# 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 gemseo import configure_logger
from gemseo import execute_post
from gemseo.algos.design_space import DesignSpace
from gemseo.algos.doe.doe_factory import DOEFactory
from gemseo.algos.opt_problem import OptimizationProblem
from gemseo.core.mdofunctions.mdo_function import MDOFunction
from numpy import sum as np_sum

LOGGER = configure_logger()


## Define the objective function¶

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

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


## Define the design space¶

Then, we define the DesignSpace with GEMSEO.

design_space = DesignSpace()


## 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

DOEFactory().execute(problem, "fullfact", n_samples=11**2)

    INFO - 08:37:06: Optimization problem:
INFO - 08:37:06:    minimize f = sum(x)
INFO - 08:37:06:    with respect to x
INFO - 08:37:06:    over the design space:
INFO - 08:37:06:    +------+-------------+-------+-------------+---------+
INFO - 08:37:06:    | name | lower_bound | value | upper_bound | type    |
INFO - 08:37:06:    +------+-------------+-------+-------------+---------+
INFO - 08:37:06:    | x[0] |      -5     |  None |      5      | integer |
INFO - 08:37:06:    | x[1] |      -5     |  None |      5      | integer |
INFO - 08:37:06:    +------+-------------+-------+-------------+---------+
INFO - 08:37:06: Solving optimization problem with algorithm fullfact:
INFO - 08:37:06: ...   0%|          | 0/121 [00:00<?, ?it]
INFO - 08:37:06: ...   1%|          | 1/121 [00:00<00:00, 3701.95 it/sec, obj=-10]
INFO - 08:37:06: ...   2%|▏         | 2/121 [00:00<00:00, 3247.62 it/sec, obj=-9]
INFO - 08:37:06: ...   2%|▏         | 3/121 [00:00<00:00, 3141.02 it/sec, obj=-8]
INFO - 08:37:06: ...   3%|▎         | 4/121 [00:00<00:00, 3107.47 it/sec, obj=-7]
INFO - 08:37:06: ...   4%|▍         | 5/121 [00:00<00:00, 2870.06 it/sec, obj=-6]
INFO - 08:37:06: ...   5%|▍         | 6/121 [00:00<00:00, 2873.47 it/sec, obj=-5]
INFO - 08:37:06: ...   6%|▌         | 7/121 [00:00<00:00, 2868.88 it/sec, obj=-4]
INFO - 08:37:06: ...   7%|▋         | 8/121 [00:00<00:00, 2889.88 it/sec, obj=-3]
INFO - 08:37:06: ...   7%|▋         | 9/121 [00:00<00:00, 2906.66 it/sec, obj=-2]
INFO - 08:37:06: ...   8%|▊         | 10/121 [00:00<00:00, 2923.27 it/sec, obj=-1]
INFO - 08:37:06: ...   9%|▉         | 11/121 [00:00<00:00, 2909.59 it/sec, obj=0]
INFO - 08:37:06: ...  10%|▉         | 12/121 [00:00<00:00, 2912.54 it/sec, obj=-9]
INFO - 08:37:06: ...  11%|█         | 13/121 [00:00<00:00, 2923.17 it/sec, obj=-8]
INFO - 08:37:06: ...  12%|█▏        | 14/121 [00:00<00:00, 2934.11 it/sec, obj=-7]
INFO - 08:37:06: ...  12%|█▏        | 15/121 [00:00<00:00, 2943.92 it/sec, obj=-6]
INFO - 08:37:06: ...  13%|█▎        | 16/121 [00:00<00:00, 2953.48 it/sec, obj=-5]
INFO - 08:37:06: ...  14%|█▍        | 17/121 [00:00<00:00, 2962.45 it/sec, obj=-4]
INFO - 08:37:06: ...  15%|█▍        | 18/121 [00:00<00:00, 2969.19 it/sec, obj=-3]
INFO - 08:37:06: ...  16%|█▌        | 19/121 [00:00<00:00, 2966.49 it/sec, obj=-2]
INFO - 08:37:06: ...  17%|█▋        | 20/121 [00:00<00:00, 2969.63 it/sec, obj=-1]
INFO - 08:37:06: ...  17%|█▋        | 21/121 [00:00<00:00, 2974.18 it/sec, obj=0]
INFO - 08:37:06: ...  18%|█▊        | 22/121 [00:00<00:00, 2979.20 it/sec, obj=1]
INFO - 08:37:06: ...  19%|█▉        | 23/121 [00:00<00:00, 2970.01 it/sec, obj=-8]
INFO - 08:37:06: ...  20%|█▉        | 24/121 [00:00<00:00, 2961.99 it/sec, obj=-7]
INFO - 08:37:06: ...  21%|██        | 25/121 [00:00<00:00, 2963.84 it/sec, obj=-6]
INFO - 08:37:06: ...  21%|██▏       | 26/121 [00:00<00:00, 2967.96 it/sec, obj=-5]
INFO - 08:37:06: ...  22%|██▏       | 27/121 [00:00<00:00, 2962.70 it/sec, obj=-4]
INFO - 08:37:06: ...  23%|██▎       | 28/121 [00:00<00:00, 2964.55 it/sec, obj=-3]
INFO - 08:37:06: ...  24%|██▍       | 29/121 [00:00<00:00, 2968.22 it/sec, obj=-2]
INFO - 08:37:06: ...  25%|██▍       | 30/121 [00:00<00:00, 2971.03 it/sec, obj=-1]
INFO - 08:37:06: ...  26%|██▌       | 31/121 [00:00<00:00, 2968.03 it/sec, obj=0]
INFO - 08:37:06: ...  26%|██▋       | 32/121 [00:00<00:00, 2971.13 it/sec, obj=1]
INFO - 08:37:06: ...  27%|██▋       | 33/121 [00:00<00:00, 2974.17 it/sec, obj=2]
INFO - 08:37:06: ...  28%|██▊       | 34/121 [00:00<00:00, 2976.92 it/sec, obj=-7]
INFO - 08:37:06: ...  29%|██▉       | 35/121 [00:00<00:00, 2971.55 it/sec, obj=-6]
INFO - 08:37:06: ...  30%|██▉       | 36/121 [00:00<00:00, 2970.12 it/sec, obj=-5]
INFO - 08:37:06: ...  31%|███       | 37/121 [00:00<00:00, 2972.23 it/sec, obj=-4]
INFO - 08:37:06: ...  31%|███▏      | 38/121 [00:00<00:00, 2973.85 it/sec, obj=-3]
INFO - 08:37:06: ...  32%|███▏      | 39/121 [00:00<00:00, 2975.66 it/sec, obj=-2]
INFO - 08:37:06: ...  33%|███▎      | 40/121 [00:00<00:00, 2976.95 it/sec, obj=-1]
INFO - 08:37:06: ...  34%|███▍      | 41/121 [00:00<00:00, 2978.86 it/sec, obj=0]
INFO - 08:37:06: ...  35%|███▍      | 42/121 [00:00<00:00, 2980.82 it/sec, obj=1]
INFO - 08:37:06: ...  36%|███▌      | 43/121 [00:00<00:00, 2977.73 it/sec, obj=2]
INFO - 08:37:06: ...  36%|███▋      | 44/121 [00:00<00:00, 2979.53 it/sec, obj=3]
INFO - 08:37:06: ...  37%|███▋      | 45/121 [00:00<00:00, 2981.31 it/sec, obj=-6]
INFO - 08:37:06: ...  38%|███▊      | 46/121 [00:00<00:00, 2983.84 it/sec, obj=-5]
INFO - 08:37:06: ...  39%|███▉      | 47/121 [00:00<00:00, 2981.03 it/sec, obj=-4]
INFO - 08:37:06: ...  40%|███▉      | 48/121 [00:00<00:00, 2980.67 it/sec, obj=-3]
INFO - 08:37:06: ...  40%|████      | 49/121 [00:00<00:00, 2982.37 it/sec, obj=-2]
INFO - 08:37:06: ...  41%|████▏     | 50/121 [00:00<00:00, 2984.21 it/sec, obj=-1]
INFO - 08:37:06: ...  42%|████▏     | 51/121 [00:00<00:00, 2986.35 it/sec, obj=0]
INFO - 08:37:06: ...  43%|████▎     | 52/121 [00:00<00:00, 2988.42 it/sec, obj=1]
INFO - 08:37:06: ...  44%|████▍     | 53/121 [00:00<00:00, 2990.13 it/sec, obj=2]
INFO - 08:37:06: ...  45%|████▍     | 54/121 [00:00<00:00, 2991.97 it/sec, obj=3]
INFO - 08:37:06: ...  45%|████▌     | 55/121 [00:00<00:00, 2990.38 it/sec, obj=4]
INFO - 08:37:06: ...  46%|████▋     | 56/121 [00:00<00:00, 2992.27 it/sec, obj=-5]
INFO - 08:37:06: ...  47%|████▋     | 57/121 [00:00<00:00, 2994.21 it/sec, obj=-4]
INFO - 08:37:06: ...  48%|████▊     | 58/121 [00:00<00:00, 2996.08 it/sec, obj=-3]
INFO - 08:37:06: ...  49%|████▉     | 59/121 [00:00<00:00, 2998.00 it/sec, obj=-2]
INFO - 08:37:06: ...  50%|████▉     | 60/121 [00:00<00:00, 2994.36 it/sec, obj=-1]
INFO - 08:37:06: ...  50%|█████     | 61/121 [00:00<00:00, 2995.90 it/sec, obj=0]
INFO - 08:37:06: ...  51%|█████     | 62/121 [00:00<00:00, 2997.83 it/sec, obj=1]
INFO - 08:37:06: ...  52%|█████▏    | 63/121 [00:00<00:00, 2999.50 it/sec, obj=2]
INFO - 08:37:06: ...  53%|█████▎    | 64/121 [00:00<00:00, 3001.19 it/sec, obj=3]
INFO - 08:37:06: ...  54%|█████▎    | 65/121 [00:00<00:00, 3002.93 it/sec, obj=4]
INFO - 08:37:06: ...  55%|█████▍    | 66/121 [00:00<00:00, 3004.42 it/sec, obj=5]
INFO - 08:37:06: ...  55%|█████▌    | 67/121 [00:00<00:00, 3003.23 it/sec, obj=-4]
INFO - 08:37:06: ...  56%|█████▌    | 68/121 [00:00<00:00, 3003.91 it/sec, obj=-3]
INFO - 08:37:06: ...  57%|█████▋    | 69/121 [00:00<00:00, 3005.20 it/sec, obj=-2]
INFO - 08:37:06: ...  58%|█████▊    | 70/121 [00:00<00:00, 3006.42 it/sec, obj=-1]
INFO - 08:37:06: ...  59%|█████▊    | 71/121 [00:00<00:00, 3007.70 it/sec, obj=0]
INFO - 08:37:06: ...  60%|█████▉    | 72/121 [00:00<00:00, 3004.28 it/sec, obj=1]
INFO - 08:37:06: ...  60%|██████    | 73/121 [00:00<00:00, 3005.14 it/sec, obj=2]
INFO - 08:37:06: ...  61%|██████    | 74/121 [00:00<00:00, 3006.38 it/sec, obj=3]
INFO - 08:37:06: ...  62%|██████▏   | 75/121 [00:00<00:00, 3007.65 it/sec, obj=4]
INFO - 08:37:06: ...  63%|██████▎   | 76/121 [00:00<00:00, 3008.94 it/sec, obj=5]
INFO - 08:37:06: ...  64%|██████▎   | 77/121 [00:00<00:00, 3010.26 it/sec, obj=6]
INFO - 08:37:06: ...  64%|██████▍   | 78/121 [00:00<00:00, 3011.60 it/sec, obj=-3]
INFO - 08:37:06: ...  65%|██████▌   | 79/121 [00:00<00:00, 3011.18 it/sec, obj=-2]
INFO - 08:37:06: ...  66%|██████▌   | 80/121 [00:00<00:00, 3011.47 it/sec, obj=-1]
INFO - 08:37:06: ...  67%|██████▋   | 81/121 [00:00<00:00, 3012.32 it/sec, obj=0]
INFO - 08:37:06: ...  68%|██████▊   | 82/121 [00:00<00:00, 3013.23 it/sec, obj=1]
INFO - 08:37:06: ...  69%|██████▊   | 83/121 [00:00<00:00, 3014.40 it/sec, obj=2]
INFO - 08:37:06: ...  69%|██████▉   | 84/121 [00:00<00:00, 3012.20 it/sec, obj=3]
INFO - 08:37:06: ...  70%|███████   | 85/121 [00:00<00:00, 3012.28 it/sec, obj=4]
INFO - 08:37:06: ...  71%|███████   | 86/121 [00:00<00:00, 3013.02 it/sec, obj=5]
INFO - 08:37:06: ...  72%|███████▏  | 87/121 [00:00<00:00, 3014.27 it/sec, obj=6]
INFO - 08:37:06: ...  73%|███████▎  | 88/121 [00:00<00:00, 3015.39 it/sec, obj=7]
INFO - 08:37:06: ...  74%|███████▎  | 89/121 [00:00<00:00, 3016.41 it/sec, obj=-2]
INFO - 08:37:06: ...  74%|███████▍  | 90/121 [00:00<00:00, 3017.32 it/sec, obj=-1]
INFO - 08:37:06: ...  75%|███████▌  | 91/121 [00:00<00:00, 3017.13 it/sec, obj=0]
INFO - 08:37:06: ...  76%|███████▌  | 92/121 [00:00<00:00, 3016.99 it/sec, obj=1]
INFO - 08:37:06: ...  77%|███████▋  | 93/121 [00:00<00:00, 3017.72 it/sec, obj=2]
INFO - 08:37:06: ...  78%|███████▊  | 94/121 [00:00<00:00, 3018.27 it/sec, obj=3]
INFO - 08:37:06: ...  79%|███████▊  | 95/121 [00:00<00:00, 3018.95 it/sec, obj=4]
INFO - 08:37:06: ...  79%|███████▉  | 96/121 [00:00<00:00, 3017.28 it/sec, obj=5]
INFO - 08:37:06: ...  80%|████████  | 97/121 [00:00<00:00, 3017.10 it/sec, obj=6]
INFO - 08:37:06: ...  81%|████████  | 98/121 [00:00<00:00, 3017.80 it/sec, obj=7]
INFO - 08:37:06: ...  82%|████████▏ | 99/121 [00:00<00:00, 3006.89 it/sec, obj=8]
INFO - 08:37:06: ...  83%|████████▎ | 100/121 [00:00<00:00, 2991.27 it/sec, obj=-1]
INFO - 08:37:06: ...  83%|████████▎ | 101/121 [00:00<00:00, 2989.08 it/sec, obj=0]
INFO - 08:37:06: ...  84%|████████▍ | 102/121 [00:00<00:00, 2989.07 it/sec, obj=1]
INFO - 08:37:06: ...  85%|████████▌ | 103/121 [00:00<00:00, 2986.98 it/sec, obj=2]
INFO - 08:37:06: ...  86%|████████▌ | 104/121 [00:00<00:00, 2987.15 it/sec, obj=3]
INFO - 08:37:06: ...  87%|████████▋ | 105/121 [00:00<00:00, 2987.70 it/sec, obj=4]
INFO - 08:37:06: ...  88%|████████▊ | 106/121 [00:00<00:00, 2988.48 it/sec, obj=5]
INFO - 08:37:06: ...  88%|████████▊ | 107/121 [00:00<00:00, 2987.06 it/sec, obj=6]
INFO - 08:37:06: ...  89%|████████▉ | 108/121 [00:00<00:00, 2987.18 it/sec, obj=7]
INFO - 08:37:06: ...  90%|█████████ | 109/121 [00:00<00:00, 2987.71 it/sec, obj=8]
INFO - 08:37:06: ...  91%|█████████ | 110/121 [00:00<00:00, 2988.32 it/sec, obj=9]
INFO - 08:37:06: ...  92%|█████████▏| 111/121 [00:00<00:00, 2989.12 it/sec, obj=0]
INFO - 08:37:06: ...  93%|█████████▎| 112/121 [00:00<00:00, 2989.91 it/sec, obj=1]
INFO - 08:37:06: ...  93%|█████████▎| 113/121 [00:00<00:00, 2990.60 it/sec, obj=2]
INFO - 08:37:06: ...  94%|█████████▍| 114/121 [00:00<00:00, 2991.47 it/sec, obj=3]
INFO - 08:37:06: ...  95%|█████████▌| 115/121 [00:00<00:00, 2990.51 it/sec, obj=4]
INFO - 08:37:06: ...  96%|█████████▌| 116/121 [00:00<00:00, 2991.12 it/sec, obj=5]
INFO - 08:37:06: ...  97%|█████████▋| 117/121 [00:00<00:00, 2991.95 it/sec, obj=6]
INFO - 08:37:06: ...  98%|█████████▊| 118/121 [00:00<00:00, 2992.83 it/sec, obj=7]
INFO - 08:37:06: ...  98%|█████████▊| 119/121 [00:00<00:00, 2991.73 it/sec, obj=8]
INFO - 08:37:06: ...  99%|█████████▉| 120/121 [00:00<00:00, 2991.75 it/sec, obj=9]
INFO - 08:37:06: ... 100%|██████████| 121/121 [00:00<00:00, 2992.40 it/sec, obj=10]
INFO - 08:37:06: Optimization result:
INFO - 08:37:06:    Optimizer info:
INFO - 08:37:06:       Status: None
INFO - 08:37:06:       Message: None
INFO - 08:37:06:       Number of calls to the objective function by the optimizer: 121
INFO - 08:37:06:    Solution:
INFO - 08:37:06:       Objective: -10.0
INFO - 08:37:06:       Design space:
INFO - 08:37:06:       +------+-------------+-------+-------------+---------+
INFO - 08:37:06:       | name | lower_bound | value | upper_bound | type    |
INFO - 08:37:06:       +------+-------------+-------+-------------+---------+
INFO - 08:37:06:       | x[0] |      -5     |   -5  |      5      | integer |
INFO - 08:37:06:       | x[1] |      -5     |   -5  |      5      | integer |
INFO - 08:37:06:       +------+-------------+-------+-------------+---------+

Optimization result:
Design variables: [-5. -5.]
Objective function: -10.0
Feasible solution: True


## Post-process the results¶

execute_post(
problem,
"ScatterPlotMatrix",
variable_names=["x", "f"],
save=False,
show=True,
)

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/5.0.1/lib/python3.9/site-packages/genson/schema/strategies/base.py:42: UserWarning: Schema incompatible. Keyword 'description' has conflicting values ('The width and height of the figure in inches, e.g. (w, h).\nIf None, use the :attr:.OptPostProcessor.DEFAULT_FIG_SIZE\nof the post-processor.' vs. 'The width and height of the figure in inches, e.g. (w, h).\nIf None, use the :attr:.OptPostProcessor.DEFAULT_FIG_SIZE\nof the post-processor.'). Using 'The width and height of the figure in inches, e.g. (w, h).\nIf None, use the :attr:.OptPostProcessor.DEFAULT_FIG_SIZE\nof the post-processor.'
warn(('Schema incompatible. Keyword {0!r} has conflicting '

<gemseo.post.scatter_mat.ScatterPlotMatrix object at 0x7f87770812b0>


Note that you can get all the optimization algorithms names:

algo_list = DOEFactory().algorithms
print("Available algorithms ", algo_list)

Available algorithms  ['CustomDOE', 'DiagonalDOE', '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', 'fullfact', 'ff2n', 'pbdesign', 'bbdesign', 'ccdesign', 'lhs']


Total running time of the script: ( 0 minutes 0.761 seconds)

Gallery generated by Sphinx-Gallery