Note

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Analytical test case # 3#

In this example, we consider a simple optimization problem to illustrate algorithms interfaces and DOE libraries integration. Integer variables are used

Imports#

from __future__ import annotations

from numpy import sum as np_sum

from gemseo import execute_algo
from gemseo import execute_post
from gemseo import get_available_doe_algorithms
from gemseo import get_available_opt_algorithms
from gemseo.algos.design_space import DesignSpace
from gemseo.algos.optimization_problem import OptimizationProblem
from gemseo.core.mdo_functions.mdo_function import MDOFunction

Define the objective function#

We define the objective function \(f(x)=\sum_{i=1}^dx_i\) using an MDOFunction.

objective = MDOFunction(np_sum, name="f", expr="sum(x)")

Define the design space#

Then, we define the DesignSpace with GEMSEO.

design_space = DesignSpace()
design_space.add_variable("x", 2, lower_bound=-5, upper_bound=5, type_="integer")

Define the optimization problem#

Then, we define the OptimizationProblem with GEMSEO.

problem = OptimizationProblem(design_space)
problem.objective = objective

Solve the optimization problem using a DOE algorithm#

We can see this optimization problem as a trade-off and solve it by means of a design of experiments (DOE), e.g. full factorial design

execute_algo(problem, algo_name="PYDOE_FULLFACT", n_samples=11**2, algo_type="doe")

INFO - 16:24:28: Optimization problem:
INFO - 16:24:28:    minimize f = sum(x)
INFO - 16:24:28:    with respect to x
INFO - 16:24:28:    over the design space:
INFO - 16:24:28:       +------+-------------+-------+-------------+---------+
INFO - 16:24:28:       | Name | Lower bound | Value | Upper bound | Type    |
INFO - 16:24:28:       +------+-------------+-------+-------------+---------+
INFO - 16:24:28:       | x[0] |      -5     |  None |      5      | integer |
INFO - 16:24:28:       | x[1] |      -5     |  None |      5      | integer |
INFO - 16:24:28:       +------+-------------+-------+-------------+---------+
INFO - 16:24:28: Solving optimization problem with algorithm PYDOE_FULLFACT:
INFO - 16:24:28:      1%|          | 1/121 [00:00<00:00, 4793.49 it/sec, feas=True, obj=-10]
INFO - 16:24:28:      2%|▏         | 2/121 [00:00<00:00, 4874.26 it/sec, feas=True, obj=-9]
INFO - 16:24:28:      2%|▏         | 3/121 [00:00<00:00, 5197.40 it/sec, feas=True, obj=-8]
INFO - 16:24:28:      3%|▎         | 4/121 [00:00<00:00, 5488.13 it/sec, feas=True, obj=-7]
INFO - 16:24:28:      4%|▍         | 5/121 [00:00<00:00, 5389.75 it/sec, feas=True, obj=-6]
INFO - 16:24:28:      5%|▍         | 6/121 [00:00<00:00, 5301.42 it/sec, feas=True, obj=-5]
INFO - 16:24:28:      6%|▌         | 7/121 [00:00<00:00, 5450.18 it/sec, feas=True, obj=-4]
INFO - 16:24:28:      7%|▋         | 8/121 [00:00<00:00, 5586.82 it/sec, feas=True, obj=-3]
INFO - 16:24:28:      7%|▋         | 9/121 [00:00<00:00, 5711.72 it/sec, feas=True, obj=-2]
INFO - 16:24:28:      8%|▊         | 10/121 [00:00<00:00, 5831.09 it/sec, feas=True, obj=-1]
INFO - 16:24:28:      9%|▉         | 11/121 [00:00<00:00, 5937.12 it/sec, feas=True, obj=0]
INFO - 16:24:28:     10%|▉         | 12/121 [00:00<00:00, 6021.97 it/sec, feas=True, obj=-9]
INFO - 16:24:28:     11%|█         | 13/121 [00:00<00:00, 6040.32 it/sec, feas=True, obj=-8]
INFO - 16:24:28:     12%|█▏        | 14/121 [00:00<00:00, 6097.01 it/sec, feas=True, obj=-7]
INFO - 16:24:28:     12%|█▏        | 15/121 [00:00<00:00, 6167.49 it/sec, feas=True, obj=-6]
INFO - 16:24:28:     13%|█▎        | 16/121 [00:00<00:00, 6224.16 it/sec, feas=True, obj=-5]
INFO - 16:24:28:     14%|█▍        | 17/121 [00:00<00:00, 6273.37 it/sec, feas=True, obj=-4]
INFO - 16:24:28:     15%|█▍        | 18/121 [00:00<00:00, 6325.19 it/sec, feas=True, obj=-3]
INFO - 16:24:28:     16%|█▌        | 19/121 [00:00<00:00, 6375.85 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     17%|█▋        | 20/121 [00:00<00:00, 6376.74 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     17%|█▋        | 21/121 [00:00<00:00, 6405.38 it/sec, feas=True, obj=0]
INFO - 16:24:28:     18%|█▊        | 22/121 [00:00<00:00, 6446.01 it/sec, feas=True, obj=1]
INFO - 16:24:28:     19%|█▉        | 23/121 [00:00<00:00, 6483.13 it/sec, feas=True, obj=-8]
INFO - 16:24:28:     20%|█▉        | 24/121 [00:00<00:00, 6498.18 it/sec, feas=True, obj=-7]
INFO - 16:24:28:     21%|██        | 25/121 [00:00<00:00, 6465.91 it/sec, feas=True, obj=-6]
INFO - 16:24:28:     21%|██▏       | 26/121 [00:00<00:00, 6436.02 it/sec, feas=True, obj=-5]
INFO - 16:24:28:     22%|██▏       | 27/121 [00:00<00:00, 6424.59 it/sec, feas=True, obj=-4]
INFO - 16:24:28:     23%|██▎       | 28/121 [00:00<00:00, 6449.23 it/sec, feas=True, obj=-3]
INFO - 16:24:28:     24%|██▍       | 29/121 [00:00<00:00, 6478.90 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     25%|██▍       | 30/121 [00:00<00:00, 6505.82 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     26%|██▌       | 31/121 [00:00<00:00, 6531.21 it/sec, feas=True, obj=0]
INFO - 16:24:28:     26%|██▋       | 32/121 [00:00<00:00, 6552.00 it/sec, feas=True, obj=1]
INFO - 16:24:28:     27%|██▋       | 33/121 [00:00<00:00, 6577.58 it/sec, feas=True, obj=2]
INFO - 16:24:28:     28%|██▊       | 34/121 [00:00<00:00, 6572.02 it/sec, feas=True, obj=-7]
INFO - 16:24:28:     29%|██▉       | 35/121 [00:00<00:00, 6584.76 it/sec, feas=True, obj=-6]
INFO - 16:24:28:     30%|██▉       | 36/121 [00:00<00:00, 6599.14 it/sec, feas=True, obj=-5]
INFO - 16:24:28:     31%|███       | 37/121 [00:00<00:00, 6615.06 it/sec, feas=True, obj=-4]
INFO - 16:24:28:     31%|███▏      | 38/121 [00:00<00:00, 6630.48 it/sec, feas=True, obj=-3]
INFO - 16:24:28:     32%|███▏      | 39/121 [00:00<00:00, 6649.78 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     33%|███▎      | 40/121 [00:00<00:00, 6664.24 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     34%|███▍      | 41/121 [00:00<00:00, 6657.88 it/sec, feas=True, obj=0]
INFO - 16:24:28:     35%|███▍      | 42/121 [00:00<00:00, 6668.21 it/sec, feas=True, obj=1]
INFO - 16:24:28:     36%|███▌      | 43/121 [00:00<00:00, 6682.79 it/sec, feas=True, obj=2]
INFO - 16:24:28:     36%|███▋      | 44/121 [00:00<00:00, 6697.49 it/sec, feas=True, obj=3]
INFO - 16:24:28:     37%|███▋      | 45/121 [00:00<00:00, 6714.71 it/sec, feas=True, obj=-6]
INFO - 16:24:28:     38%|███▊      | 46/121 [00:00<00:00, 6732.67 it/sec, feas=True, obj=-5]
INFO - 16:24:28:     39%|███▉      | 47/121 [00:00<00:00, 6745.56 it/sec, feas=True, obj=-4]
INFO - 16:24:28:     40%|███▉      | 48/121 [00:00<00:00, 6743.25 it/sec, feas=True, obj=-3]
INFO - 16:24:28:     40%|████      | 49/121 [00:00<00:00, 6747.46 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     41%|████▏     | 50/121 [00:00<00:00, 6759.77 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     42%|████▏     | 51/121 [00:00<00:00, 6773.36 it/sec, feas=True, obj=0]
INFO - 16:24:28:     43%|████▎     | 52/121 [00:00<00:00, 6778.25 it/sec, feas=True, obj=1]
INFO - 16:24:28:     44%|████▍     | 53/121 [00:00<00:00, 6781.93 it/sec, feas=True, obj=2]
INFO - 16:24:28:     45%|████▍     | 54/121 [00:00<00:00, 6793.82 it/sec, feas=True, obj=3]
INFO - 16:24:28:     45%|████▌     | 55/121 [00:00<00:00, 6791.30 it/sec, feas=True, obj=4]
INFO - 16:24:28:     46%|████▋     | 56/121 [00:00<00:00, 6793.18 it/sec, feas=True, obj=-5]
INFO - 16:24:28:     47%|████▋     | 57/121 [00:00<00:00, 6805.25 it/sec, feas=True, obj=-4]
INFO - 16:24:28:     48%|████▊     | 58/121 [00:00<00:00, 6816.95 it/sec, feas=True, obj=-3]
INFO - 16:24:28:     49%|████▉     | 59/121 [00:00<00:00, 6829.04 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     50%|████▉     | 60/121 [00:00<00:00, 6837.80 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     50%|█████     | 61/121 [00:00<00:00, 6844.27 it/sec, feas=True, obj=0]
INFO - 16:24:28:     51%|█████     | 62/121 [00:00<00:00, 6835.24 it/sec, feas=True, obj=1]
INFO - 16:24:28:     52%|█████▏    | 63/121 [00:00<00:00, 6737.92 it/sec, feas=True, obj=2]
INFO - 16:24:28:     53%|█████▎    | 64/121 [00:00<00:00, 6731.75 it/sec, feas=True, obj=3]
INFO - 16:24:28:     54%|█████▎    | 65/121 [00:00<00:00, 6735.76 it/sec, feas=True, obj=4]
INFO - 16:24:28:     55%|█████▍    | 66/121 [00:00<00:00, 6740.30 it/sec, feas=True, obj=5]
INFO - 16:24:28:     55%|█████▌    | 67/121 [00:00<00:00, 6744.87 it/sec, feas=True, obj=-4]
INFO - 16:24:28:     56%|█████▌    | 68/121 [00:00<00:00, 6750.12 it/sec, feas=True, obj=-3]
INFO - 16:24:28:     57%|█████▋    | 69/121 [00:00<00:00, 6741.06 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     58%|█████▊    | 70/121 [00:00<00:00, 6747.28 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     59%|█████▊    | 71/121 [00:00<00:00, 6754.42 it/sec, feas=True, obj=0]
INFO - 16:24:28:     60%|█████▉    | 72/121 [00:00<00:00, 6757.29 it/sec, feas=True, obj=1]
INFO - 16:24:28:     60%|██████    | 73/121 [00:00<00:00, 6765.16 it/sec, feas=True, obj=2]
INFO - 16:24:28:     61%|██████    | 74/121 [00:00<00:00, 6773.27 it/sec, feas=True, obj=3]
INFO - 16:24:28:     62%|██████▏   | 75/121 [00:00<00:00, 6779.44 it/sec, feas=True, obj=4]
INFO - 16:24:28:     63%|██████▎   | 76/121 [00:00<00:00, 6770.61 it/sec, feas=True, obj=5]
INFO - 16:24:28:     64%|██████▎   | 77/121 [00:00<00:00, 6774.80 it/sec, feas=True, obj=6]
INFO - 16:24:28:     64%|██████▍   | 78/121 [00:00<00:00, 6781.13 it/sec, feas=True, obj=-3]
INFO - 16:24:28:     65%|██████▌   | 79/121 [00:00<00:00, 6779.82 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     66%|██████▌   | 80/121 [00:00<00:00, 6783.74 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     67%|██████▋   | 81/121 [00:00<00:00, 6787.31 it/sec, feas=True, obj=0]
INFO - 16:24:28:     68%|██████▊   | 82/121 [00:00<00:00, 6794.27 it/sec, feas=True, obj=1]
INFO - 16:24:28:     69%|██████▊   | 83/121 [00:00<00:00, 6787.96 it/sec, feas=True, obj=2]
INFO - 16:24:28:     69%|██████▉   | 84/121 [00:00<00:00, 6796.19 it/sec, feas=True, obj=3]
INFO - 16:24:28:     70%|███████   | 85/121 [00:00<00:00, 6805.04 it/sec, feas=True, obj=4]
INFO - 16:24:28:     71%|███████   | 86/121 [00:00<00:00, 6812.15 it/sec, feas=True, obj=5]
INFO - 16:24:28:     72%|███████▏  | 87/121 [00:00<00:00, 6816.95 it/sec, feas=True, obj=6]
INFO - 16:24:28:     73%|███████▎  | 88/121 [00:00<00:00, 6823.28 it/sec, feas=True, obj=7]
INFO - 16:24:28:     74%|███████▎  | 89/121 [00:00<00:00, 6831.61 it/sec, feas=True, obj=-2]
INFO - 16:24:28:     74%|███████▍  | 90/121 [00:00<00:00, 6827.65 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     75%|███████▌  | 91/121 [00:00<00:00, 6835.15 it/sec, feas=True, obj=0]
INFO - 16:24:28:     76%|███████▌  | 92/121 [00:00<00:00, 6842.74 it/sec, feas=True, obj=1]
INFO - 16:24:28:     77%|███████▋  | 93/121 [00:00<00:00, 6851.87 it/sec, feas=True, obj=2]
INFO - 16:24:28:     78%|███████▊  | 94/121 [00:00<00:00, 6860.95 it/sec, feas=True, obj=3]
INFO - 16:24:28:     79%|███████▊  | 95/121 [00:00<00:00, 6868.44 it/sec, feas=True, obj=4]
INFO - 16:24:28:     79%|███████▉  | 96/121 [00:00<00:00, 6876.61 it/sec, feas=True, obj=5]
INFO - 16:24:28:     80%|████████  | 97/121 [00:00<00:00, 6874.17 it/sec, feas=True, obj=6]
INFO - 16:24:28:     81%|████████  | 98/121 [00:00<00:00, 6875.56 it/sec, feas=True, obj=7]
INFO - 16:24:28:     82%|████████▏ | 99/121 [00:00<00:00, 6882.63 it/sec, feas=True, obj=8]
INFO - 16:24:28:     83%|████████▎ | 100/121 [00:00<00:00, 6889.12 it/sec, feas=True, obj=-1]
INFO - 16:24:28:     83%|████████▎ | 101/121 [00:00<00:00, 6895.38 it/sec, feas=True, obj=0]
INFO - 16:24:28:     84%|████████▍ | 102/121 [00:00<00:00, 6902.42 it/sec, feas=True, obj=1]
INFO - 16:24:28:     85%|████████▌ | 103/121 [00:00<00:00, 6908.45 it/sec, feas=True, obj=2]
INFO - 16:24:28:     86%|████████▌ | 104/121 [00:00<00:00, 6913.07 it/sec, feas=True, obj=3]
INFO - 16:24:28:     87%|████████▋ | 105/121 [00:00<00:00, 6908.27 it/sec, feas=True, obj=4]
INFO - 16:24:28:     88%|████████▊ | 106/121 [00:00<00:00, 6913.22 it/sec, feas=True, obj=5]
INFO - 16:24:28:     88%|████████▊ | 107/121 [00:00<00:00, 6894.29 it/sec, feas=True, obj=6]
INFO - 16:24:28:     89%|████████▉ | 108/121 [00:00<00:00, 6894.22 it/sec, feas=True, obj=7]
INFO - 16:24:28:     90%|█████████ | 109/121 [00:00<00:00, 6897.38 it/sec, feas=True, obj=8]
INFO - 16:24:28:     91%|█████████ | 110/121 [00:00<00:00, 6902.45 it/sec, feas=True, obj=9]
INFO - 16:24:28:     92%|█████████▏| 111/121 [00:00<00:00, 6905.89 it/sec, feas=True, obj=0]
INFO - 16:24:28:     93%|█████████▎| 112/121 [00:00<00:00, 6899.03 it/sec, feas=True, obj=1]
INFO - 16:24:28:     93%|█████████▎| 113/121 [00:00<00:00, 6902.14 it/sec, feas=True, obj=2]
INFO - 16:24:28:     94%|█████████▍| 114/121 [00:00<00:00, 6905.50 it/sec, feas=True, obj=3]
INFO - 16:24:28:     95%|█████████▌| 115/121 [00:00<00:00, 6909.40 it/sec, feas=True, obj=4]
INFO - 16:24:28:     96%|█████████▌| 116/121 [00:00<00:00, 6909.20 it/sec, feas=True, obj=5]
INFO - 16:24:28:     97%|█████████▋| 117/121 [00:00<00:00, 6911.55 it/sec, feas=True, obj=6]
INFO - 16:24:28:     98%|█████████▊| 118/121 [00:00<00:00, 6916.17 it/sec, feas=True, obj=7]
INFO - 16:24:28:     98%|█████████▊| 119/121 [00:00<00:00, 6910.56 it/sec, feas=True, obj=8]
INFO - 16:24:28:     99%|█████████▉| 120/121 [00:00<00:00, 6914.45 it/sec, feas=True, obj=9]
INFO - 16:24:28:    100%|██████████| 121/121 [00:00<00:00, 6822.02 it/sec, feas=True, obj=10]
INFO - 16:24:28: Optimization result:
INFO - 16:24:28:    Optimizer info:
INFO - 16:24:28:       Status: None
INFO - 16:24:28:       Message: None
INFO - 16:24:28:    Solution:
INFO - 16:24:28:       Objective: -10.0
INFO - 16:24:28:       Design space:
INFO - 16:24:28:          +------+-------------+-------+-------------+---------+
INFO - 16:24:28:          | Name | Lower bound | Value | Upper bound | Type    |
INFO - 16:24:28:          +------+-------------+-------+-------------+---------+
INFO - 16:24:28:          | x[0] |      -5     |   -5  |      5      | integer |
INFO - 16:24:28:          | x[1] |      -5     |   -5  |      5      | integer |
INFO - 16:24:28:          +------+-------------+-------+-------------+---------+
Optimization result:
  • Design variables: [-5. -5.]
  • Objective function: -10.0
  • Feasible solution: True


Post-process the results#

execute_post(
    problem,
    post_name="ScatterPlotMatrix",
    variable_names=["x", "f"],
    save=False,
    show=True,
)
plot simple opt 3
<gemseo.post.scatter_plot_matrix.ScatterPlotMatrix object at 0x7c2f8f9cc920>

Note that you can get all the optimization algorithms names:

get_available_opt_algorithms()

['Augmented_Lagrangian_order_0', 'Augmented_Lagrangian_order_1', 'Scipy_MILP', 'HEXALY', 'MMA', 'MNBI', 'MultiStart', 'NLOPT_MMA', 'NLOPT_COBYLA', 'NLOPT_SLSQP', 'NLOPT_BOBYQA', 'NLOPT_BFGS', 'NLOPT_NEWUOA', 'PDFO_COBYLA', 'PDFO_BOBYQA', 'PDFO_NEWUOA', 'PYOPTSPARSE_SLSQP', 'PYOPTSPARSE_SNOPT', 'PYMOO_GA', 'PYMOO_NSGA2', 'PYMOO_NSGA3', 'PYMOO_UNSGA3', 'PYMOO_RNSGA3', 'SMT_EGO', 'DUAL_ANNEALING', 'SHGO', 'DIFFERENTIAL_EVOLUTION', 'INTERIOR_POINT', 'DUAL_SIMPLEX', 'SLSQP', 'L-BFGS-B', 'TNC', 'NELDER-MEAD', 'COBYQA']

and all the DOE algorithms names:

get_available_doe_algorithms()

['CustomDOE', 'DiagonalDOE', 'MorrisDOE', 'OATDOE', 'OT_SOBOL', 'OT_RANDOM', 'OT_HASELGROVE', 'OT_REVERSE_HALTON', 'OT_HALTON', 'OT_FAURE', 'OT_MONTE_CARLO', 'OT_FACTORIAL', 'OT_COMPOSITE', 'OT_AXIAL', 'OT_OPT_LHS', 'OT_LHS', 'OT_LHSC', 'OT_FULLFACT', 'OT_SOBOL_INDICES', 'PYDOE_BBDESIGN', 'PYDOE_CCDESIGN', 'PYDOE_FF2N', 'PYDOE_FULLFACT', 'PYDOE_LHS', 'PYDOE_PBDESIGN', 'Halton', 'LHS', 'MC', 'PoissonDisk', 'Sobol']

Total running time of the script: (0 minutes 0.308 seconds)

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