global minimum: f(0, …, 0) = 0

  • ~10^n local minima in n dimensions

  • Separable but highly multimodal

Rosenbrock Function#

A non-separable function with a narrow, curved valley leading to the global minimum.

from surfaces.test_functions.algebraic import RosenbrockFunction

func = RosenbrockFunction(n_dim=10)

# Global minimum at (1, 1, ..., 1)
result = func({f"x{i}": 1.0 for i in range(10)})  # 0.0

Properties:

  • Formula: f(x) = sum(100*(x_{i+1} - x_i^2)^2 + (1 - x_i)^2)

  • Global minimum: f(1, …, 1) = 0

  • Non-separable (adjacent variables interact)

  • Tests ability to follow narrow valleys

Griewank Function#

Combines a sum term with a product term that creates local minima.

from surfaces.test_functions.algebraic import GriewankFunction

func = GriewankFunction(n_dim=10)
result = func({f"x{i}": 0.0 for i in range(10)})  # 0.0

Properties:

  • Global minimum: f(0, …, 0) = 0

  • Number of local minima increases with dimension

  • Product term becomes less significant at high dimensions

Styblinski-Tang Function#

An asymmetric multimodal function.

from surfaces.test_functions.algebraic import StyblinskiTangFunction

func = StyblinskiTangFunction(n_dim=10)

# Global minimum at x_i = -2.903534
x_opt = {f"x{i}": -2.903534 for i in range(10)}
result = func(x_opt)

Properties:

  • Global minimum: x_i = -2.903534

  • Minimum value: f(x*) = -39.16599 * n

  • Asymmetric landscape

Scaling Behavior#

N-D functions are crucial for understanding how optimizers scale:

from surfaces.test_functions.algebraic import RastriginFunction
import time

for n_dim in [2, 5, 10, 20, 50, 100]:
    func = RastriginFunction(n_dim=n_dim)

    # Measure evaluation time
    start = time.time()
    for _ in range(1000):
        func(func.search_space_sample())
    elapsed = time.time() - start

    print(f"n_dim={n_dim:3d}: {elapsed:.3f}s for 1000 evaluations")

Choosing Dimensions#

Dimension

Use Case

2-3

Visualization, debugging

5-10

Standard benchmarking

20-50

Testing scaling behavior

100+

High-dimensional optimization research

Next Steps#