Ackley Function

Ackley Function

The Ackley Function is a widely used benchmark function for testing the performance of optimization algorithms. Proposed by David Ackley in 1987, this function has many local minima, making it suitable for evaluating the escape capability of global optimization algorithms.

Mathematical Definition

Parameters:

• $a = 20$ (commonly used value)
• $b = 0.2$
• $c = 2\pi$
• $d$: number of dimensions
• $e$: Euler’s number (≈ 2.71828)

Characteristics

Python Implementation

import numpy as np
np.random.seed(0)

def ackley(X, a=20, b=0.2, c=2*np.pi):
    """
    Compute the Ackley function.
    Parameters:
      X: A NumPy array of shape (n, d) where each row is a n-dimensional point.
      a, b, c: Parameters of the Ackley function.
    Returns:
      A NumPy array of shape (n,) of function values
    """
    X = np.atleast_2d(X)
    d = X.shape[1]
    sum_sq = np.sum(X ** 2, axis=1)
    term1 = -a * np.exp(-b * np.sqrt(sum_sq / d))
    term2 = -np.exp(np.sum(np.cos(c * X), axis=1) / d)
    return term1 + term2 + a + np.e

Function Structure Analysis

The Ackley Function consists of two main terms:

1. First term ($-a \cdot \exp(-b\sqrt{\cdot})$): Creates a deep “well” at the origin
2. Second term ($-\exp(\cos(\cdot))$): Generates many local minima through the cosine function

The combination of these two terms results in:

• A single global minimum at the center
• Numerous local minima distributed across the nearly flat outer region
• Gradient-based optimization algorithms easily getting trapped in local minima

Applications

• Performance evaluation of Evolutionary Algorithms
• Testing Particle Swarm Optimization (PSO)
• Benchmarking Bayesian Optimization
• Comparing Gradient-free optimization algorithms

Usage Example

# Single point evaluation
x = np.array([[0, 0]])
print(f"f(0,0) = {ackley(x)[0]:.6f}")  # Value close to 0

# Multiple points evaluation
points = np.random.uniform(-5, 5, size=(100, 2))
values = ackley(points)
print(f"Min value: {values.min():.4f}")
print(f"Max value: {values.max():.4f}")

References

• Ackley, D. H. (1987). A Connectionist Machine for Genetic Hillclimbing. Kluwer Academic Publishers.
• Virtual Library of Simulation Experiments