This repository provides a simple implementation of popular optimization benchmark functions. You can use them to test optimization algorithms and visualize the functions.
| Function Name | Formula | Range | Optimal Value |
|---|---|---|---|
| ackley | -20 * exp(-0.2 * sqrt(0.5 * (x^2 + y^2))) - exp(0.5 * (cos(2pix) + cos(2piy))) + 20 + exp(1) | [-32.768, 32.768] | 0 at (0, 0) |
| beale | (1.5 - x + xy)^2 + (2.25 - x + xy^2)^2 + (2.625 - x + x*y^3)^2 | [-4.5, 4.5] | 0 at (3, 0.5) |
| rastrigin | 20 + x^2 - 10 * cos(2pix) + y^2 - 10 * cos(2piy) | [-5.12, 5.12] | 0 at (0, 0) |
| sphere | x^2 + y^2 | [-5.12, 5.12] | 0 at (0, 0) |
| rosenbrock | 100 * (y - x^2)^2 + (1 - x)^2 | [-5, 10] | 0 at (1, 1) |
| eggholder | -(y + 47) * sin(sqrt(abs(y + x/2 + 47))) - x * sin(sqrt(abs(x - (y + 47)))) | [-512, 512] | -959.6407 at (512, 404.2319) |
| schwefel | 418.9829 * 2 - x * sin(sqrt(abs(x))) - y * sin(sqrt(abs(y))) | [-500, 500] | 0 at (420.9687, 420.9687) |
| goldstein_price | (1 + (x + y + 1)^2 * (19 - 14x + 3x^2 - 14y + 6xy + 3y^2)) * (30 + (2x - 3y)^2 * (18 - 32x + 12x^2 + 48y - 36xy + 27y^2)) | [-2, 2] | 3 at (0, -1) |
| levi | sin(3pix)^2 + (x - 1)^2 * (1 + sin(3piy)^2) + (y - 1)^2 * (1 + sin(2piy)^2) | [-10, 10] | 0 at (1, 1) |
| himmelblau | (x^2 + y - 11)^2 + (x + y^2 - 7)^2 | [-5, 5] | 0 at (3, 2), (-2.8051, 3.1313), (-3.7793, -3.2832), (3.5844, -1.8481) |
| booth | (x + 2y - 7)^2 + (2x + y - 5)^2 | [-10, 10] | 0 at (1, 3) |
| three_hump_camel | 2x^2 - 1.05x^4 + x^6/6 + x*y + y^2 | [-5, 5] | 0 at (0, 0) |
| bukin | 100 * sqrt(abs(y - 0.01*x^2)) + 0.01 * abs(x + 10) | [-15, -3], [-5, 3] | 0 at (-10, 1) |
| matyas | 0.26 * (x^2 + y^2) - 0.48 * x * y | [-10, 10] | 0 at (0, 0) |
| six_hump_camel | (4 - 2.1x^2 + x^4/3) * x^2 + xy + (-4 + 4*y^2) * y^2 | [-3, 3], [-2, 2] | -1.0316 at (0.0898, -0.7126) and (-0.0898, 0.7126) |
| styblinski_tang | (x^4 - 16x^2 + 5x + y^4 - 16y^2 + 5y) / 2 | [-5, 5] | -78.3323 at (-2.9035, -2.9035) |
| michalewicz | -sin(x) * (sin(x^2/pi))^20 - sin(y) * (sin(2*y^2/pi))^20 | [0, pi], [0, pi] | -1.8013 at (2.2029, 1.5707) |
| easom | -cos(x) * cos(y) * exp(-(x - pi)^2 - (y - pi)^2) | [-100, 100] | -1 at (pi, pi) |
| cross_in_tray | -0.0001 * (abs(sin(x) * sin(y) * exp(abs(100 - sqrt(x^2 + y^2)/pi))) + 1)^0.1 | [-10, 10] | -2.06261 at (1.3494, 1.3494), (-1.3494, 1.3494), (1.3494, -1.3494), (-1.3494, -1.3494) |
| griewank | 1 + x^2/4000 - cos(x) + y^2/4000 - cos(y) | [-600, 600] | 0 at (0, 0) |
| drop_wave | -(1 + cos(12 * sqrt(x^2 + y^2))) / (0.5 * (x^2 + y^2) + 2) | [-5.12, 5.12] | -1 at (0, 0) |
| dixon_price | (x - 1)^2 + 2 * (2 * y^2 - x)^2 | [-10, 10] | 0 at (1, 1) |
| zakharov | x^2 + y^2 + (0.5x + 0.5y)^2 + (0.5x + 0.5y)^4 | [-5, 10] | 0 at (0, 0) |
| salomon | 1 - cos(2pisqrt(x^2 + y^2)) + 0.1 * sqrt(x^2 + y^2) | [-100, 100] | 0 at (0, 0) |
| alpine | sin(x) * sqrt(abs(x)) + sin(y) * sqrt(abs(y)) | [0, 10] | 0 at (0, 0) |
| xin_she_yang | -exp(-sum((sin(x))^2) - sum((sin(y))^2)) | [-2pi, 2pi] | -1 at (0, 0) |
| brown | (x^2)^((y^2) + 1) + (y^2)^((x^2) + 1) | [-1, 4] | 0 at (0, 0) |
| langermann | -exp(sum(cos(2pi(x+y))) / 2) * cos(pix) * cos(piy) | [0, 10] | -1 at (1, 1) |
| bent_cigar | x^2 + 10^6 * y^2 | [-100, 100] | 0 at (0, 0) |
| katsuura | 1 - (10 / n) * sum(sin(pi * (1 + (i * (x_i + 1)) / (1 + sum((x_i)^2)))) | [0, 100] | 0 at (0, 0) |
| schaffer_n4 | 0.5 + (cos(sin(abs(x^2 - y^2)))^2 - 0.5) / (1 + 0.001 * (x^2 + y^2))^2 | [-100, 100] | 0 at (0, 1.2531) |
| happy_cat | ((x^2 + y^2 - 1)^2)^0.125 + 0.5 * (x + y) + 0.5 | [-2, 2] | 0 at (-1, -1) |
| periodic | 1 + sin(x)^2 * sin(y)^2 - 0.1 * exp(-x^2 - y^2) | [-10, 10] | 0.9 at (0, 0) |
| shubert | sum(i * cos((i + 1) * x + i)) * sum(i * cos((i + 1) * y + i)) | [-10, 10] | -186.7309 at (4.8581, 4.8581) |
| schwefel_221 | max(abs(x), abs(y)) | [-100, 100] | 0 at (0, 0) |
| sum_squares | x^2 + 2 * y^2 | [-10, 10] | 0 at (0, 0) |
| powell | (x + 10y)^2 + 5(x - y)^2 + (x - 2y)^4 + 10(x^2 + y^2)^2 | [-10, 10] | 0 at (0, 0) |
| sum_of_different_powers | abs(x)^3/3 + abs(y)^5/5 | [-1, 1] | 0 at (0, 0) |
| trid | (x - 1)^2 + (y - 1)^2 - (x * y) | [-2, 2], [-5, 5] | 2 at (1, 1) |
| colville | 100*(x^2 - y)^2 + (x - 1)^2 + (y - 1)^2 + 90*(y^2 - z)^2 + 10.1*((y - 1)^2 + (z - 1)^2) + 19.8*(y - 1)*(z - 1) | [-10, 10] | 0 at (1, 1, 1) |
To use the benchmark functions, import the BF class from benchmark.py and create an instance with the desired function name. You can then call the calc method with an input array, use plot to visualize the function, or retrieve the function itself with get_function.
import numpy as np
from benchmark import BF
# Create an instance of the benchmark function
bf = BF("ackley")
# Calculate the function value at a given point
y = bf.evaluate(np.array([1, 1]))
print(y)
# Plot the function
bf.plot()
# or specify the number of points for the plot
bf.plot(num_points=32)
# Get the function itself
f = bf.get_function()
y = f(np.array([1, 1]))
print(y)You can also use the provided benchmark.py script to calculate and visualize benchmark functions from the command line.
> python benchmark.py --function ackley --x0 0 --x1 1Or just visualize the function:
> python benchmark.py --function ackleyIf you find any mistakes or inaccuracies, please feel free to open an issue on the repository.