# 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 division, unicode_literals

from numpy import sum as np_sum

from gemseo.algos.design_space import DesignSpace
from gemseo.algos.doe.doe_factory import DOEFactory
from gemseo.algos.opt_problem import OptimizationProblem
from gemseo.api import configure_logger, execute_post
from gemseo.core.mdofunctions.mdo_function import MDOFunction

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()
design_space.add_variable("x", 2, l_b=-5, u_b=5, var_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

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


Out:

    INFO - 14:41:36: Optimization problem:
INFO - 14:41:36:    Minimize: f = sum(x)
INFO - 14:41:36:    With respect to: x
INFO - 14:41:36: Full factorial design required. Number of samples along each direction for a design vector of size 2 with 121 samples: 11
INFO - 14:41:36: Final number of samples for DOE = 121 vs 121 requested
INFO - 14:41:36: DOE sampling:   0%|          | 0/121 [00:00<?, ?it]
INFO - 14:41:36: DOE sampling: 100%|██████████| 121/121 [00:00<00:00, 2932.91 it/sec, obj=10]
INFO - 14:41:36: Optimization result:
INFO - 14:41:36: Objective value = -10.0
INFO - 14:41:36: The result is feasible.
INFO - 14:41:36: Status: None
INFO - 14:41:36: Optimizer message: None
INFO - 14:41:36: Number of calls to the objective function by the optimizer: 121
INFO - 14:41:36: Design space:
INFO - 14:41:36: +------+-------------+-------+-------------+---------+
INFO - 14:41:36: | name | lower_bound | value | upper_bound | type    |
INFO - 14:41:36: +------+-------------+-------+-------------+---------+
INFO - 14:41:36: | x    |      -5     |   -5  |      5      | integer |
INFO - 14:41:36: | x    |      -5     |   -5  |      5      | integer |
INFO - 14:41:36: +------+-------------+-------+-------------+---------+

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


## Post-process the results¶

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


Out:

<gemseo.post.scatter_mat.ScatterPlotMatrix object at 0x7fcaac540be0>


Note that you can get all the optimization algorithms names:

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


Out:

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 0.478 seconds)

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