Algebraic Functions#

Algebraic functions are classic mathematical test functions from the optimization literature. They have analytical formulas, known global optima, and well-studied properties.

Why Algebraic Functions?#

These functions are the foundation of optimization benchmarking:

  • Well-understood: Decades of research on their properties

  • Fast evaluation: Simple formulas, microsecond evaluation

  • Visualizable: 2D functions can be plotted as surfaces

  • Scalable: Many extend to arbitrary dimensions

Function Dimensions#

1D Functions

Univariate test functions for simple benchmarks and visualization of optimizer behavior.

  • Gramacy & Lee

  • Forrester

  • And more…

1D Functions
2D Functions

Two-dimensional functions that can be visualized as 3D surfaces and contour plots.

  • Ackley, Rastrigin

  • Himmelblau, Rosenbrock

  • And more…

2D Functions
N-D Functions

Scalable functions that work in any dimension. Test optimizer scaling behavior.

  • Sphere, Rastrigin

  • Rosenbrock, Griewank

  • And more…

N-D Functions

Common Properties#

Unimodal vs Multimodal#

  • Unimodal: Single global optimum (e.g., Sphere)

  • Multimodal: Multiple local optima (e.g., Rastrigin, Ackley)

Multimodal functions test an optimizer’s ability to escape local optima.

Separable vs Non-separable#

  • Separable: Variables can be optimized independently

  • Non-separable: Variables interact (e.g., Rosenbrock)

Non-separable functions test an optimizer’s ability to handle variable interactions.

Quick Start#

from surfaces.test_functions.algebraic import (
    # 2D only
    AckleyFunction,
    HimmelblausFunction,
    # N-D (scalable)
    SphereFunction,
    RastriginFunction,
    RosenbrockFunction,
)

# 2D function
ackley = AckleyFunction()
result = ackley({"x0": 0.0, "x1": 0.0})

# N-D function with custom dimensions
sphere = SphereFunction(n_dim=10)
result = sphere({f"x{i}": 0.0 for i in range(10)})