optimization-benchmarks

A comprehensive collection of 55+ standard mathematical benchmark functions for testing and evaluating optimization algorithms.

🎯 Features

• 55+ Benchmark Functions: Complete collection of standard test functions
• Rich Metadata: Bounds, dimensions, known minima, and optimal points
• Visualization Tools: 2D/3D plots, convergence tracking, heatmaps
• Progress Tracking: Real-time progress bars with tqdm integration (v0.3.0)
• Benchmarking Utilities: Automated testing and comparison tools
• Multi-Format Export: Save plots as PNG, SVG, PDF, EPS (v0.3.0)
• Batch Processing: Generate multiple plots efficiently (v0.3.0)
• Type Hints: Full type annotation support
• Zero Core Dependencies: Only NumPy required (matplotlib optional)

📦 Installation

From PyPI

pip install optimization-benchmarks

From Source

git clone https://github.com/ak-rahul/optimization-benchmarks.git
cd optimization-benchmarks
pip install -e .

---

🚀 Quick Start

import numpy as np
from optimization_benchmarks import ackley, rastrigin, rosenbrock

Test Ackley function

x = np.zeros(5)
result = ackley(x)
print(f"Ackley(0) = {result}") # Should be close to 0

Test Rosenbrock function

x = np.ones(10)
result = rosenbrock(x)
print(f"Rosenbrock(1) = {result}") # Should be 0

Test Rastrigin function

x = np.random.randn(5)
result = rastrigin(x)
print(f"Rastrigin(x) = {result}")

---

📊 Usage Examples

Benchmarking an Optimization Algorithm

import numpy as np
from optimization_benchmarks import ackley, rastrigin, sphere

def my_optimizer(func, bounds, max_iter=1000):
"""Your optimization algorithm here."""
# ... implementation ...
pass

test_functions = {
'Sphere': (sphere, [(-5.12, 5.12)] * 10),
'Ackley': (ackley, [(-32, 32)] * 10),
'Rastrigin': (rastrigin, [(-5.12, 5.12)] * 10),
}

for name, (func, bounds) in test_functions.items():
best_x, best_f = my_optimizer(func, bounds)
print(f"{name}: f(x*) = {best_f}")

---

🎯 Using Benchmark Metadata (v0.1.1+)

Version 0.1.1 introduces comprehensive metadata for all 55 functions, eliminating the need to manually specify bounds and known minima:

from optimization_benchmarks import BENCHMARK_SUITE, get_function_info
import numpy as np

Get all available functions

from optimization_benchmarks import get_all_functions
print(f"Total functions: {len(get_all_functions())}") # 55

Get metadata for a specific function

info = get_function_info('ackley')
func = info['function']
bounds = info['bounds'] * info['default_dim'] # 10D by default
known_min = info['known_minimum']

Test at known minimum

x = np.zeros(info['default_dim'])
result = func(x)
print(f"Ackley(0) = {result:.6f}, Expected: {known_min}")

Simple Benchmarking with Metadata

from optimization_benchmarks import BENCHMARK_SUITE
import numpy as np

def simple_random_search(func, bounds, n_iter=1000):
"""Simple random search optimizer."""
best_x = None
best_cost = float('inf')

for _ in range(n_iter):
    x = np.array([np.random.uniform(b, b) for b in bounds])[1]
    cost = func(x)
    if cost < best_cost:
        best_cost = cost
        best_x = x

return best_x, best_cost

Benchmark on all functions - no manual bounds needed!

for name, meta in BENCHMARK_SUITE.items():
func = meta['function']
bounds = meta['bounds'] * meta['default_dim']
known_min = meta['known_minimum']

best_x, best_cost = simple_random_search(func, bounds)
error = abs(best_cost - known_min)

print(f"{name:20s} | Found: {best_cost:12.6f} | "
      f"Expected: {known_min:12.6f} | Error: {error:10.6f}")

Metadata Helper Functions

Function	Description
BENCHMARK_SUITE	Dictionary with all 55 functions and metadata
get_all_functions()	Returns list of all function names
get_function_info(name)	Returns metadata for specific function
get_bounds(name, dim=None)	Returns bounds for given dimension
get_function_list()	Returns formatted string with all functions

Metadata Fields

Each entry in BENCHMARK_SUITE contains:

• function: The callable function
• bounds: List of (min, max) tuples for each dimension
• default_dim: Recommended test dimension
• known_minimum: Known global minimum value
• optimal_point: Location(s) of the global minimum

---

🎨 Visualization Features (v0.2.0+)

Installation with Visualization

Install with visualization support

pip install optimization-benchmarks[viz]

Or install all optional features

pip install optimization-benchmarks[all]

2D Contour Plots

from optimization_benchmarks.visualization import plot_function_2d
import matplotlib.pyplot as plt

Create 2D contour plot

fig = plot_function_2d('ackley', show_optimum=True, resolution=100)
plt.savefig('ackley_2d.png')
plt.show()

Custom bounds

fig = plot_function_2d('sphere', bounds=[(-10, 10), (-10, 10)])
plt.show()

3D Surface Plots

from optimization_benchmarks.visualization import plot_function_3d

Create 3D surface plot

fig = plot_function_3d('rastrigin', resolution=50, elevation=30, azimuth=45)
plt.show()

Different colormap

fig = plot_function_3d('griewank', cmap='plasma')
plt.show()

Convergence Visualization

from optimization_benchmarks.visualization import plot_convergence

Simple convergence plot

history = [100, 50, 25, 10, 5, 1, 0.5, 0.1, 0.01]
fig = plot_convergence(history, function_name='sphere', known_minimum=0.0)
plt.show()

With logarithmic scale

fig = plot_convergence(history, log_scale=True)
plt.show()

With multiple series (best and current)

history_dict = {
'best': [10, 5, 2, 1, 0.5, 0.1],
'current': [10, 7, 3, 2, 1, 0.5],
'iterations': range(6)
}
fig = plot_convergence(history_dict, function_name='ackley', known_minimum=0.0)
plt.show()

Optimization Trajectory

from optimization_benchmarks.visualization import plot_trajectory_2d
import numpy as np

Your optimization path (must be 2D)

trajectory = np.array([
[5.0, 5.0], # Starting point
[3.0, 3.0],
[1.0, 1.0],
[0.1, 0.1],
[0.0, 0.0] # End point
])

fig = plot_trajectory_2d('sphere', trajectory)
plt.show()

Algorithm Comparison

from optimization_benchmarks.visualization import plot_algorithm_comparison

Results from multiple algorithms

results = {
'Simulated Annealing': {
'sphere': {'error': 0.001, 'time': 1.2},
'ackley': {'error': 0.01, 'time': 1.5},
'rastrigin': {'error': 0.1, 'time': 2.0}
},
'Genetic Algorithm': {
'sphere': {'error': 0.01, 'time': 0.8},
'ackley': {'error': 0.05, 'time': 1.0},
'rastrigin': {'error': 0.5, 'time': 1.5}
}
}

Compare by error

fig = plot_algorithm_comparison(results, metric='error')
plt.show()

Compare by time

fig = plot_algorithm_comparison(results, metric='time')
plt.show()

Benchmark Summary Dashboard

from optimization_benchmarks.visualization import plot_benchmark_summary

Your benchmark results

results = [
{'function': 'sphere', 'error': 0.001, 'time': 1.0},
{'function': 'ackley', 'error': 0.01, 'time': 1.5},
{'function': 'rastrigin', 'error': 0.1, 'time': 2.0},
{'function': 'rosenbrock', 'error': 1.0, 'time': 2.5},
{'function': 'griewank', 'error': 0.05, 'time': 1.8}
]

Creates 4-panel summary: error bars, time bars, error distribution, success rates

fig = plot_benchmark_summary(results)
plt.savefig('summary.png', dpi=300)
plt.show()

---

🆕 What's New in v0.3.0

Progress Bars with tqdm

Real-time progress tracking during benchmarking:

from optimization_benchmarks import BenchmarkRunner

runner = BenchmarkRunner(
my_optimizer,
algorithm_name='MyOptimizer',
n_runs=10,
show_progress=True # 🆕 NEW in v0.3.0
)

results = runner.run_suite(functions=['sphere', 'ackley', 'rastrigin'])

Shows: Benchmarking: 33%|███▍ | 1/3 [00:05<00:10, 5.2s/function]

Heatmap Visualization

Visualize where your optimization algorithm searches:

from optimization_benchmarks import plot_search_heatmap
import numpy as np

Your algorithm's search points

points = np.array([,, [0.5, 1], [0.1, 0.2], ])​

Create heatmap showing search density

fig = plot_search_heatmap('rastrigin', points, bins=30, cmap='hot')
plt.savefig('search_heatmap.png')
plt.show()

Multi-Format Plot Export

Save plots in multiple formats simultaneously:

from optimization_benchmarks import plot_function_2d, save_plot

Create plot

fig = plot_function_2d('ackley')

Save in multiple formats at once

save_plot(fig, 'ackley_function', formats=['png', 'svg', 'pdf', 'eps'], dpi=300)

Creates: ackley_function.png, ackley_function.svg, ackley_function.pdf, ackley_function.eps

Or directly in plot functions:

fig = plot_function_2d('sphere', formats=['png', 'svg']) # Auto-saves both formats

Enhanced Colormaps

Choose from 9 beautiful colormaps:

from optimization_benchmarks import COLORMAPS

print(COLORMAPS)  # Colours are : ['viridis', 'plasma', 'inferno', 'magma', 'cividis', 'coolwarm', 'jet', 'rainbow', 'turbo']

Use any colormap in visualization functions

fig = plot_function_2d('rastrigin', cmap='plasma')
fig = plot_function_3d('ackley', cmap='inferno')
fig = plot_search_heatmap('sphere', points, cmap='turbo'

Batch Plotting

Generate multiple plots efficiently:

from optimization_benchmarks import batch_plot_functions

Plot multiple functions in one call

batch_plot_functions(
function_names=['sphere', 'ackley', 'rastrigin', 'rosenbrock'],
plot_types=['2d', '3d'], # Generate both 2D and 3D plots
output_dir='plots',
formats=['png', 'svg'], # Save in multiple formats
resolution=100,
cmap='viridis'
)

---

🔬 Systematic Benchmarking (v0.2.0+)

Quick Benchmarking

from optimization_benchmarks.benchmarking import quick_benchmark

Your optimization algorithm

def my_optimizer(func, bounds, max_iter=1000):
"""Your optimization algorithm."""
# ... your implementation ...
return best_x, best_cost

Quick test on common functions

results = quick_benchmark(
my_optimizer,
function_names=['sphere', 'ackley', 'rastrigin'],
n_runs=5,
max_iter=1000
)

Detailed Benchmarking with BenchmarkRunner

from optimization_benchmarks.benchmarking import BenchmarkRunner

Create benchmark runner

runner = BenchmarkRunner(
algorithm=my_optimizer,
algorithm_name='MyOptimizer',
n_runs=10, # 10 independent runs per function
seed=42, # For reproducibility
verbose=True, # Show progress
show_progress=True # 🆕 Progress bars (v0.3.0)
)

Run on all 55+ functions

results = runner.run_suite(max_iter=5000)

Or test specific functions

results = runner.run_suite(
functions=['sphere', 'ackley', 'rastrigin', 'rosenbrock', 'griewank'],
max_iter=2000
)

Custom dimensions

results = runner.run_suite(
functions=['sphere', 'ackley'],
dimensions={'sphere': 10, 'ackley': 5},
max_iter=3000
)

Save results

runner.save_results('results.csv', format='csv')
runner.save_results('results.json', format='json')

Get summary statistics

stats = runner.get_summary_stats()
print(f"Success rate: {stats['success_rate']*100:.1f}%")
print(f"Mean error: {stats['error_mean']:.6f}")
print(f"Total time: {stats['time_total']:.2f}s")

Testing Multiple Algorithms

from optimization_benchmarks.benchmarking import BenchmarkRunner

algorithms = {
'SimulatedAnnealing': simulated_annealing,
'GeneticAlgorithm': genetic_algorithm,
'ParticleSwarm': particle_swarm
}

test_functions = ['sphere', 'ackley', 'rastrigin', 'rosenbrock']
all_results = {}

for name, algo in algorithms.items():
print(f"\nTesting {name}...")
runner = BenchmarkRunner(algo, algorithm_name=name, n_runs=10, show_progress=True)
results = runner.run_suite(functions=test_functions)
all_results[name] = results
runner.save_results(f'{name}_results.csv')

Compare algorithms

from optimization_benchmarks.visualization import plot_algorithm_comparison
fig = plot_algorithm_comparison(all_results, metric='error')
plt.savefig('algorithm_comparison.png')

---

🛠️ Utility Functions (v0.2.0+)

Bounds Normalization

from optimization_benchmarks.utils import normalize_bounds

Replicate single bound to all dimensions

bounds = normalize_bounds([(-5, 5)], dim=10)

Result: [(-5, 5), (-5, 5), ..., (-5, 5)] # 10 times

Different bounds per dimension

bounds = normalize_bounds([(-5, 5), (-10, 10), (0, 1)], dim=3)

Result: [(-5, 5), (-10, 10), (0, 1)]

From simple tuple

bounds = normalize_bounds((-5, 5), dim=5)

Result: [(-5, 5)] * 5

Random Point Generation

from optimization_benchmarks.utils import generate_random_point

bounds = [(-5, 5), (-10, 10)]

Uniform random

point = generate_random_point(bounds, method='uniform')

Normal distribution (centered, 99.7% within bounds)

point = generate_random_point(bounds, method='normal')

Center-biased (beta distribution)

point = generate_random_point(bounds, method='center_biased')

Bounds Checking and Clipping

from optimization_benchmarks.utils import check_bounds, clip_to_bounds

bounds = [(-5, 5), (-5, 5)]
point = np.array([10, -10])

Check if within bounds

is_valid = check_bounds(point, bounds) # False

Clip to bounds

clipped = clip_to_bounds(point, bounds)

Result: [5, -5]

Coordinate Transformations

from optimization_benchmarks.utils import scale_to_unit, scale_from_unit

bounds = [(-10, 10), (-5, 5)]
point = np.array()

Scale to unit hypercube ^n​

unit_point = scale_to_unit(point, bounds)

Result: [0.5, 0.5]

Scale back to original bounds

original = scale_from_unit(unit_point, bounds)

Bounds Information

from optimization_benchmarks.utils import (
get_bounds_range,
get_bounds_center,
generate_grid_points
)

bounds = [(-5, 5), (-10, 10)]

Get range of each dimension

ranges = get_bounds_range(bounds)

Get center point

center = get_bounds_center(bounds)

Generate grid of points

grid = generate_grid_points(bounds, points_per_dim=10)

Distance to Optimum

from optimization_benchmarks.utils import calculate_distance_to_optimum

current_point = np.array()​
optimal_point = np.array()

Euclidean distance

distance = calculate_distance_to_optimum(current_point, optimal_point)

Result: 1.4142135623730951

Multiple optima (returns minimum distance)

optimal_points = [np.array(), np.array(), np.array()]​
distance = calculate_distance_to_optimum(current_point, optimal_points)

Result: 0.0

---

💡 Complete Usage Example

import numpy as np
import matplotlib.pyplot as plt
from optimization_benchmarks import (
BENCHMARK_SUITE,
BenchmarkRunner,
normalize_bounds,
generate_random_point,
clip_to_bounds,
plot_function_2d,
plot_convergence,
plot_trajectory_2d,
plot_benchmark_summary
)

1. Define your optimizer with history tracking

def my_optimizer(func, bounds, max_iter=1000):
bounds = normalize_bounds(bounds, len(bounds))

Initialize

current = generate_random_point(bounds)
current_cost = func(current)
best = current.copy()
best_cost = current_cost

history = [best_cost]
trajectory = [best.copy()]

Optimization loop

for i in range(max_iter):
    # Generate neighbor
    neighbor = current + np.random.randn(len(bounds)) * 0.1
    neighbor = clip_to_bounds(neighbor, bounds)
    cost = func(neighbor)

    # Update if better
    if cost < current_cost:
        current = neighbor
        current_cost = cost
        if cost < best_cost:
            best = current.copy()
            best_cost = cost
            trajectory.append(best.copy())

    history.append(best_cost)

return best, best_cost

2. Visualize a test function

plot_function_2d('ackley', show_optimum=True)
plt.savefig('test_function.png')
plt.close()

3. Run benchmark suite

runner = BenchmarkRunner(
my_optimizer,
algorithm_name='MyOptimizer',
n_runs=10,
seed=42,
show_progress=True # v0.3.0 feature
)

results = runner.run_suite(
functions=['sphere', 'ackley', 'rastrigin', 'rosenbrock', 'griewank'],
max_iter=5000
)

4. Save and visualize results

runner.save_results('my_results.csv')
plot_benchmark_summary(results)
plt.savefig('benchmark_summary.png')
plt.show()

5. Print statistics

stats = runner.get_summary_stats()
print(f"\nResults:")
print(f" Success rate: {stats['success_rate']*100:.1f}%")
print(f" Mean error: {stats['error_mean']:.6f}")
print(f" Total time: {stats['time_total']:.2f}s")

---

📊 Supported Functions

The package supports 55+ benchmark functions including:

Multimodal Functions

ackley, rastrigin, rastrigin2, griewank, levy, michalewicz, schwefel2_26, katsuura, langerman

Unimodal Functions

sphere, sphere2, rosenbrock, rosenbrock_ext1, rosenbrock_ext2, sum_squares, hyperellipsoid, schwefel1_2, schwefel2_21, schwefel2_22, schwefel3_2

2D Test Functions

beale, booth, matyas, himmelblau, easom, goldstein_price, branin, branin2, camel3, camel6, bohachevsky1, bohachevsky2, schaffer1, schaffer2, leon, trecanni, mccormick, eggholder, chichinadze, hosaki, zettl

Special Functions

box_betts, colville, corana, kowalik, exp2, gear, holzman1, holzman2, stretched_v, trefethen4, step, step2, maxmod, multimod

All functions work automatically with the new utilities!

---

🔧 Installation & Requirements

Basic (only numpy required)

pip install optimization-benchmarks

With visualization

pip install optimization-benchmarks[viz]

Everything

pip install optimization-benchmarks[all]

Requirements

• Core: Python 3.8+, NumPy ≥1.20.0
• Visualization: matplotlib ≥3.3.0 (optional)
• Progress: tqdm ≥4.65.0 (v0.3.0+)
• Development: pytest, pytest-cov, black, flake8, mypy, isort (optional)

---

🎮 Command-Line Interface

The package includes a CLI for quick function evaluation:

List all available functions

optbench --list

Get function information

optbench --info ackley

Evaluate a function

optbench --function rastrigin --values 0 0 0 0 0

Batch evaluation from CSV

optbench --function sphere --input points.csv --output results.json

---

🔬 Function Properties

Each function includes:

• Domain: Valid input ranges
• Dimension: Number of variables (n for arbitrary dimensions)
• Global Minimum: Known optimal value and location
• Mathematical Formula: Documented in docstrings

---

📚 Academic Citations

This package implements benchmark functions based on these authoritative sources:

Primary References

1. Jamil, M., & Yang, X. S. (2013). "A literature survey of benchmark functions for global optimization problems." International Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150-194. DOI: 10.1504/IJMMNO.2013.055204

2. Surjanovic, S., & Bingham, D. (2013). "Virtual Library of Simulation Experiments: Test Functions and Datasets." Simon Fraser University. URL: http://www.sfu.ca/~ssurjano

3. Adorio, E. P., & Diliman, U. (2005). "MVF-Multivariate Test Functions Library in C for Unconstrained Global Optimization." University of the Philippines.

How to Cite This Package

If you use this package in your research, please cite:

@software{optimization_benchmarks,
author = {AK Rahul},
title = {optimization-benchmarks: A Python Package for Optimization Algorithm Evaluation},
year = {2025},
url = {https://github.com/ak-rahul/optimization-benchmarks},
version = {0.3.0}
}

---

🎓 Academic Use

This package is perfect for:

• Algorithm Development: Test new optimization algorithms
• Comparative Studies: Benchmark against existing methods
• Academic Research: Reproduce published results
• Teaching: Demonstrate optimization concepts
• Thesis Projects: Comprehensive evaluation suite

---

🤝 Contributing

Contributions are welcome! Please see CONTRIBUTING.md for guidelines.

Quick Contribution Guide

1. Fork the repository
2. Create your feature branch (git checkout -b feature/new-function)
3. Add your function to functions.py
4. Add tests to tests/test_functions.py
5. Run tests: pytest
6. Submit a pull request

---

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

---

🙏 Acknowledgments

• Mathematical formulations based on the MVF C library by E.P. Adorio
• Function definitions from Virtual Library of Simulation Experiments
• Inspired by the optimization research community

---

📞 Support

• Issues: GitHub Issues
• Discussions: GitHub Discussions

---

🔗 Related Projects

• SciPy Optimize - Optimization algorithms
• PyGMO - Massively parallel optimization
• DEAP - Evolutionary algorithms

---

Made with ❤️ for the optimization community