Note

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Multi-start optimization#

The optimization algorithm multistart generates starting points using a DOE algorithm and run a sub-optimization algorithm from each starting point.

from __future__ import annotations

from gemseo import configure_logger
from gemseo import create_design_space
from gemseo import create_discipline
from gemseo import create_scenario
from gemseo import execute_post
from gemseo.algos.opt.multi_start.settings.multi_start_settings import (
    MultiStart_Settings,
)

configure_logger()

<RootLogger root (INFO)>

First, we create the disciplines

objective = create_discipline("AnalyticDiscipline", expressions={"obj": "x**3-x+1"})
constraint = create_discipline(
    "AnalyticDiscipline", expressions={"cstr": "x**2+obj**2-1.5"}
)

and the design space

design_space = create_design_space()
design_space.add_variable("x", lower_bound=-1.5, upper_bound=1.5, value=1.5)

Then, we define the MDO scenario

scenario = create_scenario(
    [objective, constraint],
    "obj",
    design_space,
    formulation_name="DisciplinaryOpt",
)

Note that the formulation settings passed to create_scenario() can be provided via a Pydantic model. For more information, see Formulation Settings.

scenario.add_constraint("cstr", constraint_type="ineq")

and execute it with the MultiStart optimization algorithm combining the local optimization algorithm SLSQP and the full-factorial DOE algorithm:

multistart_settings = MultiStart_Settings(
    max_iter=100,
    opt_algo_name="SLSQP",
    doe_algo_name="PYDOE_FULLFACT",
    n_start=10,
    # Set multistart_file_path to save the history of the local optima.
    multistart_file_path="multistart.hdf5",
)
scenario.execute(multistart_settings)

   INFO - 20:38:20: *** Start MDOScenario execution ***
   INFO - 20:38:20: MDOScenario
   INFO - 20:38:20:    Disciplines: AnalyticDiscipline AnalyticDiscipline
   INFO - 20:38:20:    MDO formulation: DisciplinaryOpt
   INFO - 20:38:20: Optimization problem:
   INFO - 20:38:20:    minimize obj(x)
   INFO - 20:38:20:    with respect to x
   INFO - 20:38:20:    under the inequality constraints
   INFO - 20:38:20:       cstr(x) <= 0
   INFO - 20:38:20:    over the design space:
   INFO - 20:38:20:       +------+-------------+-------+-------------+-------+
   INFO - 20:38:20:       | Name | Lower bound | Value | Upper bound | Type  |
   INFO - 20:38:20:       +------+-------------+-------+-------------+-------+
   INFO - 20:38:20:       | x    |     -1.5    |  1.5  |     1.5     | float |
   INFO - 20:38:20:       +------+-------------+-------+-------------+-------+
   INFO - 20:38:20: Solving optimization problem with algorithm MultiStart:
   INFO - 20:38:20:      1%|          | 1/100 [00:00<00:00, 329.87 it/sec, obj=2.88]
   INFO - 20:38:20:      2%|▏         | 2/100 [00:00<00:00, 424.95 it/sec, obj=-0.875]
   INFO - 20:38:20:      3%|▎         | 3/100 [00:00<00:00, 464.59 it/sec, obj=-0.267]
   INFO - 20:38:20:      4%|▍         | 4/100 [00:00<00:00, 571.63 it/sec, obj=-0.809]
   INFO - 20:38:20:      5%|▌         | 5/100 [00:00<00:00, 660.50 it/sec, obj=-0.868]
   INFO - 20:38:20:      6%|▌         | 6/100 [00:00<00:00, 741.46 it/sec, obj=-0.872]
   INFO - 20:38:20:      7%|▋         | 7/100 [00:00<00:00, 775.37 it/sec, obj=-0.265]
   INFO - 20:38:20:      8%|▊         | 8/100 [00:00<00:00, 801.68 it/sec, obj=0.136]
   INFO - 20:38:20:      9%|▉         | 9/100 [00:00<00:00, 824.26 it/sec, obj=0.579]
   INFO - 20:38:20:     10%|█         | 10/100 [00:00<00:00, 870.53 it/sec, obj=0.289]
   INFO - 20:38:20:     11%|█         | 11/100 [00:00<00:00, 885.43 it/sec, obj=1.25]
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
   INFO - 20:38:20:     12%|█▏        | 12/100 [00:00<00:00, 853.50 it/sec, obj=0.579]
   INFO - 20:38:20:     13%|█▎        | 13/100 [00:00<00:00, 838.13 it/sec, obj=-0.875]
   INFO - 20:38:20:     14%|█▍        | 14/100 [00:00<00:00, 850.14 it/sec, obj=-0.267]
   INFO - 20:38:20:     15%|█▌        | 15/100 [00:00<00:00, 881.54 it/sec, obj=-0.809]
   INFO - 20:38:20:     16%|█▌        | 16/100 [00:00<00:00, 910.42 it/sec, obj=-0.868]
   INFO - 20:38:20:     17%|█▋        | 17/100 [00:00<00:00, 939.21 it/sec, obj=-0.874]
   INFO - 20:38:20:     18%|█▊        | 18/100 [00:00<00:00, 944.32 it/sec, obj=-0.266]
   INFO - 20:38:20:     19%|█▉        | 19/100 [00:00<00:00, 949.55 it/sec, obj=0.135]
   INFO - 20:38:20:     20%|██        | 20/100 [00:00<00:00, 953.03 it/sec, obj=0.577]
   INFO - 20:38:20:     21%|██        | 21/100 [00:00<00:00, 973.90 it/sec, obj=0.288]
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
   INFO - 20:38:20:     22%|██▏       | 22/100 [00:00<00:00, 930.92 it/sec, obj=1.25]
   INFO - 20:38:20:     23%|██▎       | 23/100 [00:00<00:00, 922.20 it/sec, obj=-0.875]
   INFO - 20:38:20:     24%|██▍       | 24/100 [00:00<00:00, 942.06 it/sec, obj=1.01]
   INFO - 20:38:20:     25%|██▌       | 25/100 [00:00<00:00, 946.04 it/sec, obj=-0.875]
   INFO - 20:38:20:     26%|██▌       | 26/100 [00:00<00:00, 964.34 it/sec, obj=0.819]
   INFO - 20:38:20:     27%|██▋       | 27/100 [00:00<00:00, 968.50 it/sec, obj=-0.875]
   INFO - 20:38:20:     28%|██▊       | 28/100 [00:00<00:00, 984.87 it/sec, obj=0.693]
   INFO - 20:38:20:     29%|██▉       | 29/100 [00:00<00:00, 982.28 it/sec, obj=0.584]
   INFO - 20:38:20:     30%|███       | 30/100 [00:00<00:00, 985.68 it/sec, obj=-0.875]
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
   INFO - 20:38:20:     31%|███       | 31/100 [00:00<00:00, 962.53 it/sec, obj=1.38]
   INFO - 20:38:20:     32%|███▏      | 32/100 [00:00<00:00, 950.65 it/sec, obj=1.23]
   INFO - 20:38:20:     33%|███▎      | 33/100 [00:00<00:00, 953.00 it/sec, obj=2.87]
   INFO - 20:38:20:     34%|███▍      | 34/100 [00:00<00:00, 965.97 it/sec, obj=0.848]
   INFO - 20:38:20:     35%|███▌      | 35/100 [00:00<00:00, 967.58 it/sec, obj=0.658]
   INFO - 20:38:20:     36%|███▌      | 36/100 [00:00<00:00, 960.08 it/sec, obj=0.643]
   INFO - 20:38:20:     37%|███▋      | 37/100 [00:00<00:00, 954.73 it/sec, obj=0.616]
   INFO - 20:38:20:     38%|███▊      | 38/100 [00:00<00:00, 949.42 it/sec, obj=0.615]
   INFO - 20:38:20:     39%|███▉      | 39/100 [00:00<00:00, 944.43 it/sec, obj=0.615]
   INFO - 20:38:20:     40%|████      | 40/100 [00:00<00:00, 915.65 it/sec, obj=1.16]
   INFO - 20:38:20:     41%|████      | 41/100 [00:00<00:00, 902.72 it/sec, obj=2.87]
   INFO - 20:38:20:     42%|████▏     | 42/100 [00:00<00:00, 909.73 it/sec, obj=0.785]
   INFO - 20:38:20:     43%|████▎     | 43/100 [00:00<00:00, 898.80 it/sec, obj=0.644]
   INFO - 20:38:20:     44%|████▍     | 44/100 [00:00<00:00, 894.98 it/sec, obj=0.624]
   INFO - 20:38:20:     45%|████▌     | 45/100 [00:00<00:00, 891.95 it/sec, obj=0.615]
   INFO - 20:38:20:     46%|████▌     | 46/100 [00:00<00:00, 889.02 it/sec, obj=0.615]
   INFO - 20:38:20:     47%|████▋     | 47/100 [00:00<00:00, 886.55 it/sec, obj=0.615]
   INFO - 20:38:20:     48%|████▊     | 48/100 [00:00<00:00, 884.04 it/sec, obj=0.615]
   INFO - 20:38:20:     49%|████▉     | 49/100 [00:00<00:00, 863.90 it/sec, obj=0.838]
   INFO - 20:38:20:     50%|█████     | 50/100 [00:00<00:00, 856.39 it/sec, obj=0.656]
   INFO - 20:38:20:     51%|█████     | 51/100 [00:00<00:00, 854.35 it/sec, obj=0.639]
   INFO - 20:38:20:     52%|█████▏    | 52/100 [00:00<00:00, 852.77 it/sec, obj=0.616]
   INFO - 20:38:20:     53%|█████▎    | 53/100 [00:00<00:00, 851.43 it/sec, obj=0.615]
   INFO - 20:38:20:     54%|█████▍    | 54/100 [00:00<00:00, 849.88 it/sec, obj=0.615]
   INFO - 20:38:20:     55%|█████▌    | 55/100 [00:00<00:00, 848.98 it/sec, obj=0.615]
   INFO - 20:38:20:     56%|█████▌    | 56/100 [00:00<00:00, 847.84 it/sec, obj=0.615]
   INFO - 20:38:20:     57%|█████▋    | 57/100 [00:00<00:00, 846.54 it/sec, obj=0.615]
   INFO - 20:38:20:     58%|█████▊    | 58/100 [00:00<00:00, 836.14 it/sec, obj=0.625]
   INFO - 20:38:20:     59%|█████▉    | 59/100 [00:00<00:00, 831.79 it/sec, obj=2.87]
   INFO - 20:38:20:     60%|██████    | 60/100 [00:00<00:00, 837.16 it/sec, obj=0.616]
   INFO - 20:38:20:     61%|██████    | 61/100 [00:00<00:00, 836.19 it/sec, obj=0.615]
   INFO - 20:38:20:     62%|██████▏   | 62/100 [00:00<00:00, 834.98 it/sec, obj=0.615]
   INFO - 20:38:20:     63%|██████▎   | 63/100 [00:00<00:00, 834.27 it/sec, obj=0.615]
   INFO - 20:38:20:     64%|██████▍   | 64/100 [00:00<00:00, 833.93 it/sec, obj=0.615]
   INFO - 20:38:20:     65%|██████▌   | 65/100 [00:00<00:00, 833.55 it/sec, obj=0.615]
   INFO - 20:38:20:     66%|██████▌   | 66/100 [00:00<00:00, 836.99 it/sec, obj=0.615]
   INFO - 20:38:20:     67%|██████▋   | 67/100 [00:00<00:00, 833.20 it/sec, obj=0.745]
   INFO - 20:38:20:     68%|██████▊   | 68/100 [00:00<00:00, 828.46 it/sec, obj=-0.875]
   INFO - 20:38:20:     69%|██████▉   | 69/100 [00:00<00:00, 828.98 it/sec, obj=-0.267]
   INFO - 20:38:20:     70%|███████   | 70/100 [00:00<00:00, 832.19 it/sec, obj=-0.809]
   INFO - 20:38:20:     71%|███████   | 71/100 [00:00<00:00, 838.33 it/sec, obj=-0.868]
   INFO - 20:38:20:     72%|███████▏  | 72/100 [00:00<00:00, 844.56 it/sec, obj=-0.874]
   INFO - 20:38:20:     73%|███████▎  | 73/100 [00:00<00:00, 847.05 it/sec, obj=-0.266]
   INFO - 20:38:20:     74%|███████▍  | 74/100 [00:00<00:00, 849.91 it/sec, obj=0.135]
   INFO - 20:38:20:     75%|███████▌  | 75/100 [00:00<00:00, 852.52 it/sec, obj=0.577]
   INFO - 20:38:20:     76%|███████▌  | 76/100 [00:00<00:00, 858.65 it/sec, obj=0.288]
   INFO - 20:38:20:     77%|███████▋  | 77/100 [00:00<00:00, 851.01 it/sec, obj=1.42]
   INFO - 20:38:20:     78%|███████▊  | 78/100 [00:00<00:00, 849.57 it/sec, obj=-0.875]
   INFO - 20:38:20:     79%|███████▉  | 79/100 [00:00<00:00, 852.05 it/sec, obj=-0.267]
   INFO - 20:38:20:     80%|████████  | 80/100 [00:00<00:00, 857.58 it/sec, obj=-0.809]
   INFO - 20:38:20:     81%|████████  | 81/100 [00:00<00:00, 863.48 it/sec, obj=-0.868]
   INFO - 20:38:20:     82%|████████▏ | 82/100 [00:00<00:00, 869.21 it/sec, obj=-0.874]
   INFO - 20:38:20:     83%|████████▎ | 83/100 [00:00<00:00, 871.18 it/sec, obj=-0.266]
   INFO - 20:38:20:     84%|████████▍ | 84/100 [00:00<00:00, 873.38 it/sec, obj=0.135]
   INFO - 20:38:20:     85%|████████▌ | 85/100 [00:00<00:00, 875.68 it/sec, obj=0.577]
   INFO - 20:38:20:     86%|████████▌ | 86/100 [00:00<00:00, 880.96 it/sec, obj=0.288]
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
   INFO - 20:38:20:     87%|████████▋ | 87/100 [00:00<00:00, 859.96 it/sec, obj=-0.267]
   INFO - 20:38:20:     88%|████████▊ | 88/100 [00:00<00:00, 864.81 it/sec, obj=-0.809]
   INFO - 20:38:20:     89%|████████▉ | 89/100 [00:00<00:00, 869.88 it/sec, obj=-0.868]
   INFO - 20:38:20:     90%|█████████ | 90/100 [00:00<00:00, 875.19 it/sec, obj=-0.874]
   INFO - 20:38:20:     91%|█████████ | 91/100 [00:00<00:00, 877.03 it/sec, obj=-0.266]
   INFO - 20:38:20:     92%|█████████▏| 92/100 [00:00<00:00, 878.89 it/sec, obj=0.135]
   INFO - 20:38:20:     93%|█████████▎| 93/100 [00:00<00:00, 880.61 it/sec, obj=0.577]
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
WARNING - 20:38:20: Optimization found no feasible point; the least infeasible point is selected.
   INFO - 20:38:20: Exporting the optimization problem to the file multistart.hdf5
   INFO - 20:38:20: Optimization result:
   INFO - 20:38:20:    Optimizer info:
   INFO - 20:38:20:       Status: None
   INFO - 20:38:20:       Message: None
   INFO - 20:38:20:       Number of calls to the objective function by the optimizer: 0
   INFO - 20:38:20:    Solution:
   INFO - 20:38:20:       The solution is feasible.
   INFO - 20:38:20:       Objective: 0.6150998205402495
   INFO - 20:38:20:       Standardized constraints:
   INFO - 20:38:20:          cstr = -0.7883188793606977
   INFO - 20:38:20:       Design space:
   INFO - 20:38:20:          +------+-------------+--------------------+-------------+-------+
   INFO - 20:38:20:          | Name | Lower bound |       Value        | Upper bound | Type  |
   INFO - 20:38:20:          +------+-------------+--------------------+-------------+-------+
   INFO - 20:38:20:          | x    |     -1.5    | 0.5773502675245377 |     1.5     | float |
   INFO - 20:38:20:          +------+-------------+--------------------+-------------+-------+
   INFO - 20:38:20: *** End MDOScenario execution ***

Lastly, we can plot the history of the objective, either by concatenating the 10 sub-optimization histories:

execute_post(
    scenario, post_name="BasicHistory", variable_names=["obj"], save=False, show=True
)
History plot
<gemseo.post.basic_history.BasicHistory object at 0x78590dddd730>

or by filtering the local optima (one per starting point):

execute_post(
    "multistart.hdf5",
    post_name="BasicHistory",
    variable_names=["obj"],
    save=False,
    show=True,
)
History plot
    INFO - 20:38:20: Importing the optimization problem from the file multistart.hdf5

<gemseo.post.basic_history.BasicHistory object at 0x78590cf22b70>

Total running time of the script: (0 minutes 0.369 seconds)

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