Note
Go to the end to download the full example code.
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 import configure_logger
from gemseo.algos.doe.factory import DOELibraryFactory
from gemseo.post.factory import PostFactory
from gemseo.problems.multiobjective_optimization.binh_korn import BinhKorn
Import#
The first step is to import a high-level function for logging.
configure_logger()
<RootLogger root (INFO)>
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 - 20:39:11: Optimization problem:
INFO - 20:39:11: minimize compute_binhkorn(x, y) = (4*x**2+ 4*y**2, (x-5.)**2 + (y-5.)**2)
INFO - 20:39:11: with respect to x, y
INFO - 20:39:11: under the inequality constraints
INFO - 20:39:11: ineq1(x, y): (x-5.)**2 + y**2 <= 25. <= 0.0
INFO - 20:39:11: ineq2(x, y): (x-8.)**2 + (y+3)**2 >= 7.7 <= 0.0
INFO - 20:39:11: over the design space:
INFO - 20:39:11: +------+-------------+-------+-------------+-------+
INFO - 20:39:11: | Name | Lower bound | Value | Upper bound | Type |
INFO - 20:39:11: +------+-------------+-------+-------------+-------+
INFO - 20:39:11: | x | 0 | 1 | 5 | float |
INFO - 20:39:11: | y | 0 | 1 | 3 | float |
INFO - 20:39:11: +------+-------------+-------+-------------+-------+
INFO - 20:39:11: Solving optimization problem with algorithm OT_OPT_LHS:
INFO - 20:39:11: 1%| | 1/100 [00:00<00:00, 4040.76 it/sec, obj=[98.50728236 17.96432773]]
INFO - 20:39:11: 2%|▏ | 2/100 [00:00<00:00, 3463.50 it/sec, obj=[ 5.10086961 36.30607322]]
INFO - 20:39:11: 3%|▎ | 3/100 [00:00<00:00, 3653.57 it/sec, obj=[ 5.73344945 34.53104096]]
INFO - 20:39:11: 4%|▍ | 4/100 [00:00<00:00, 3808.68 it/sec, obj=[ 2.99274814 38.89733089]]
INFO - 20:39:11: 5%|▌ | 5/100 [00:00<00:00, 3916.98 it/sec, obj=[ 1.45355254 43.72252794]]
INFO - 20:39:11: 6%|▌ | 6/100 [00:00<00:00, 3943.87 it/sec, obj=[26.6936315 22.26390516]]
INFO - 20:39:11: 7%|▋ | 7/100 [00:00<00:00, 3980.49 it/sec, obj=[95.06326723 19.21865528]]
INFO - 20:39:11: 8%|▊ | 8/100 [00:00<00:00, 4039.30 it/sec, obj=[11.37121324 33.49445422]]
INFO - 20:39:11: 9%|▉ | 9/100 [00:00<00:00, 4088.02 it/sec, obj=[33.52756721 17.72645069]]
INFO - 20:39:11: 10%|█ | 10/100 [00:00<00:00, 4112.87 it/sec, obj=[ 4.39737722 39.0292696 ]]
INFO - 20:39:11: 11%|█ | 11/100 [00:00<00:00, 4152.40 it/sec, obj=[83.81080473 6.44933463]]
INFO - 20:39:11: 12%|█▏ | 12/100 [00:00<00:00, 4195.70 it/sec, obj=[64.12902701 21.25106868]]
INFO - 20:39:11: 13%|█▎ | 13/100 [00:00<00:00, 4230.10 it/sec, obj=[71.23914138 9.90898899]]
INFO - 20:39:11: 14%|█▍ | 14/100 [00:00<00:00, 4264.05 it/sec, obj=[58.61362754 10.51836124]]
INFO - 20:39:11: 15%|█▌ | 15/100 [00:00<00:00, 4271.48 it/sec, obj=[17.1313131 28.27257144]]
INFO - 20:39:11: 16%|█▌ | 16/100 [00:00<00:00, 4295.52 it/sec, obj=[38.07815452 16.26141839]]
INFO - 20:39:11: 17%|█▋ | 17/100 [00:00<00:00, 4322.98 it/sec, obj=[45.29614269 15.87407947]]
INFO - 20:39:11: 18%|█▊ | 18/100 [00:00<00:00, 4347.43 it/sec, obj=[78.26474617 13.97242459]]
INFO - 20:39:11: 19%|█▉ | 19/100 [00:00<00:00, 4367.63 it/sec, obj=[58.47852866 16.87161502]]
INFO - 20:39:11: 20%|██ | 20/100 [00:00<00:00, 4353.42 it/sec, obj=[ 7.27923171 33.31157275]]
INFO - 20:39:11: 21%|██ | 21/100 [00:00<00:00, 4372.75 it/sec, obj=[15.973784 25.86458657]]
INFO - 20:39:11: 22%|██▏ | 22/100 [00:00<00:00, 4391.73 it/sec, obj=[47.66727367 15.96717933]]
INFO - 20:39:11: 23%|██▎ | 23/100 [00:00<00:00, 4408.60 it/sec, obj=[88.93580164 6.7712213 ]]
INFO - 20:39:11: 24%|██▍ | 24/100 [00:00<00:00, 4409.26 it/sec, obj=[22.10070032 24.30208587]]
INFO - 20:39:11: 25%|██▌ | 25/100 [00:00<00:00, 4421.57 it/sec, obj=[16.24176378 31.32352362]]
INFO - 20:39:11: 26%|██▌ | 26/100 [00:00<00:00, 4433.54 it/sec, obj=[29.38105263 25.45614728]]
INFO - 20:39:11: 27%|██▋ | 27/100 [00:00<00:00, 4445.56 it/sec, obj=[27.91051427 19.91664614]]
INFO - 20:39:11: 28%|██▊ | 28/100 [00:00<00:00, 4459.15 it/sec, obj=[37.28915469 18.0103703 ]]
INFO - 20:39:11: 29%|██▉ | 29/100 [00:00<00:00, 4457.61 it/sec, obj=[ 1.01177424 43.45819442]]
INFO - 20:39:11: 30%|███ | 30/100 [00:00<00:00, 4462.50 it/sec, obj=[85.96675181 10.67473104]]
INFO - 20:39:11: 31%|███ | 31/100 [00:00<00:00, 4474.16 it/sec, obj=[87.34794862 6.40751572]]
INFO - 20:39:11: 32%|███▏ | 32/100 [00:00<00:00, 4487.09 it/sec, obj=[25.57943091 20.74700556]]
INFO - 20:39:11: 33%|███▎ | 33/100 [00:00<00:00, 4498.28 it/sec, obj=[76.39229092 11.95231974]]
INFO - 20:39:11: 34%|███▍ | 34/100 [00:00<00:00, 4498.20 it/sec, obj=[51.64657249 21.13171808]]
INFO - 20:39:11: 35%|███▌ | 35/100 [00:00<00:00, 4508.76 it/sec, obj=[95.8196038 9.31540942]]
INFO - 20:39:11: 36%|███▌ | 36/100 [00:00<00:00, 4519.45 it/sec, obj=[117.01496061 6.96792478]]
INFO - 20:39:11: 37%|███▋ | 37/100 [00:00<00:00, 4529.88 it/sec, obj=[78.37371654 21.95951365]]
INFO - 20:39:11: 38%|███▊ | 38/100 [00:00<00:00, 4465.53 it/sec, obj=[38.21112297 20.46923673]]
INFO - 20:39:11: 39%|███▉ | 39/100 [00:00<00:00, 4459.96 it/sec, obj=[77.87964266 17.24338363]]
INFO - 20:39:11: 40%|████ | 40/100 [00:00<00:00, 4465.59 it/sec, obj=[59.79323468 22.48208172]]
INFO - 20:39:11: 41%|████ | 41/100 [00:00<00:00, 4472.94 it/sec, obj=[34.56674439 21.30358144]]
INFO - 20:39:11: 42%|████▏ | 42/100 [00:00<00:00, 4482.01 it/sec, obj=[ 0.46290462 45.68910522]]
INFO - 20:39:11: 43%|████▎ | 43/100 [00:00<00:00, 4477.98 it/sec, obj=[100.73843409 11.99829359]]
INFO - 20:39:11: 44%|████▍ | 44/100 [00:00<00:00, 4487.30 it/sec, obj=[21.18253098 23.10192252]]
INFO - 20:39:11: 45%|████▌ | 45/100 [00:00<00:00, 4497.32 it/sec, obj=[51.0570421 12.66142788]]
INFO - 20:39:11: 46%|████▌ | 46/100 [00:00<00:00, 4502.85 it/sec, obj=[90.5172837 14.59877358]]
INFO - 20:39:11: 47%|████▋ | 47/100 [00:00<00:00, 4504.03 it/sec, obj=[19.5728568 23.86163594]]
INFO - 20:39:11: 48%|████▊ | 48/100 [00:00<00:00, 4509.70 it/sec, obj=[47.52592344 19.80950276]]
INFO - 20:39:11: 49%|████▉ | 49/100 [00:00<00:00, 4516.45 it/sec, obj=[113.47621648 6.0816867 ]]
INFO - 20:39:11: 50%|█████ | 50/100 [00:00<00:00, 4523.53 it/sec, obj=[37.3534796 16.14867587]]
INFO - 20:39:11: 51%|█████ | 51/100 [00:00<00:00, 4531.79 it/sec, obj=[41.42991507 25.64561402]]
INFO - 20:39:11: 52%|█████▏ | 52/100 [00:00<00:00, 4532.97 it/sec, obj=[11.19222105 30.06624599]]
INFO - 20:39:11: 53%|█████▎ | 53/100 [00:00<00:00, 4538.64 it/sec, obj=[86.67968625 15.70216699]]
INFO - 20:39:11: 54%|█████▍ | 54/100 [00:00<00:00, 4545.67 it/sec, obj=[90.96123503 7.41344494]]
INFO - 20:39:11: 55%|█████▌ | 55/100 [00:00<00:00, 4553.36 it/sec, obj=[34.69157131 20.23247304]]
INFO - 20:39:11: 56%|█████▌ | 56/100 [00:00<00:00, 4558.14 it/sec, obj=[56.73716821 11.35918483]]
INFO - 20:39:11: 57%|█████▋ | 57/100 [00:00<00:00, 4550.96 it/sec, obj=[58.33058905 10.70125973]]
INFO - 20:39:11: 58%|█████▊ | 58/100 [00:00<00:00, 4553.56 it/sec, obj=[19.48697889 32.41188233]]
INFO - 20:39:11: 59%|█████▉ | 59/100 [00:00<00:00, 4558.19 it/sec, obj=[ 9.51315338 32.51410675]]
INFO - 20:39:11: 60%|██████ | 60/100 [00:00<00:00, 4559.52 it/sec, obj=[14.29503909 27.34400969]]
INFO - 20:39:11: 61%|██████ | 61/100 [00:00<00:00, 4562.28 it/sec, obj=[21.01471588 31.54351738]]
INFO - 20:39:11: 62%|██████▏ | 62/100 [00:00<00:00, 4555.75 it/sec, obj=[32.62517737 26.61449355]]
INFO - 20:39:11: 63%|██████▎ | 63/100 [00:00<00:00, 4560.36 it/sec, obj=[54.44405197 15.56114393]]
INFO - 20:39:11: 64%|██████▍ | 64/100 [00:00<00:00, 4566.32 it/sec, obj=[31.67860311 19.0346702 ]]
INFO - 20:39:11: 65%|██████▌ | 65/100 [00:00<00:00, 4572.25 it/sec, obj=[12.95434975 27.79409607]]
INFO - 20:39:11: 66%|██████▌ | 66/100 [00:00<00:00, 4570.32 it/sec, obj=[33.53586239 22.39126054]]
INFO - 20:39:11: 67%|██████▋ | 67/100 [00:00<00:00, 4573.64 it/sec, obj=[13.77290554 32.89122016]]
INFO - 20:39:11: 68%|██████▊ | 68/100 [00:00<00:00, 4579.60 it/sec, obj=[24.55394276 22.04978627]]
INFO - 20:39:11: 69%|██████▉ | 69/100 [00:00<00:00, 4584.74 it/sec, obj=[38.26118694 15.93322028]]
INFO - 20:39:11: 70%|███████ | 70/100 [00:00<00:00, 4591.04 it/sec, obj=[69.87762034 24.69083953]]
INFO - 20:39:11: 71%|███████ | 71/100 [00:00<00:00, 4590.30 it/sec, obj=[117.708931 4.51806144]]
INFO - 20:39:11: 72%|███████▏ | 72/100 [00:00<00:00, 4593.77 it/sec, obj=[24.18294227 29.58378487]]
INFO - 20:39:11: 73%|███████▎ | 73/100 [00:00<00:00, 4599.22 it/sec, obj=[65.41388231 10.29184587]]
INFO - 20:39:11: 74%|███████▍ | 74/100 [00:00<00:00, 4603.86 it/sec, obj=[12.24819086 35.25245724]]
INFO - 20:39:11: 75%|███████▌ | 75/100 [00:00<00:00, 4608.99 it/sec, obj=[25.31003032 24.72346394]]
INFO - 20:39:11: 76%|███████▌ | 76/100 [00:00<00:00, 4605.46 it/sec, obj=[49.90530196 13.18421478]]
INFO - 20:39:11: 77%|███████▋ | 77/100 [00:00<00:00, 4608.80 it/sec, obj=[13.58752584 29.15968355]]
INFO - 20:39:11: 78%|███████▊ | 78/100 [00:00<00:00, 4609.39 it/sec, obj=[ 7.8547085 33.00337593]]
INFO - 20:39:11: 79%|███████▉ | 79/100 [00:00<00:00, 4612.27 it/sec, obj=[34.10323875 25.61801713]]
INFO - 20:39:11: 80%|████████ | 80/100 [00:00<00:00, 4617.56 it/sec, obj=[50.66487602 26.86160736]]
INFO - 20:39:11: 81%|████████ | 81/100 [00:00<00:00, 4616.64 it/sec, obj=[19.81349957 25.42966204]]
INFO - 20:39:11: 82%|████████▏ | 82/100 [00:00<00:00, 4620.83 it/sec, obj=[ 7.4546377 34.97394601]]
INFO - 20:39:11: 83%|████████▎ | 83/100 [00:00<00:00, 4625.91 it/sec, obj=[106.96361831 5.44850984]]
INFO - 20:39:11: 84%|████████▍ | 84/100 [00:00<00:00, 4631.00 it/sec, obj=[ 5.36344422 38.7517519 ]]
INFO - 20:39:11: 85%|████████▌ | 85/100 [00:00<00:00, 4636.40 it/sec, obj=[49.116045 14.30366519]]
INFO - 20:39:11: 86%|████████▌ | 86/100 [00:00<00:00, 4634.23 it/sec, obj=[65.46370949 9.53265669]]
INFO - 20:39:11: 87%|████████▋ | 87/100 [00:00<00:00, 4635.24 it/sec, obj=[34.43204876 18.46542147]]
INFO - 20:39:11: 88%|████████▊ | 88/100 [00:00<00:00, 4639.25 it/sec, obj=[89.05896719 23.04880076]]
INFO - 20:39:11: 89%|████████▉ | 89/100 [00:00<00:00, 4642.20 it/sec, obj=[28.30965373 24.49805668]]
INFO - 20:39:11: 90%|█████████ | 90/100 [00:00<00:00, 4645.37 it/sec, obj=[39.10277528 23.64680578]]
INFO - 20:39:11: 91%|█████████ | 91/100 [00:00<00:00, 4642.54 it/sec, obj=[117.48545685 8.18526365]]
INFO - 20:39:11: 92%|█████████▏| 92/100 [00:00<00:00, 4645.53 it/sec, obj=[66.02068516 15.50045449]]
INFO - 20:39:11: 93%|█████████▎| 93/100 [00:00<00:00, 4649.12 it/sec, obj=[67.39952092 12.92818287]]
INFO - 20:39:11: 94%|█████████▍| 94/100 [00:00<00:00, 4651.49 it/sec, obj=[55.28167693 13.5771757 ]]
INFO - 20:39:11: 95%|█████████▌| 95/100 [00:00<00:00, 4648.54 it/sec, obj=[17.335399 26.84155459]]
INFO - 20:39:11: 96%|█████████▌| 96/100 [00:00<00:00, 4650.11 it/sec, obj=[ 4.93473731 35.93049995]]
INFO - 20:39:11: 97%|█████████▋| 97/100 [00:00<00:00, 4653.51 it/sec, obj=[25.16386975 26.89754332]]
INFO - 20:39:11: 98%|█████████▊| 98/100 [00:00<00:00, 4656.33 it/sec, obj=[81.61206359 10.43246637]]
INFO - 20:39:11: 99%|█████████▉| 99/100 [00:00<00:00, 4657.83 it/sec, obj=[40.83406128 15.59204339]]
INFO - 20:39:11: 100%|██████████| 100/100 [00:00<00:00, 4654.13 it/sec, obj=[52.02690192 12.02601654]]
INFO - 20:39:11: Optimization result:
INFO - 20:39:11: Optimizer info:
INFO - 20:39:11: Status: None
INFO - 20:39:11: Message: None
INFO - 20:39:11: Number of calls to the objective function by the optimizer: 0
INFO - 20:39:11: Solution:
INFO - 20:39:11: The solution is feasible.
INFO - 20:39:11: Objective: 30.39964825035985
INFO - 20:39:11: Standardized constraints:
INFO - 20:39:11: ineq1 = [-11.77907222]
INFO - 20:39:11: ineq2 = [-38.26307397]
INFO - 20:39:11: Design space:
INFO - 20:39:11: +------+-------------+-------------------+-------------+-------+
INFO - 20:39:11: | Name | Lower bound | Value | Upper bound | Type |
INFO - 20:39:11: +------+-------------+-------------------+-------------+-------+
INFO - 20:39:11: | x | 0 | 1.542975634014225 | 5 | float |
INFO - 20:39:11: | y | 0 | 1.269910308585573 | 3 | float |
INFO - 20:39:11: +------+-------------+-------------------+-------------+-------+
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,
)
<gemseo.post.pareto_front.ParetoFront object at 0x785916bd4140>
Total running time of the script: (0 minutes 0.265 seconds)