Note

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Pareto front on a multi-objective problem#

In this example, we illustrate the use of the ParetoFront plot on the Binh and Korn multi-objective problem.

from __future__ import annotations

from gemseo.algos.doe.factory import DOELibraryFactory
from gemseo.post.factory import PostFactory
from gemseo.problems.multiobjective_optimization.binh_korn import BinhKorn

Import the optimization problem#

Then, we instantiate the Binh and Korn optimization problem (see BinhKorn).

problem = BinhKorn()

Create and execute scenario#

Then, we instantiate the design of experiment factory, and we request the execution of a 100-length LHS optimized by simulated annealing.

doe_factory = DOELibraryFactory()
doe_factory.execute(problem, algo_name="OT_OPT_LHS", n_samples=100)

INFO - 16:25:16: Optimization problem:
INFO - 16:25:16:    minimize compute_binhkorn(x, y) = (4*x**2+ 4*y**2, (x-5.)**2 + (y-5.)**2)
INFO - 16:25:16:    with respect to x, y
INFO - 16:25:16:    under the inequality constraints
INFO - 16:25:16:       ineq1(x, y): (x-5.)**2 + y**2 <= 25. <= 0.0
INFO - 16:25:16:       ineq2(x, y): (x-8.)**2 + (y+3)**2 >= 7.7 <= 0.0
INFO - 16:25:16:    over the design space:
INFO - 16:25:16:       +------+-------------+-------+-------------+-------+
INFO - 16:25:16:       | Name | Lower bound | Value | Upper bound | Type  |
INFO - 16:25:16:       +------+-------------+-------+-------------+-------+
INFO - 16:25:16:       | x    |      0      |   1   |      5      | float |
INFO - 16:25:16:       | y    |      0      |   1   |      3      | float |
INFO - 16:25:16:       +------+-------------+-------+-------------+-------+
INFO - 16:25:16: Solving optimization problem with algorithm OT_OPT_LHS:
INFO - 16:25:16:      1%|          | 1/100 [00:00<00:00, 2966.27 it/sec, feas=True, obj=[98.50728236 17.96432773]]
INFO - 16:25:16:      2%|▏         | 2/100 [00:00<00:00, 2952.70 it/sec, feas=True, obj=[ 5.10086961 36.30607322]]
INFO - 16:25:16:      3%|▎         | 3/100 [00:00<00:00, 3096.19 it/sec, feas=True, obj=[ 5.73344945 34.53104096]]
INFO - 16:25:16:      4%|▍         | 4/100 [00:00<00:00, 3182.93 it/sec, feas=True, obj=[ 2.99274814 38.89733089]]
INFO - 16:25:16:      5%|▌         | 5/100 [00:00<00:00, 3299.48 it/sec, feas=True, obj=[ 1.45355254 43.72252794]]
INFO - 16:25:16:      6%|▌         | 6/100 [00:00<00:00, 3407.69 it/sec, feas=True, obj=[26.6936315  22.26390516]]
INFO - 16:25:16:      7%|▋         | 7/100 [00:00<00:00, 3484.88 it/sec, feas=True, obj=[95.06326723 19.21865528]]
INFO - 16:25:16:      8%|▊         | 8/100 [00:00<00:00, 3498.53 it/sec, feas=True, obj=[11.37121324 33.49445422]]
INFO - 16:25:16:      9%|▉         | 9/100 [00:00<00:00, 3552.82 it/sec, feas=True, obj=[33.52756721 17.72645069]]
INFO - 16:25:16:     10%|█         | 10/100 [00:00<00:00, 3602.12 it/sec, feas=True, obj=[ 4.39737722 39.0292696 ]]
INFO - 16:25:16:     11%|█         | 11/100 [00:00<00:00, 3636.58 it/sec, feas=True, obj=[83.81080473  6.44933463]]
INFO - 16:25:16:     12%|█▏        | 12/100 [00:00<00:00, 3638.78 it/sec, feas=True, obj=[64.12902701 21.25106868]]
INFO - 16:25:16:     13%|█▎        | 13/100 [00:00<00:00, 3667.34 it/sec, feas=True, obj=[71.23914138  9.90898899]]
INFO - 16:25:16:     14%|█▍        | 14/100 [00:00<00:00, 3697.52 it/sec, feas=True, obj=[58.61362754 10.51836124]]
INFO - 16:25:16:     15%|█▌        | 15/100 [00:00<00:00, 3724.96 it/sec, feas=True, obj=[17.1313131  28.27257144]]
INFO - 16:25:16:     16%|█▌        | 16/100 [00:00<00:00, 3727.65 it/sec, feas=True, obj=[38.07815452 16.26141839]]
INFO - 16:25:16:     17%|█▋        | 17/100 [00:00<00:00, 3747.87 it/sec, feas=True, obj=[45.29614269 15.87407947]]
INFO - 16:25:16:     18%|█▊        | 18/100 [00:00<00:00, 3764.15 it/sec, feas=True, obj=[78.26474617 13.97242459]]
INFO - 16:25:16:     19%|█▉        | 19/100 [00:00<00:00, 3788.89 it/sec, feas=True, obj=[58.47852866 16.87161502]]
INFO - 16:25:16:     20%|██        | 20/100 [00:00<00:00, 3794.89 it/sec, feas=True, obj=[ 7.27923171 33.31157275]]
INFO - 16:25:16:     21%|██        | 21/100 [00:00<00:00, 3810.20 it/sec, feas=True, obj=[15.973784   25.86458657]]
INFO - 16:25:16:     22%|██▏       | 22/100 [00:00<00:00, 3825.65 it/sec, feas=True, obj=[47.66727367 15.96717933]]
INFO - 16:25:16:     23%|██▎       | 23/100 [00:00<00:00, 3838.19 it/sec, feas=True, obj=[88.93580164  6.7712213 ]]
INFO - 16:25:16:     24%|██▍       | 24/100 [00:00<00:00, 3842.55 it/sec, feas=True, obj=[22.10070032 24.30208587]]
INFO - 16:25:16:     25%|██▌       | 25/100 [00:00<00:00, 3850.67 it/sec, feas=True, obj=[16.24176378 31.32352362]]
INFO - 16:25:16:     26%|██▌       | 26/100 [00:00<00:00, 3859.70 it/sec, feas=True, obj=[29.38105263 25.45614728]]
INFO - 16:25:16:     27%|██▋       | 27/100 [00:00<00:00, 3871.40 it/sec, feas=True, obj=[27.91051427 19.91664614]]
INFO - 16:25:16:     28%|██▊       | 28/100 [00:00<00:00, 3871.07 it/sec, feas=True, obj=[37.28915469 18.0103703 ]]
INFO - 16:25:16:     29%|██▉       | 29/100 [00:00<00:00, 3876.68 it/sec, feas=True, obj=[ 1.01177424 43.45819442]]
INFO - 16:25:16:     30%|███       | 30/100 [00:00<00:00, 3885.29 it/sec, feas=True, obj=[85.96675181 10.67473104]]
INFO - 16:25:16:     31%|███       | 31/100 [00:00<00:00, 3897.00 it/sec, feas=True, obj=[87.34794862  6.40751572]]
INFO - 16:25:16:     32%|███▏      | 32/100 [00:00<00:00, 3899.30 it/sec, feas=True, obj=[25.57943091 20.74700556]]
INFO - 16:25:16:     33%|███▎      | 33/100 [00:00<00:00, 3907.52 it/sec, feas=True, obj=[76.39229092 11.95231974]]
INFO - 16:25:16:     34%|███▍      | 34/100 [00:00<00:00, 3915.39 it/sec, feas=True, obj=[51.64657249 21.13171808]]
INFO - 16:25:16:     35%|███▌      | 35/100 [00:00<00:00, 3924.84 it/sec, feas=True, obj=[95.8196038   9.31540942]]
INFO - 16:25:16:     36%|███▌      | 36/100 [00:00<00:00, 3926.84 it/sec, feas=True, obj=[117.01496061   6.96792478]]
INFO - 16:25:16:     37%|███▋      | 37/100 [00:00<00:00, 3933.72 it/sec, feas=True, obj=[78.37371654 21.95951365]]
INFO - 16:25:16:     38%|███▊      | 38/100 [00:00<00:00, 3905.50 it/sec, feas=True, obj=[38.21112297 20.46923673]]
INFO - 16:25:16:     39%|███▉      | 39/100 [00:00<00:00, 3906.90 it/sec, feas=True, obj=[77.87964266 17.24338363]]
INFO - 16:25:16:     40%|████      | 40/100 [00:00<00:00, 3901.32 it/sec, feas=True, obj=[59.79323468 22.48208172]]
INFO - 16:25:16:     41%|████      | 41/100 [00:00<00:00, 3905.13 it/sec, feas=True, obj=[34.56674439 21.30358144]]
INFO - 16:25:16:     42%|████▏     | 42/100 [00:00<00:00, 3910.43 it/sec, feas=True, obj=[ 0.46290462 45.68910522]]
INFO - 16:25:16:     43%|████▎     | 43/100 [00:00<00:00, 3915.40 it/sec, feas=True, obj=[100.73843409  11.99829359]]
INFO - 16:25:16:     44%|████▍     | 44/100 [00:00<00:00, 3915.83 it/sec, feas=True, obj=[21.18253098 23.10192252]]
INFO - 16:25:16:     45%|████▌     | 45/100 [00:00<00:00, 3919.75 it/sec, feas=True, obj=[51.0570421  12.66142788]]
INFO - 16:25:16:     46%|████▌     | 46/100 [00:00<00:00, 3925.49 it/sec, feas=True, obj=[90.5172837  14.59877358]]
INFO - 16:25:16:     47%|████▋     | 47/100 [00:00<00:00, 3932.42 it/sec, feas=True, obj=[19.5728568  23.86163594]]
INFO - 16:25:16:     48%|████▊     | 48/100 [00:00<00:00, 3932.01 it/sec, feas=True, obj=[47.52592344 19.80950276]]
INFO - 16:25:16:     49%|████▉     | 49/100 [00:00<00:00, 3936.43 it/sec, feas=True, obj=[113.47621648   6.0816867 ]]
INFO - 16:25:16:     50%|█████     | 50/100 [00:00<00:00, 3940.16 it/sec, feas=True, obj=[37.3534796  16.14867587]]
INFO - 16:25:16:     51%|█████     | 51/100 [00:00<00:00, 3947.76 it/sec, feas=True, obj=[41.42991507 25.64561402]]
INFO - 16:25:16:     52%|█████▏    | 52/100 [00:00<00:00, 3948.51 it/sec, feas=True, obj=[11.19222105 30.06624599]]
INFO - 16:25:16:     53%|█████▎    | 53/100 [00:00<00:00, 3952.53 it/sec, feas=True, obj=[86.67968625 15.70216699]]
INFO - 16:25:16:     54%|█████▍    | 54/100 [00:00<00:00, 3959.10 it/sec, feas=True, obj=[90.96123503  7.41344494]]
INFO - 16:25:16:     55%|█████▌    | 55/100 [00:00<00:00, 3963.35 it/sec, feas=True, obj=[34.69157131 20.23247304]]
INFO - 16:25:16:     56%|█████▌    | 56/100 [00:00<00:00, 3962.56 it/sec, feas=True, obj=[56.73716821 11.35918483]]
INFO - 16:25:16:     57%|█████▋    | 57/100 [00:00<00:00, 3963.78 it/sec, feas=True, obj=[58.33058905 10.70125973]]
INFO - 16:25:16:     58%|█████▊    | 58/100 [00:00<00:00, 3960.82 it/sec, feas=False, obj=[19.48697889 32.41188233]]
INFO - 16:25:16:     59%|█████▉    | 59/100 [00:00<00:00, 3964.82 it/sec, feas=True, obj=[ 9.51315338 32.51410675]]
INFO - 16:25:16:     60%|██████    | 60/100 [00:00<00:00, 3963.56 it/sec, feas=True, obj=[14.29503909 27.34400969]]
INFO - 16:25:16:     61%|██████    | 61/100 [00:00<00:00, 3965.97 it/sec, feas=True, obj=[21.01471588 31.54351738]]
INFO - 16:25:16:     62%|██████▏   | 62/100 [00:00<00:00, 3971.15 it/sec, feas=True, obj=[32.62517737 26.61449355]]
INFO - 16:25:16:     63%|██████▎   | 63/100 [00:00<00:00, 3976.84 it/sec, feas=True, obj=[54.44405197 15.56114393]]
INFO - 16:25:16:     64%|██████▍   | 64/100 [00:00<00:00, 3977.88 it/sec, feas=True, obj=[31.67860311 19.0346702 ]]
INFO - 16:25:16:     65%|██████▌   | 65/100 [00:00<00:00, 3978.43 it/sec, feas=True, obj=[12.95434975 27.79409607]]
INFO - 16:25:16:     66%|██████▌   | 66/100 [00:00<00:00, 3982.05 it/sec, feas=True, obj=[33.53586239 22.39126054]]
INFO - 16:25:16:     67%|██████▋   | 67/100 [00:00<00:00, 3982.80 it/sec, feas=True, obj=[13.77290554 32.89122016]]
INFO - 16:25:16:     68%|██████▊   | 68/100 [00:00<00:00, 3985.64 it/sec, feas=True, obj=[24.55394276 22.04978627]]
INFO - 16:25:16:     69%|██████▉   | 69/100 [00:00<00:00, 3984.13 it/sec, feas=True, obj=[38.26118694 15.93322028]]
INFO - 16:25:16:     70%|███████   | 70/100 [00:00<00:00, 3987.52 it/sec, feas=True, obj=[69.87762034 24.69083953]]
INFO - 16:25:16:     71%|███████   | 71/100 [00:00<00:00, 3990.61 it/sec, feas=True, obj=[117.708931     4.51806144]]
INFO - 16:25:16:     72%|███████▏  | 72/100 [00:00<00:00, 3991.04 it/sec, feas=False, obj=[24.18294227 29.58378487]]
INFO - 16:25:16:     73%|███████▎  | 73/100 [00:00<00:00, 3988.07 it/sec, feas=True, obj=[65.41388231 10.29184587]]
INFO - 16:25:16:     74%|███████▍  | 74/100 [00:00<00:00, 3991.75 it/sec, feas=True, obj=[12.24819086 35.25245724]]
INFO - 16:25:16:     75%|███████▌  | 75/100 [00:00<00:00, 3994.27 it/sec, feas=True, obj=[25.31003032 24.72346394]]
INFO - 16:25:16:     76%|███████▌  | 76/100 [00:00<00:00, 3997.83 it/sec, feas=True, obj=[49.90530196 13.18421478]]
INFO - 16:25:16:     77%|███████▋  | 77/100 [00:00<00:00, 3995.46 it/sec, feas=True, obj=[13.58752584 29.15968355]]
INFO - 16:25:16:     78%|███████▊  | 78/100 [00:00<00:00, 3997.36 it/sec, feas=True, obj=[ 7.8547085  33.00337593]]
INFO - 16:25:16:     79%|███████▉  | 79/100 [00:00<00:00, 4001.14 it/sec, feas=True, obj=[34.10323875 25.61801713]]
INFO - 16:25:16:     80%|████████  | 80/100 [00:00<00:00, 4002.96 it/sec, feas=True, obj=[50.66487602 26.86160736]]
INFO - 16:25:16:     81%|████████  | 81/100 [00:00<00:00, 4001.73 it/sec, feas=True, obj=[19.81349957 25.42966204]]
INFO - 16:25:16:     82%|████████▏ | 82/100 [00:00<00:00, 4003.64 it/sec, feas=True, obj=[ 7.4546377  34.97394601]]
INFO - 16:25:16:     83%|████████▎ | 83/100 [00:00<00:00, 4003.86 it/sec, feas=True, obj=[106.96361831   5.44850984]]
INFO - 16:25:16:     84%|████████▍ | 84/100 [00:00<00:00, 4003.61 it/sec, feas=False, obj=[ 5.36344422 38.7517519 ]]
INFO - 16:25:16:     85%|████████▌ | 85/100 [00:00<00:00, 4002.06 it/sec, feas=True, obj=[49.116045   14.30366519]]
INFO - 16:25:16:     86%|████████▌ | 86/100 [00:00<00:00, 4003.00 it/sec, feas=True, obj=[65.46370949  9.53265669]]
INFO - 16:25:16:     87%|████████▋ | 87/100 [00:00<00:00, 4005.63 it/sec, feas=True, obj=[34.43204876 18.46542147]]
INFO - 16:25:16:     88%|████████▊ | 88/100 [00:00<00:00, 4008.37 it/sec, feas=True, obj=[89.05896719 23.04880076]]
INFO - 16:25:16:     89%|████████▉ | 89/100 [00:00<00:00, 4007.57 it/sec, feas=True, obj=[28.30965373 24.49805668]]
INFO - 16:25:16:     90%|█████████ | 90/100 [00:00<00:00, 4009.68 it/sec, feas=True, obj=[39.10277528 23.64680578]]
INFO - 16:25:16:     91%|█████████ | 91/100 [00:00<00:00, 4011.49 it/sec, feas=True, obj=[117.48545685   8.18526365]]
INFO - 16:25:16:     92%|█████████▏| 92/100 [00:00<00:00, 4013.94 it/sec, feas=True, obj=[66.02068516 15.50045449]]
INFO - 16:25:16:     93%|█████████▎| 93/100 [00:00<00:00, 4012.66 it/sec, feas=True, obj=[67.39952092 12.92818287]]
INFO - 16:25:16:     94%|█████████▍| 94/100 [00:00<00:00, 4014.06 it/sec, feas=True, obj=[55.28167693 13.5771757 ]]
INFO - 16:25:16:     95%|█████████▌| 95/100 [00:00<00:00, 4016.97 it/sec, feas=True, obj=[17.335399   26.84155459]]
INFO - 16:25:16:     96%|█████████▌| 96/100 [00:00<00:00, 4018.29 it/sec, feas=True, obj=[ 4.93473731 35.93049995]]
INFO - 16:25:16:     97%|█████████▋| 97/100 [00:00<00:00, 4016.94 it/sec, feas=True, obj=[25.16386975 26.89754332]]
INFO - 16:25:16:     98%|█████████▊| 98/100 [00:00<00:00, 4018.40 it/sec, feas=True, obj=[81.61206359 10.43246637]]
INFO - 16:25:16:     99%|█████████▉| 99/100 [00:00<00:00, 4019.09 it/sec, feas=True, obj=[40.83406128 15.59204339]]
INFO - 16:25:16:    100%|██████████| 100/100 [00:00<00:00, 3852.90 it/sec, feas=True, obj=[52.02690192 12.02601654]]
INFO - 16:25:16: Optimization result:
INFO - 16:25:16:    Optimizer info:
INFO - 16:25:16:       Status: None
INFO - 16:25:16:       Message: None
INFO - 16:25:16:    Solution:
INFO - 16:25:16:       The solution is feasible.
INFO - 16:25:16:       Objective: 30.39964825035985
INFO - 16:25:16:       Standardized constraints:
INFO - 16:25:16:          ineq1 = [-11.77907222]
INFO - 16:25:16:          ineq2 = [-38.26307397]
INFO - 16:25:16:       Design space:
INFO - 16:25:16:          +------+-------------+-------------------+-------------+-------+
INFO - 16:25:16:          | Name | Lower bound |       Value       | Upper bound | Type  |
INFO - 16:25:16:          +------+-------------+-------------------+-------------+-------+
INFO - 16:25:16:          | x    |      0      | 1.542975634014225 |      5      | float |
INFO - 16:25:16:          | y    |      0      | 1.269910308585573 |      3      | float |
INFO - 16:25:16:          +------+-------------+-------------------+-------------+-------+
Optimization result:
  • Design variables: [1.54297563 1.26991031]
  • Objective function: 30.39964825035985
  • Feasible solution: True


Post-processing#

Lastly, we post-process the scenario by means of the ParetoFront plot which generates a plot or a matrix of plots if there are more than 2 objectives, plots in blue the locally non dominated points for the current two objectives, plots in green the globally (all objectives) Pareto optimal points. The plots in green denote non-feasible points. Note that the user can avoid the display of the non-feasible points.

PostFactory().execute(
    problem,
    post_name="ParetoFront",
    show_non_feasible=False,
    objectives=["compute_binhkorn"],
    objectives_labels=["f1", "f2"],
    save=False,
    show=True,
)

PostFactory().execute(
    problem,
    post_name="ParetoFront",
    objectives=["compute_binhkorn"],
    objectives_labels=["f1", "f2"],
    save=False,
    show=True,
)
  • Pareto front
  • Pareto front
<gemseo.post.pareto_front.ParetoFront object at 0x7c2f71e7edb0>

Total running time of the script: (0 minutes 0.258 seconds)

Download Jupyter notebook: plot_pareto_front_binhkorn.ipynb

Download Python source code: plot_pareto_front_binhkorn.py

Download zipped: plot_pareto_front_binhkorn.zip

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