This framework provides multi-objective optimization algorithms for python. Either define your own problems to be solved or use our test problems defined in pymop.

The test problems are uploaded to the PyPi Repository.

pip install pymoo

import time

from matplotlib import animation
import matplotlib.pyplot as plt
import numpy as np


if __name__ == '__main__':

    # load the problem instance
    from pymop.zdt import ZDT1
    problem = ZDT1(n_var=30)

    # create the algorithm instance by specifying the intended parameters
    from pymoo.algorithms.NSGAII import NSGAII
    algorithm = NSGAII("real", pop_size=100, verbose=True)

    start_time = time.time()

    # save the history in an object to observe the convergence over generations
    history = []

    # number of generations to run it
    n_gen = 200

    # solve the problem and return the results
    X, F, G = algorithm.solve(problem,
                              evaluator=(100 * n_gen),
                              seed=2,
                              return_only_feasible=False,
                              return_only_non_dominated=False,
                              history=history)

    print("--- %s seconds ---" % (time.time() - start_time))

    scatter_plot = True
    save_animation = True

    # get the problem dimensionality
    is_2d = problem.n_obj == 2
    is_3d = problem.n_obj == 3

    if scatter_plot and is_2d:
        plt.scatter(F[:, 0], F[:, 1])
        plt.show()

    if scatter_plot and is_3d:
        fig = plt.figure()
        from mpl_toolkits.mplot3d import Axes3D
        ax = fig.add_subplot(111, projection='3d')
        ax.scatter(F[:, 0], F[:, 1], F[:, 2])
        plt.show()

    # create an animation to watch the convergence over time
    if is_2d and save_animation:

        fig = plt.figure()
        ax = plt.gca()

        _F = history[0]['F']
        pf = problem.pareto_front()
        plt.scatter(pf[:,0], pf[:,1], label='Pareto Front', s=60, facecolors='none', edgecolors='r')
        scat = plt.scatter(_F[:, 0], _F[:, 1])


        def update(frame_number):
            _F = history[frame_number]['F']
            scat.set_offsets(_F)

            # get the bounds for plotting and add padding
            min = np.min(_F, axis=0) - 0.1
            max = np.max(_F, axis=0) + 0.

            # set the scatter object with padding
            ax.set_xlim(min[0], max[0])
            ax.set_ylim(min[1], max[1])


        # create the animation
        ani = animation.FuncAnimation(fig, update, frames=range(n_gen))

        # write the file
        Writer = animation.writers['ffmpeg']
        writer = Writer(fps=6, bitrate=1800)
        ani.save('%s.mp4' % problem.name(), writer=writer)

Genetic Algorithm: A simple genetic algorithm to solve single-objective problems.

NSGA-II [1]: Non-dominated sorting genetic algorithm for bi-objective problems. The mating selection is done using the binary tournament by comparing the rank and the crowding distance. The crowding distance is a niching measure in a two-dimensional space which sums up the difference to the neighbours in each dimension. The non-dominated sorting considers the rank determined by being in the ith front and the crowding distance to achieve a good diversity when converging.

NSGA-III [2][3]: A referenced-based algorithm used to solve many-objective problems. The survival selection uses the perpendicular distance to the reference directions. As normalization the boundary intersection method is used [5].

Simulated Binary Crossover [4]: This crossover simulates a single-point crossover in a binary representation by using an exponential distribution for real values. The polynomial mutation is defined accordingly which performs basically a binary bitflip for real numbers.

[1] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan. 2002. A fast and elitist multiobjective genetic algorithm: NSGA-II. Trans. Evol. Comp 6, 2 (April 2002), 182-197.

[2] K. Deb and H. Jain, "An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints," in IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577-601, Aug. 2014. doi: 10.1109/TEVC.2013.2281535

[3] H. Jain and K. Deb. An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach. IEEE Trans. Evolutionary Computation 18(4): 602-622 (2014)

[4] Kalyanmoy Deb, Karthik Sindhya, and Tatsuya Okabe. 2007. Self-adaptive simulated binary crossover for real-parameter optimization. In Proceedings of the 9th annual conference on Genetic and evolutionary computation (GECCO '07). ACM, New York, NY, USA, 1187-1194.

[5] Indraneel Das and J. E. Dennis. 1998. Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems. SIAM J. on Optimization 8, 3 (March 1998), 631-657.

Feel free to contact me if you have any question:

Julian Blank (blankjul@egr.msu.edu)
Michigan State University
Computational Optimization and Innovation Laboratory (COIN)
East Lansing, MI 48824, USA