Optimization Test Functions

Collection of optimization test functions and some useful methods for working with them

pip install OptimizationTestFunctions

• Optimization Test Functions
  • Test function object
  • Available test functions
    • Sphere
    • Ackley
    • AckleyTest
    • Rosenbrock
    • Fletcher
    • Griewank
    • Penalty2
    • Quartic
    • Rastrigin
    • SchwefelDouble
    • SchwefelMax
    • SchwefelAbs
    • SchwefelSin
    • Stairs
    • Abs
    • Michalewicz
    • Scheffer
    • Eggholder
    • Weierstrass
  • Plotting tools
    • Structure
    • How to use
  • Transformation tools
    • Structure
    • How to use

Test function object

Each test function is the callable object of some class with next fields at least:

• bounds -- tuple with structure (xmin, xmax, ymin, ymax). It is recommended borders for 3D plotting and 2D optimization for this function
• x_best -- global minimum argument of function in bounds area as numpy array. If unknown, it's None
• f_best -- function value at x_best if x_best exists and None otherwise

A lot of function objects need determined dim argument in constructor.

U can call these "functions" like usual functions with structure numpy 1D-array -> float value.

Available test functions

Checklist:

• Sphere(dim, degree = 2)
• Ackley(dim)
• AckleyTest(dim)
• Rosenbrock(dim)
• Fletcher(dim, seed = None)
• Griewank(dim)
• Penalty2(dim, a=5, k=100, m=4)
• Quartic(dim)
• Rastrigin(dim)
• SchwefelDouble(dim)
• SchwefelMax(dim)
• SchwefelAbs(dim)
• SchwefelSin(dim)
• Stairs(dim)
• Abs(dim)
• Michalewicz(m = 10)
• Scheffer(dim)
• Eggholder(dim)
• Weierstrass(dim, a = 0.5, b = 3, kmax = 20)

U imports them using code:

from OptimizationTestFunctions import Sphere, Ackley, AckleyTest, Rosenbrock, Fletcher, Griewank, Penalty2, Quartic, Rastrigin, SchwefelDouble, SchwefelMax, SchwefelAbs, SchwefelSin, Stairs, Abs, Michalewicz, Scheffer, Eggholder, Weierstrass

And plot them using code

Sphere

Ackley

AckleyTest

Rosenbrock

Fletcher

Griewank

Penalty2

Quartic

Rastrigin

SchwefelDouble

SchwefelMax

SchwefelAbs

SchwefelSin

Stairs

Abs

Michalewicz

Scheffer

Eggholder

Weierstrass

Plotting tools

Structure

There are plot_3d function for 3D-plotting:

plot_3d(func, points_by_dim = 50, title = '', bounds = None, show_best_if_exists = True, save_as = None, cmap = 'twilight', plot_surface = True, plot_heatmap = True)

with arguments:

• func : class callable object; Object which can be called as function.
• points_by_dim : int, optional; points for each dimension of plotting (50x50, 100x100...). The default is 50.
• title : str, optional; title of plot with LaTeX notation. The default is ''.
• bounds : tuple, optional; space bounds with structure (xmin, xmax, ymin, ymax). The default is None.
• show_best_if_exists : boolean, optional; point best solution by arrow if x_best exists. The default is True.
• save_as : str/None, optional; file path to save image (None if not needed). The default is None.
• cmap : str, optional; color map of plot. The default is 'twilight'. See another cmaps examples here
• plot_surface : boolean, optional; plot 3D surface. The default is True.
• plot_heatmap : boolean, optional; plot 2D heatmap. The default is True.

How to use

from OptimizationTestFunctions import Fletcher, plot_3d

# dim should be 2 for plotting 3D
dim = 2

# Fletcher is good function depends on random seed!

seed = 1
f1 = Fletcher(dim, seed)

# full available functional of plotting

plot_3d(f1, 
        points_by_dim = 70, 
        title = fr"{type(f1).__name__}\ with\ seed = {seed}", # LaTeX formula notation
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Fletcher1.png",
        cmap = 'twilight',
        plot_surface = True,
        plot_heatmap = True)

# disable arrow

plot_3d(f1, 
        points_by_dim = 70, 
        title = fr"{type(f1).__name__}\ with\ seed = {seed}",
        bounds = None, 
        show_best_if_exists = False, 
        save_as = "Fletcher2.png",
        cmap = 'twilight',
        plot_surface = True,
        plot_heatmap = True)

# select another bounds

plot_3d(f1, 
        points_by_dim = 70, 
        title = fr"{type(f1).__name__}\ with\ seed = {seed}",
        bounds = (-2, 6, -8, 10), 
        show_best_if_exists = False, 
        save_as = "Fletcher3.png",
        cmap = 'twilight',
        plot_surface = True,
        plot_heatmap = True)

# Create another Fletcher function

seed = 33

f2 = Fletcher(dim, seed)

# use another cmap

plot_3d(f2, 
        points_by_dim = 70, 
        title = fr"{type(f1).__name__}\ with\ seed = {seed}",
        bounds = None, 
        show_best_if_exists = False, 
        save_as = "Fletcher4.png",
        cmap = 'inferno',
        plot_surface = True,
        plot_heatmap = True)

# plot only 3D

plot_3d(f2, 
        points_by_dim = 70, 
        title = fr"{type(f1).__name__}\ with\ seed = {seed}",
        bounds = None, 
        show_best_if_exists = False, 
        save_as = "Fletcher5.png",
        cmap = 'inferno',
        plot_surface = True,
        plot_heatmap = False)

# plot only heatmap

plot_3d(f2, 
        points_by_dim = 70, 
        title = fr"{type(f1).__name__}\ with\ seed = {seed}",
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Fletcher6.png",
        cmap = 'inferno',
        plot_surface = False,
        plot_heatmap = True)

Transformation tools

Structure

Transformation object is the callable object like "functions" of this package. It performs next useful transformations:

• parallel transfer (shift)
• rotation
• add noises

U can create Transformation object using code:

transform = Transformation(transformed_function, shift_step = None, rotation_matrix = None, noise_generator = None, seed = None)

where:

• transformed_function : function or class callable object; transformed function.
• shift_step : numpy 1D array/None, optional; array of shifts by each dimension or None. The default is None.
• rotation_matrix : 2D-array/int/None, optional; 2D ortogonal rotation matrix or dimension for creating random rotation matrix or None if no rotate. The default is None.
• noise_generator : function, optional; function gets current value and returns value with some noise. The default is None.
• seed : int, optional; random seed for rotation matrix if needed reproduce. The default is None.

U also can create noises by using Noises static class.

How to use

import numpy as np

from OptimizationTestFunctions import Weierstrass, plot_3d, Transformation, Noises

# dim should be 2 for plotting 3D
dim = 2

# Let's create Weierstrass function

f = Weierstrass(dim, a = 0.5, b = 5, kmax = 20)

# show it

plot_3d(f, 
        points_by_dim = 70, 
        title = f"{type(f).__name__}",
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Trans1.png",
        cmap = 'hot',
        plot_surface = True,
        plot_heatmap = True)

# transformation with shift

shifted_func = Transformation(f, shift_step=np.array([3, 4]))

# show it

plot_3d(shifted_func, 
        points_by_dim = 70, 
        title = "shifted",
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Trans2.png",
        cmap = 'hot',
        plot_surface = True,
        plot_heatmap = True)

# transformation with rotation

rotated_func = Transformation(f, rotation_matrix = dim, seed = 2) # random rotation matrix with dim 2

# show it

plot_3d(rotated_func, 
        points_by_dim = 70, 
        title = "rotated",
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Trans3.png",
        cmap = 'hot',
        plot_surface = True,
        plot_heatmap = True)

# transformation with noise

noised_func = Transformation(f, noise_generator = Noises.normal(center = 0, sd = 0.5)) 

# show it

plot_3d(noised_func, 
        points_by_dim = 70, 
        title = "noised",
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Trans4.png",
        cmap = 'hot',
        plot_surface = True,
        plot_heatmap = True)

# U can specify your noise behavior

def add_noise(current_val):
    if current_val > 5:
        return 0

    return current_val + np.random.random()/10

noised_func = Transformation(f, noise_generator = add_noise) 

plot_3d(noised_func, 
        points_by_dim = 70, 
        title = "noised",
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Trans5.png",
        cmap = 'hot',
        plot_surface = True,
        plot_heatmap = True)

# Also u can combine all these transformations 

new_func = Transformation(f,
                          shift_step= np.array([10, -10]),
                          rotation_matrix = 2, seed = 3,
                          noise_generator = Noises.uniform(-0.1, 0.5)
                          ) 

plot_3d(new_func, 
        points_by_dim = 70, 
        title = "mixed",
        bounds = None, 
        show_best_if_exists = True, 
        save_as = "Trans6.png",
        cmap = 'hot',
        plot_surface = True,
        plot_heatmap = True)