.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/multi_objective/plot_mnbi_fonseca.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_multi_objective_plot_mnbi_fonseca.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_multi_objective_plot_mnbi_fonseca.py:


Multi-objective Fonseca-Fleming example with the mNBI algorithm
===============================================================

In this example, the modified Normal Boundary Intersection algorithm (mNBI) is used to
solve the :class:`.FonsecaFleming` optimization problem :cite:`fonseca1995overview`:

.. math::

   \begin{aligned}
   \text{minimize the objective function }
   & f_1(x) = 1 - exp(-\sum_{i=1}^{d}((x_i - 1 / sqrt(d)) ^ 2)) \\
   & f_2(x) = 1 + exp(-\sum_{i=1}^{d}((x_i + 1 / sqrt(d)) ^ 2)) \\
   \text{with respect to the design variables }&x\\
   \text{subject to the bound constraint}
   & x\in[-4,4]^d
   \end{aligned}

We also show how the Pareto front can be refined.

.. GENERATED FROM PYTHON SOURCE LINES 40-52

.. code-block:: Python


    from __future__ import annotations

    from gemseo import configure_logger
    from gemseo import execute_algo
    from gemseo import execute_post
    from gemseo.algos.opt.mnbi.settings.mnbi_settings import MNBI_Settings
    from gemseo.problems.multiobjective_optimization.fonseca_fleming import FonsecaFleming

    configure_logger()






.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <RootLogger root (INFO)>



.. GENERATED FROM PYTHON SOURCE LINES 53-59

Solve the Fonseca-Fleming optimization problem
----------------------------------------------
The 3 sub-optimization problems of mNBI are solved with SLSQP,
a gradient-based optimization algorithm from the NLOPT library,
with a maximum of 100 iterations.
The analytic gradients are provided.

.. GENERATED FROM PYTHON SOURCE LINES 59-67

.. code-block:: Python

    opt_problem = FonsecaFleming()
    mnbi_settings = MNBI_Settings(
        max_iter=1000,
        sub_optim_max_iter=100,
        n_sub_optim=3,
        sub_optim_algo="NLOPT_SLSQP",
    )
    result = execute_algo(opt_problem, settings_model=mnbi_settings)




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

        INFO - 08:35:47: Optimization problem:
        INFO - 08:35:47:    minimize FonsecaFleming
        INFO - 08:35:47:    with respect to x
        INFO - 08:35:47:    over the design space:
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47: Solving optimization problem with algorithm MNBI:
        INFO - 08:35:47: Searching for the individual optimum of each objective
        INFO - 08:35:47: Optimization problem:
        INFO - 08:35:47:    minimize f_0
        INFO - 08:35:47:    with respect to x
        INFO - 08:35:47:    over the design space:
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:      1%|          | 6/1000 [00:00<00:01, 715.55 it/sec, obj=[2.04753293e-04 9.82706845e-01]]
        INFO - 08:35:47:      1%|          | 7/1000 [00:00<00:01, 734.41 it/sec, obj=[1.10060811e-06 9.81607360e-01]]
        INFO - 08:35:47:      1%|          | 8/1000 [00:00<00:01, 751.38 it/sec, obj=[9.01501096e-14 9.81684376e-01]]
        INFO - 08:35:47:      1%|          | 9/1000 [00:00<00:01, 763.46 it/sec, obj=[8.15628010e-10 9.81684362e-01]]
        INFO - 08:35:47:      1%|          | 10/1000 [00:00<00:01, 827.05 it/sec, obj=[7.07334191e-12 9.81684374e-01]]
        INFO - 08:35:47:      1%|          | 11/1000 [00:00<00:01, 828.48 it/sec, obj=[4.27435864e-14 9.81684376e-01]]
        INFO - 08:35:47:      1%|          | 12/1000 [00:00<00:01, 777.20 it/sec, obj=[0.         0.98168436]]
        INFO - 08:35:47:      1%|▏         | 13/1000 [00:00<00:01, 784.87 it/sec, obj=[0.         0.98168436]]
        INFO - 08:35:47:      1%|▏         | 14/1000 [00:00<00:01, 830.12 it/sec, obj=[0.         0.98168436]]
        INFO - 08:35:47: Optimization result:
        INFO - 08:35:47:    Optimizer info:
        INFO - 08:35:47:       Status: None
        INFO - 08:35:47:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:47:       Number of calls to the objective function by the optimizer: 17
        INFO - 08:35:47:    Solution:
        INFO - 08:35:47:       Objective: 0.0
        INFO - 08:35:47:       Design space:
        INFO - 08:35:47:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:47:          | Name | Lower bound |       Value        | Upper bound | Type  |
        INFO - 08:35:47:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:47:          | x[0] |      -4     | 0.5773502684895009 |      4      | float |
        INFO - 08:35:47:          | x[1] |      -4     | 0.5773502684994094 |      4      | float |
        INFO - 08:35:47:          | x[2] |      -4     | 0.5773502684993135 |      4      | float |
        INFO - 08:35:47:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:47: Optimization problem:
        INFO - 08:35:47:    minimize f_1
        INFO - 08:35:47:    with respect to x
        INFO - 08:35:47:    over the design space:
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:      2%|▏         | 15/1000 [00:00<00:01, 664.24 it/sec, obj=[1. 1.]]
        INFO - 08:35:47:      2%|▏         | 16/1000 [00:00<00:01, 699.01 it/sec, obj=[0.99999998 0.9931056 ]]
        INFO - 08:35:47:      2%|▏         | 17/1000 [00:00<00:01, 705.02 it/sec, obj=[0.99688636 0.14955784]]
        INFO - 08:35:47:      2%|▏         | 18/1000 [00:00<00:01, 684.40 it/sec, obj=[0.9492493  0.07206897]]
        INFO - 08:35:47:      2%|▏         | 19/1000 [00:00<00:01, 692.79 it/sec, obj=[9.82706845e-01 2.04753293e-04]]
        INFO - 08:35:47:      2%|▏         | 20/1000 [00:00<00:01, 699.98 it/sec, obj=[9.81607360e-01 1.10060811e-06]]
        INFO - 08:35:47:      2%|▏         | 21/1000 [00:00<00:01, 707.23 it/sec, obj=[9.81684376e-01 9.24815780e-14]]
        INFO - 08:35:47:      2%|▏         | 22/1000 [00:00<00:01, 714.28 it/sec, obj=[9.81684362e-01 8.52572346e-10]]
        INFO - 08:35:47:      2%|▏         | 23/1000 [00:00<00:01, 739.45 it/sec, obj=[9.81684374e-01 7.39219796e-12]]
        INFO - 08:35:47:      2%|▏         | 24/1000 [00:00<00:01, 742.36 it/sec, obj=[9.81684376e-01 4.29656311e-14]]
        INFO - 08:35:47:      2%|▎         | 25/1000 [00:00<00:01, 724.06 it/sec, obj=[0.98168436 0.        ]]
        INFO - 08:35:47:      3%|▎         | 26/1000 [00:00<00:01, 729.40 it/sec, obj=[0.98168436 0.        ]]
        INFO - 08:35:47:      3%|▎         | 27/1000 [00:00<00:01, 751.03 it/sec, obj=[0.98168436 0.        ]]
        INFO - 08:35:47: Optimization result:
        INFO - 08:35:47:    Optimizer info:
        INFO - 08:35:47:       Status: None
        INFO - 08:35:47:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:47:       Number of calls to the objective function by the optimizer: 17
        INFO - 08:35:47:    Solution:
        INFO - 08:35:47:       Objective: 0.0
        INFO - 08:35:47:       Design space:
        INFO - 08:35:47:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:47:          | Name | Lower bound |        Value        | Upper bound | Type  |
        INFO - 08:35:47:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:47:          | x[0] |      -4     | -0.5773502685200489 |      4      | float |
        INFO - 08:35:47:          | x[1] |      -4     | -0.5773502685300462 |      4      | float |
        INFO - 08:35:47:          | x[2] |      -4     | -0.5773502685281993 |      4      | float |
        INFO - 08:35:47:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:47: Solving mNBI sub-problem for phi_beta = [0.49084218 0.49084218]
        INFO - 08:35:47: Optimization problem:
        INFO - 08:35:47:    minimize -t_extraction
        INFO - 08:35:47:    with respect to t, x
        INFO - 08:35:47:    subject to constraints:
        INFO - 08:35:47:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:47:    over the design space:
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:47:       | t    |     -inf    |   0   |     inf     | float |
        INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:47:      3%|▎         | 28/1000 [00:00<00:01, 636.05 it/sec, obj=[0.63212056 0.63212056]]
        INFO - 08:35:47: Optimization result:
        INFO - 08:35:47:    Optimizer info:
        INFO - 08:35:47:       Status: None
        INFO - 08:35:47:       Message: Successive iterates of the design variables are closer than xtol_rel or xtol_abs. GEMSEO stopped the driver.
        INFO - 08:35:47:       Number of calls to the objective function by the optimizer: 4
        INFO - 08:35:47:    Solution:
        INFO - 08:35:47:       The solution is feasible.
        INFO - 08:35:47:       Objective: 0.14391425994444806
        INFO - 08:35:47:       Standardized constraints:
        INFO - 08:35:47:          beta_sub_optim_constraint = [ 8.16013923e-15 -4.96824804e-15]
        INFO - 08:35:47:       Design space:
        INFO - 08:35:47:          +------+-------------+------------------------+-------------+-------+
        INFO - 08:35:47:          | Name | Lower bound |         Value          | Upper bound | Type  |
        INFO - 08:35:47:          +------+-------------+------------------------+-------------+-------+
        INFO - 08:35:47:          | x[0] |      -4     | -9.676703882632864e-13 |      4      | float |
        INFO - 08:35:47:          | x[1] |      -4     | -9.676703882632864e-13 |      4      | float |
        INFO - 08:35:47:          | x[2] |      -4     | -9.676703882632864e-13 |      4      | float |
        INFO - 08:35:47:          | t    |     -inf    |  -0.1439142599444481   |     inf     | float |
        INFO - 08:35:47:          +------+-------------+------------------------+-------------+-------+
        INFO - 08:35:47:      3%|▎         | 29/1000 [00:00<00:02, 453.36 it/sec, obj=[0.63212056 0.63212056]]
        INFO - 08:35:47: Optimization result:
        INFO - 08:35:47:    Optimizer info:
        INFO - 08:35:47:       Status: None
        INFO - 08:35:47:       Message: None
        INFO - 08:35:47:       Number of calls to the objective function by the optimizer: 36
        INFO - 08:35:47:    Solution:
        INFO - 08:35:47:       Objective: 0.8939534673502061
        INFO - 08:35:47:    Pareto efficient solutions:
        INFO - 08:35:47:       Pareto optimal points : 9 / 29
        INFO - 08:35:47:       Utopia point : [0. 0.]
        INFO - 08:35:47:       Compromise solution (closest to utopia) : [[0.63212056 0.63212056]]
        INFO - 08:35:47:       Distance from utopia : 0.8939534673502061
        INFO - 08:35:47:       Objective values:
        INFO - 08:35:47:          +------+----------+----------+----------+
        INFO - 08:35:47:          | name |        1 |        2 |        3 |
        INFO - 08:35:47:          +------+----------+----------+----------+
        INFO - 08:35:47:          |    1 |        0 | 0.981684 | 0.632121 |
        INFO - 08:35:47:          |    2 | 0.981684 |        0 | 0.632121 |
        INFO - 08:35:47:          +------+----------+----------+----------+
        INFO - 08:35:47:       Design space:
        INFO - 08:35:47:          +-------+-------------+---------+----------+---+-------------+-------+
        INFO - 08:35:47:          |  name | lower_bound |       1 |        2 | 3 | upper_bound |  type |
        INFO - 08:35:47:          +-------+-------------+---------+----------+---+-------------+-------+
        INFO - 08:35:47:          | x (1) |          -4 | 0.57735 | -0.57735 | 0 |           4 | float |
        INFO - 08:35:47:          | x (2) |          -4 | 0.57735 | -0.57735 | 0 |           4 | float |
        INFO - 08:35:47:          | x (3) |          -4 | 0.57735 | -0.57735 | 0 |           4 | float |
        INFO - 08:35:47:          +-------+-------------+---------+----------+---+-------------+-------+




.. GENERATED FROM PYTHON SOURCE LINES 68-74

Display the Pareto front
^^^^^^^^^^^^^^^^^^^^^^^^
|g| detects the Pareto optimal points and the dominated ones.
The Fonseca-Fleming problem is interesting because
its Pareto front is not convex.
The mNBI algorithm successfully computes it.

.. GENERATED FROM PYTHON SOURCE LINES 74-77

.. code-block:: Python


    execute_post(opt_problem, post_name="ParetoFront", save=False, show=True)




.. image-sg:: /examples/multi_objective/images/sphx_glr_plot_mnbi_fonseca_001.png
   :alt: Pareto front
   :srcset: /examples/multi_objective/images/sphx_glr_plot_mnbi_fonseca_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <gemseo.post.pareto_front.ParetoFront object at 0x7f6e05c436d0>



.. GENERATED FROM PYTHON SOURCE LINES 78-81

Solve the Fonseca-Fleming optimization problem more finely
----------------------------------------------------------
The Pareto front is then refined with 10 sub-optimizations instead of 3.

.. GENERATED FROM PYTHON SOURCE LINES 81-84

.. code-block:: Python

    opt_problem = FonsecaFleming()
    mnbi_settings.n_sub_optim = 10
    result = execute_algo(opt_problem, settings_model=mnbi_settings)




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize FonsecaFleming
        INFO - 08:35:48:    with respect to x
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48: Solving optimization problem with algorithm MNBI:
        INFO - 08:35:48: Searching for the individual optimum of each objective
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize f_0
        INFO - 08:35:48:    with respect to x
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:      1%|          | 6/1000 [00:00<00:01, 738.02 it/sec, obj=[2.04753293e-04 9.82706845e-01]]
        INFO - 08:35:48:      1%|          | 7/1000 [00:00<00:01, 754.93 it/sec, obj=[1.10060811e-06 9.81607360e-01]]
        INFO - 08:35:48:      1%|          | 8/1000 [00:00<00:01, 770.25 it/sec, obj=[9.01501096e-14 9.81684376e-01]]
        INFO - 08:35:48:      1%|          | 9/1000 [00:00<00:01, 782.78 it/sec, obj=[8.15628010e-10 9.81684362e-01]]
        INFO - 08:35:48:      1%|          | 10/1000 [00:00<00:01, 847.61 it/sec, obj=[7.07334191e-12 9.81684374e-01]]
        INFO - 08:35:48:      1%|          | 11/1000 [00:00<00:01, 847.63 it/sec, obj=[4.27435864e-14 9.81684376e-01]]
        INFO - 08:35:48:      1%|          | 12/1000 [00:00<00:01, 791.09 it/sec, obj=[0.         0.98168436]]
        INFO - 08:35:48:      1%|▏         | 13/1000 [00:00<00:01, 800.08 it/sec, obj=[0.         0.98168436]]
        INFO - 08:35:48:      1%|▏         | 14/1000 [00:00<00:01, 845.32 it/sec, obj=[0.         0.98168436]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 17
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       Objective: 0.0
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |       Value        | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     | 0.5773502684895009 |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | 0.5773502684994094 |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | 0.5773502684993135 |      4      | float |
        INFO - 08:35:48:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize f_1
        INFO - 08:35:48:    with respect to x
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:      2%|▏         | 15/1000 [00:00<00:01, 667.42 it/sec, obj=[1. 1.]]
        INFO - 08:35:48:      2%|▏         | 16/1000 [00:00<00:01, 702.37 it/sec, obj=[0.99999998 0.9931056 ]]
        INFO - 08:35:48:      2%|▏         | 17/1000 [00:00<00:01, 708.39 it/sec, obj=[0.99688636 0.14955784]]
        INFO - 08:35:48:      2%|▏         | 18/1000 [00:00<00:01, 686.73 it/sec, obj=[0.9492493  0.07206897]]
        INFO - 08:35:48:      2%|▏         | 19/1000 [00:00<00:01, 693.59 it/sec, obj=[9.82706845e-01 2.04753293e-04]]
        INFO - 08:35:48:      2%|▏         | 20/1000 [00:00<00:01, 701.42 it/sec, obj=[9.81607360e-01 1.10060811e-06]]
        INFO - 08:35:48:      2%|▏         | 21/1000 [00:00<00:01, 708.20 it/sec, obj=[9.81684376e-01 9.24815780e-14]]
        INFO - 08:35:48:      2%|▏         | 22/1000 [00:00<00:01, 715.40 it/sec, obj=[9.81684362e-01 8.52572346e-10]]
        INFO - 08:35:48:      2%|▏         | 23/1000 [00:00<00:01, 740.62 it/sec, obj=[9.81684374e-01 7.39219796e-12]]
        INFO - 08:35:48:      2%|▏         | 24/1000 [00:00<00:01, 743.56 it/sec, obj=[9.81684376e-01 4.29656311e-14]]
        INFO - 08:35:48:      2%|▎         | 25/1000 [00:00<00:01, 725.30 it/sec, obj=[0.98168436 0.        ]]
        INFO - 08:35:48:      3%|▎         | 26/1000 [00:00<00:01, 731.02 it/sec, obj=[0.98168436 0.        ]]
        INFO - 08:35:48:      3%|▎         | 27/1000 [00:00<00:01, 752.79 it/sec, obj=[0.98168436 0.        ]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 17
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       Objective: 0.0
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value        | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     | -0.5773502685200489 |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | -0.5773502685300462 |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | -0.5773502685281993 |      4      | float |
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.87260832 0.10907604]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound | Value | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     |   0   |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    |   0   |     inf     | float |
        INFO - 08:35:48:       +------+-------------+-------+-------------+-------+
        INFO - 08:35:48:      3%|▎         | 28/1000 [00:00<00:01, 636.44 it/sec, obj=[0.90043834 0.20664322]]
        INFO - 08:35:48:      3%|▎         | 29/1000 [00:00<00:01, 637.68 it/sec, obj=[0.91873293 0.15869784]]
        INFO - 08:35:48:      3%|▎         | 30/1000 [00:00<00:01, 638.80 it/sec, obj=[0.91966962 0.15614908]]
        INFO - 08:35:48:      3%|▎         | 31/1000 [00:00<00:01, 639.75 it/sec, obj=[0.91967277 0.15614049]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the design variables are closer than xtol_rel or xtol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 7
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.047942550573600556
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [ 6.93889390e-16 -1.83880688e-15]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     |  -0.339464261305511  |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | -0.3394642613042733  |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | -0.3394642613061056  |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    | -0.04794255057360056 |     inf     | float |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.76353228 0.21815208]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     |  -0.339464261305511  |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     | -0.3394642613042733  |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     | -0.3394642613061056  |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    | -0.04794255057360056 |     inf     | float |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:      3%|▎         | 32/1000 [00:00<00:01, 575.19 it/sec, obj=[0.91967277 0.15614049]]
        INFO - 08:35:48:      3%|▎         | 33/1000 [00:00<00:01, 579.10 it/sec, obj=[0.84203912 0.33739501]]
        INFO - 08:35:48:      3%|▎         | 34/1000 [00:00<00:01, 583.01 it/sec, obj=[0.85527677 0.31046303]]
        INFO - 08:35:48:      4%|▎         | 35/1000 [00:00<00:01, 584.87 it/sec, obj=[0.85546004 0.31007996]]
        INFO - 08:35:48:      4%|▎         | 36/1000 [00:00<00:01, 587.12 it/sec, obj=[0.85546004 0.31007995]]
        INFO - 08:35:48:      4%|▎         | 37/1000 [00:00<00:01, 589.39 it/sec, obj=[0.85546009 0.31007984]]
        INFO - 08:35:48:      4%|▍         | 38/1000 [00:00<00:01, 590.91 it/sec, obj=[0.8554601  0.31007982]]
        INFO - 08:35:48:      4%|▍         | 39/1000 [00:00<00:01, 603.49 it/sec, obj=[0.8554601  0.31007983]]
        INFO - 08:35:48:      4%|▍         | 40/1000 [00:00<00:01, 606.37 it/sec, obj=[0.8554601  0.31007983]]
        INFO - 08:35:48:      4%|▍         | 41/1000 [00:00<00:01, 608.64 it/sec, obj=[0.85546009 0.31007984]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 12
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.09364289795630283
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [ 5.50654545e-08 -3.31157324e-08]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     | -0.2256025551311831  |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | -0.2256025557747829  |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | -0.2256025548218257  |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    | -0.09364289795630283 |     inf     | float |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.65445624 0.32722812]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     | -0.2256025551311831  |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     | -0.2256025557747829  |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     | -0.2256025548218257  |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    | -0.09364289795630283 |     inf     | float |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:      4%|▍         | 42/1000 [00:00<00:01, 555.78 it/sec, obj=[0.77161691 0.45984452]]
        INFO - 08:35:48:      4%|▍         | 43/1000 [00:00<00:01, 557.77 it/sec, obj=[0.77767797 0.45048948]]
        INFO - 08:35:48:      4%|▍         | 44/1000 [00:00<00:01, 559.58 it/sec, obj=[0.77769345 0.45046534]]
        INFO - 08:35:48:      4%|▍         | 45/1000 [00:00<00:01, 561.55 it/sec, obj=[0.77769475 0.45046867]]
        INFO - 08:35:48:      5%|▍         | 46/1000 [00:00<00:01, 571.87 it/sec, obj=[0.77769346 0.45046536]]
        INFO - 08:35:48:      5%|▍         | 47/1000 [00:00<00:01, 574.55 it/sec, obj=[0.77769345 0.45046534]]
        INFO - 08:35:48:      5%|▍         | 48/1000 [00:00<00:01, 566.03 it/sec, obj=[0.77769345 0.45046533]]
       ERROR - 08:35:48: NLopt run failed: NLopt roundoff-limited, RoundoffLimited
    Traceback (most recent call last):
      File "/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/6.0.0/lib/python3.9/site-packages/gemseo/algos/opt/nlopt/nlopt.py", line 384, in _run
        nlopt_problem.optimize(x_0.real)
      File "/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/6.0.0/lib/python3.9/site-packages/nlopt/nlopt.py", line 335, in optimize
        return _nlopt.opt_optimize(self, *args)
    nlopt.RoundoffLimited: NLopt roundoff-limited
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 11
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.1255363970177465
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [1.38732549e-06 3.42942337e-06]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value        | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     | -0.1302316085444764 |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | -0.1325189233805686 |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | -0.1291321707661721 |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    | -0.1255363970177465 |     inf     | float |
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.5453802  0.43630416]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound |        Value        | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     | -0.1302316085444764 |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     | -0.1325189233805686 |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     | -0.1291321707661721 |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    | -0.1255363970177465 |     inf     | float |
        INFO - 08:35:48:       +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:      5%|▍         | 49/1000 [00:00<00:01, 532.37 it/sec, obj=[0.77769345 0.45046533]]
        INFO - 08:35:48:      5%|▌         | 50/1000 [00:00<00:01, 535.15 it/sec, obj=[0.68555698 0.58426668]]
        INFO - 08:35:48:      5%|▌         | 51/1000 [00:00<00:01, 538.11 it/sec, obj=[0.68399938 0.57650637]]
        INFO - 08:35:48:      5%|▌         | 52/1000 [00:00<00:01, 540.98 it/sec, obj=[0.68462482 0.57558952]]
        INFO - 08:35:48:      5%|▌         | 53/1000 [00:00<00:01, 542.89 it/sec, obj=[0.68463858 0.57556374]]
        INFO - 08:35:48:      5%|▌         | 54/1000 [00:00<00:01, 545.05 it/sec, obj=[0.68463912 0.57556308]]
        INFO - 08:35:48:      6%|▌         | 55/1000 [00:00<00:01, 546.84 it/sec, obj=[0.68463912 0.57556308]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 10
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.14185306201349607
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [3.45183683e-06 4.64490810e-06]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     | -0.04291047733084286 |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | -0.04270203738455658 |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | -0.04301066762956651 |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    | -0.1418530620134961  |     inf     | float |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.43630416 0.5453802 ]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     | -0.04291047733084286 |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     | -0.04270203738455658 |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     | -0.04301066762956651 |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    | -0.1418530620134961  |     inf     | float |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:      6%|▌         | 56/1000 [00:00<00:01, 516.77 it/sec, obj=[0.68463912 0.57556308]]
        INFO - 08:35:48:      6%|▌         | 57/1000 [00:00<00:01, 518.72 it/sec, obj=[0.57517744 0.68504627]]
        INFO - 08:35:48:      6%|▌         | 58/1000 [00:00<00:01, 520.69 it/sec, obj=[0.64468964 0.73529265]]
        INFO - 08:35:48:      6%|▌         | 59/1000 [00:00<00:01, 528.25 it/sec, obj=[0.57567735 0.68526336]]
        INFO - 08:35:48:      6%|▌         | 60/1000 [00:00<00:01, 523.40 it/sec, obj=[0.57562684 0.68458418]]
        INFO - 08:35:48:      6%|▌         | 61/1000 [00:00<00:01, 525.12 it/sec, obj=[0.57556309 0.68463912]]
        INFO - 08:35:48:      6%|▌         | 62/1000 [00:00<00:01, 527.00 it/sec, obj=[0.57556308 0.68463912]]
        INFO - 08:35:48:      6%|▋         | 63/1000 [00:00<00:01, 528.96 it/sec, obj=[0.57556308 0.68463912]]
        INFO - 08:35:48:      6%|▋         | 64/1000 [00:00<00:01, 530.60 it/sec, obj=[0.57556308 0.68463912]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 13
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.141857085596892
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [4.92134715e-08 3.53502475e-08]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value        | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     | 0.04285487868216808 |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | 0.04297029133272012 |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | 0.04279940357728407 |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    |  -0.141857085596892 |     inf     | float |
        INFO - 08:35:48:          +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.32722812 0.65445624]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound |        Value        | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     | 0.04285487868216808 |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     | 0.04297029133272012 |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     | 0.04279940357728407 |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    |  -0.141857085596892 |     inf     | float |
        INFO - 08:35:48:       +------+-------------+---------------------+-------------+-------+
        INFO - 08:35:48:      6%|▋         | 65/1000 [00:00<00:01, 504.24 it/sec, obj=[0.57556308 0.68463912]]
        INFO - 08:35:48:      7%|▋         | 66/1000 [00:00<00:01, 506.01 it/sec, obj=[0.45296521 0.77610508]]
        INFO - 08:35:48:      7%|▋         | 67/1000 [00:00<00:01, 507.48 it/sec, obj=[0.4857272  0.79138788]]
        INFO - 08:35:48:      7%|▋         | 68/1000 [00:00<00:01, 513.85 it/sec, obj=[0.45359005 0.77673074]]
        INFO - 08:35:48:      7%|▋         | 69/1000 [00:00<00:01, 509.47 it/sec, obj=[0.45080811 0.77747394]]
        INFO - 08:35:48:      7%|▋         | 70/1000 [00:00<00:01, 510.97 it/sec, obj=[0.45046544 0.77769341]]
        INFO - 08:35:48:      7%|▋         | 71/1000 [00:00<00:01, 512.77 it/sec, obj=[0.45046534 0.77769345]]
        INFO - 08:35:48:      7%|▋         | 72/1000 [00:00<00:01, 514.42 it/sec, obj=[0.45046533 0.77769345]]
        INFO - 08:35:48:      7%|▋         | 73/1000 [00:00<00:01, 516.03 it/sec, obj=[0.45046533 0.77769345]]
        INFO - 08:35:48:      7%|▋         | 74/1000 [00:00<00:01, 517.50 it/sec, obj=[0.45046533 0.77769345]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 14
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.12553645235059005
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [ 1.49882953e-07 -4.26679669e-09]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |       Value        | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     | 0.1305933450637484 |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     | 0.130790914197771  |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     | 0.1304983800119963 |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    | -0.12553645235059  |     inf     | float |
        INFO - 08:35:48:          +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.21815208 0.76353228]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound |       Value        | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     | 0.1305933450637484 |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     | 0.130790914197771  |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     | 0.1304983800119963 |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    | -0.12553645235059  |     inf     | float |
        INFO - 08:35:48:       +------+-------------+--------------------+-------------+-------+
        INFO - 08:35:48:      8%|▊         | 75/1000 [00:00<00:01, 499.27 it/sec, obj=[0.45046533 0.77769345]]
        INFO - 08:35:48:      8%|▊         | 76/1000 [00:00<00:01, 500.83 it/sec, obj=[0.31703362 0.85215238]]
        INFO - 08:35:48:      8%|▊         | 77/1000 [00:00<00:01, 502.46 it/sec, obj=[0.40288985 0.87338417]]
        INFO - 08:35:48:      8%|▊         | 78/1000 [00:00<00:01, 507.99 it/sec, obj=[0.31715667 0.85249332]]
        INFO - 08:35:48:      8%|▊         | 79/1000 [00:00<00:01, 505.66 it/sec, obj=[0.31040011 0.85530776]]
        INFO - 08:35:48:      8%|▊         | 80/1000 [00:00<00:01, 507.16 it/sec, obj=[0.31008035 0.85545991]]
        INFO - 08:35:48:      8%|▊         | 81/1000 [00:00<00:01, 508.70 it/sec, obj=[0.3100799  0.85546007]]
        INFO - 08:35:48:      8%|▊         | 82/1000 [00:00<00:01, 510.26 it/sec, obj=[0.31007988 0.85546007]]
        INFO - 08:35:48:      8%|▊         | 83/1000 [00:00<00:01, 511.80 it/sec, obj=[0.31007988 0.85546008]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 13
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.09364266739626721
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [7.23490535e-07 8.72958074e-08]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     |  0.2255448712436445  |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     |  0.2258763306714222  |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     |  0.2253855494856154  |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    | -0.09364266739626721 |     inf     | float |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48: Solving mNBI sub-problem for phi_beta = [0.10907604 0.87260832]
        INFO - 08:35:48: Optimization problem:
        INFO - 08:35:48:    minimize -t_extraction
        INFO - 08:35:48:    with respect to t, x
        INFO - 08:35:48:    subject to constraints:
        INFO - 08:35:48:       beta_sub_optim_constraint <= 0.0
        INFO - 08:35:48:    over the design space:
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:       | x[0] |      -4     |  0.2255448712436445  |      4      | float |
        INFO - 08:35:48:       | x[1] |      -4     |  0.2258763306714222  |      4      | float |
        INFO - 08:35:48:       | x[2] |      -4     |  0.2253855494856154  |      4      | float |
        INFO - 08:35:48:       | t    |     -inf    | -0.09364266739626721 |     inf     | float |
        INFO - 08:35:48:       +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:      8%|▊         | 84/1000 [00:00<00:01, 496.15 it/sec, obj=[0.31007988 0.85546008]]
        INFO - 08:35:48:      8%|▊         | 85/1000 [00:00<00:01, 498.12 it/sec, obj=[0.17162737 0.91400636]]
        INFO - 08:35:48:      9%|▊         | 86/1000 [00:00<00:01, 500.41 it/sec, obj=[0.29857451 0.93068206]]
        INFO - 08:35:48:      9%|▊         | 87/1000 [00:00<00:01, 505.34 it/sec, obj=[0.17152252 0.91447953]]
        INFO - 08:35:48:      9%|▉         | 88/1000 [00:00<00:01, 503.26 it/sec, obj=[0.15696325 0.91937307]]
        INFO - 08:35:48:      9%|▉         | 89/1000 [00:00<00:01, 504.57 it/sec, obj=[0.15614412 0.91967146]]
        INFO - 08:35:48:      9%|▉         | 90/1000 [00:00<00:01, 506.01 it/sec, obj=[0.15614051 0.91967277]]
        INFO - 08:35:48:      9%|▉         | 91/1000 [00:00<00:01, 506.77 it/sec, obj=[0.15614049 0.91967277]]
        INFO - 08:35:48:      9%|▉         | 92/1000 [00:00<00:01, 508.11 it/sec, obj=[0.15614049 0.91967277]]
        INFO - 08:35:48:      9%|▉         | 93/1000 [00:00<00:01, 509.48 it/sec, obj=[0.15614049 0.91967277]]
        INFO - 08:35:48:      9%|▉         | 94/1000 [00:00<00:01, 510.71 it/sec, obj=[0.15614049 0.91967277]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: Successive iterates of the objective function are closer than ftol_rel or ftol_abs. GEMSEO stopped the driver.
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 15
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       The solution is feasible.
        INFO - 08:35:48:       Objective: 0.047941264984527124
        INFO - 08:35:48:       Standardized constraints:
        INFO - 08:35:48:          beta_sub_optim_constraint = [ 4.88540472e-06 -4.82444740e-08]
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | Name | Lower bound |        Value         | Upper bound | Type  |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:          | x[0] |      -4     |  0.3394255200896286  |      4      | float |
        INFO - 08:35:48:          | x[1] |      -4     |  0.3396321336864006  |      4      | float |
        INFO - 08:35:48:          | x[2] |      -4     |  0.3393262076587185  |      4      | float |
        INFO - 08:35:48:          | t    |     -inf    | -0.04794126498452712 |     inf     | float |
        INFO - 08:35:48:          +------+-------------+----------------------+-------------+-------+
        INFO - 08:35:48:     10%|▉         | 95/1000 [00:00<00:01, 460.34 it/sec, obj=[0.15614049 0.91967277]]
        INFO - 08:35:48: Optimization result:
        INFO - 08:35:48:    Optimizer info:
        INFO - 08:35:48:       Status: None
        INFO - 08:35:48:       Message: None
        INFO - 08:35:48:       Number of calls to the objective function by the optimizer: 123
        INFO - 08:35:48:    Solution:
        INFO - 08:35:48:       Objective: 0.8939534673502061
        INFO - 08:35:48:    Pareto efficient solutions:
        INFO - 08:35:48:       Pareto optimal points : 61 / 95
        INFO - 08:35:48:       Utopia point : [0. 0.]
        INFO - 08:35:48:       Compromise solution (closest to utopia) : [[0.63212056 0.63212056]]
        INFO - 08:35:48:       Distance from utopia : 0.8939534673502061
        INFO - 08:35:48:       Objective values:
        INFO - 08:35:48:          +------+----------+----------+----------+
        INFO - 08:35:48:          | name |        1 |        2 |        3 |
        INFO - 08:35:48:          +------+----------+----------+----------+
        INFO - 08:35:48:          |    1 |        0 | 0.981684 | 0.632121 |
        INFO - 08:35:48:          |    2 | 0.981684 |        0 | 0.632121 |
        INFO - 08:35:48:          +------+----------+----------+----------+
        INFO - 08:35:48:       Design space:
        INFO - 08:35:48:          +-------+-------------+---------+----------+---+-------------+-------+
        INFO - 08:35:48:          |  name | lower_bound |       1 |        2 | 3 | upper_bound |  type |
        INFO - 08:35:48:          +-------+-------------+---------+----------+---+-------------+-------+
        INFO - 08:35:48:          | x (1) |          -4 | 0.57735 | -0.57735 | 0 |           4 | float |
        INFO - 08:35:48:          | x (2) |          -4 | 0.57735 | -0.57735 | 0 |           4 | float |
        INFO - 08:35:48:          | x (3) |          -4 | 0.57735 | -0.57735 | 0 |           4 | float |
        INFO - 08:35:48:          +-------+-------------+---------+----------+---+-------------+-------+




.. GENERATED FROM PYTHON SOURCE LINES 85-88

Display the Pareto front
^^^^^^^^^^^^^^^^^^^^^^^^
We can clearly see the effect of the refinement.

.. GENERATED FROM PYTHON SOURCE LINES 88-90

.. code-block:: Python


    execute_post(opt_problem, post_name="ParetoFront", save=False, show=True)



.. image-sg:: /examples/multi_objective/images/sphx_glr_plot_mnbi_fonseca_002.png
   :alt: Pareto front
   :srcset: /examples/multi_objective/images/sphx_glr_plot_mnbi_fonseca_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <gemseo.post.pareto_front.ParetoFront object at 0x7f6e05c5af10>




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.770 seconds)


.. _sphx_glr_download_examples_multi_objective_plot_mnbi_fonseca.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_mnbi_fonseca.ipynb <plot_mnbi_fonseca.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_mnbi_fonseca.py <plot_mnbi_fonseca.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_mnbi_fonseca.zip <plot_mnbi_fonseca.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_