# Analytical test case # 2¶

In this example, we consider a simple optimization problem to illustrate algorithms interfaces and optimization libraries integration.

## Imports¶

from __future__ import annotations

from gemseo.algos.design_space import DesignSpace
from gemseo.algos.doe.doe_factory import DOEFactory
from gemseo.algos.opt.opt_factory import OptimizersFactory
from gemseo.algos.opt_problem import OptimizationProblem
from gemseo.api import configure_logger
from gemseo.api import execute_post
from gemseo.core.mdofunctions.mdo_function import MDOFunction
from numpy import cos
from numpy import exp
from numpy import ones
from numpy import sin

configure_logger()

<RootLogger root (INFO)>


## Define the objective function¶

We define the objective function $$f(x)=\sin(x)-\exp(x)$$ using a MDOFunction defined by the sum of MDOFunction objects.

f_1 = MDOFunction(sin, name="f_1", jac=cos, expr="sin(x)")
f_2 = MDOFunction(exp, name="f_2", jac=exp, expr="exp(x)")
objective = f_1 - f_2


The following operators are implemented: addition, subtraction and multiplication. The minus operator is also defined.

## Define the design space¶

Then, we define the DesignSpace with GEMSEO.

design_space = DesignSpace()
design_space.add_variable("x", l_b=-2.0, u_b=2.0, value=-0.5 * ones(1))


## Define the optimization problem¶

Then, we define the OptimizationProblem with GEMSEO.

problem = OptimizationProblem(design_space)
problem.objective = objective


## Solve the optimization problem using an optimization algorithm¶

Finally, we solve the optimization problems with GEMSEO interface.

### Solve the problem¶

opt = OptimizersFactory().execute(problem, "L-BFGS-B", normalize_design_space=True)

print("Optimum = ", opt)

    INFO - 16:58:57: Optimization problem:
INFO - 16:58:57:    minimize f_1-f_2 = sin(x)-exp(x)
INFO - 16:58:57:    with respect to x
INFO - 16:58:57:    over the design space:
INFO - 16:58:57:    +------+-------------+-------+-------------+-------+
INFO - 16:58:57:    | name | lower_bound | value | upper_bound | type  |
INFO - 16:58:57:    +------+-------------+-------+-------------+-------+
INFO - 16:58:57:    | x    |      -2     |  -0.5 |      2      | float |
INFO - 16:58:57:    +------+-------------+-------+-------------+-------+
INFO - 16:58:57: Solving optimization problem with algorithm L-BFGS-B:
INFO - 16:58:57: ...   0%|          | 0/999 [00:00<?, ?it]
INFO - 16:58:57: ...   0%|          | 1/999 [00:00<00:00, 2427.26 it/sec, obj=-1.09]
INFO - 16:58:57: ...   0%|          | 2/999 [00:00<00:00, 1348.00 it/sec, obj=-1.04]
INFO - 16:58:57: ...   0%|          | 3/999 [00:00<00:00, 1512.37 it/sec, obj=-1.24]
INFO - 16:58:57: ...   0%|          | 4/999 [00:00<00:00, 1371.25 it/sec, obj=-1.23]
INFO - 16:58:57: ...   1%|          | 5/999 [00:00<00:00, 1270.31 it/sec, obj=-1.24]
INFO - 16:58:57: ...   1%|          | 6/999 [00:00<00:00, 1235.31 it/sec, obj=-1.24]
INFO - 16:58:57: ...   1%|          | 7/999 [00:00<00:00, 1223.08 it/sec, obj=-1.24]
INFO - 16:58:57: Optimization result:
INFO - 16:58:57:    Optimizer info:
INFO - 16:58:57:       Status: 0
INFO - 16:58:57:       Message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
INFO - 16:58:57:       Number of calls to the objective function by the optimizer: 8
INFO - 16:58:57:    Solution:
INFO - 16:58:57:       Objective: -1.236108341859242
INFO - 16:58:57:       Design space:
INFO - 16:58:57:       +------+-------------+--------------------+-------------+-------+
INFO - 16:58:57:       | name | lower_bound |       value        | upper_bound | type  |
INFO - 16:58:57:       +------+-------------+--------------------+-------------+-------+
INFO - 16:58:57:       | x    |      -2     | -1.292695718944152 |      2      | float |
INFO - 16:58:57:       +------+-------------+--------------------+-------------+-------+
Optimum =  Optimization result:
Optimizer info:
Status: 0
Number of calls to the objective function by the optimizer: 8
Solution:
Objective: -1.236108341859242


Note that you can get all the optimization algorithms names:

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

Available algorithms  ['MMA', 'NLOPT_MMA', 'NLOPT_COBYLA', 'NLOPT_SLSQP', 'NLOPT_BOBYQA', 'NLOPT_BFGS', 'NLOPT_NEWUOA', 'PDFO_COBYLA', 'PDFO_BOBYQA', 'PDFO_NEWUOA', 'PSEVEN', 'PSEVEN_FD', 'PSEVEN_MOM', 'PSEVEN_NCG', 'PSEVEN_NLS', 'PSEVEN_POWELL', 'PSEVEN_QP', 'PSEVEN_SQP', 'PSEVEN_SQ2P', 'PYMOO_GA', 'PYMOO_NSGA2', 'PYMOO_NSGA3', 'PYMOO_UNSGA3', 'PYMOO_RNSGA3', 'DUAL_ANNEALING', 'SHGO', 'DIFFERENTIAL_EVOLUTION', 'LINEAR_INTERIOR_POINT', 'REVISED_SIMPLEX', 'SIMPLEX', 'SLSQP', 'L-BFGS-B', 'TNC', 'SNOPTB']


### Save the optimization results¶

We can serialize the results for further exploitation.

problem.export_hdf("my_optim.hdf5")

INFO - 16:58:57: Export optimization problem to file: my_optim.hdf5


### Post-process the results¶

execute_post(problem, "OptHistoryView", show=True, save=False)

<gemseo.post.opt_history_view.OptHistoryView object at 0x7fbc39b1e190>


Note

We can also save this plot using the arguments save=False and file_path='file_path'.

## Solve the optimization problem using a DOE algorithm¶

We can also see this optimization problem as a trade-off and solve it by means of a design of experiments (DOE).

opt = DOEFactory().execute(problem, "lhs", n_samples=10, normalize_design_space=True)
print("Optimum = ", opt)

    INFO - 16:58:58: Optimization problem:
INFO - 16:58:58:    minimize f_1-f_2 = sin(x)-exp(x)
INFO - 16:58:58:    with respect to x
INFO - 16:58:58:    over the design space:
INFO - 16:58:58:    +------+-------------+--------------------+-------------+-------+
INFO - 16:58:58:    | name | lower_bound |       value        | upper_bound | type  |
INFO - 16:58:58:    +------+-------------+--------------------+-------------+-------+
INFO - 16:58:58:    | x    |      -2     | -1.292695718944152 |      2      | float |
INFO - 16:58:58:    +------+-------------+--------------------+-------------+-------+
INFO - 16:58:58: Solving optimization problem with algorithm lhs:
INFO - 16:58:58: ...   0%|          | 0/10 [00:00<?, ?it]
INFO - 16:58:58: ...  10%|█         | 1/10 [00:00<00:00, 3721.65 it/sec, obj=-5.17]
INFO - 16:58:58: ...  20%|██        | 2/10 [00:00<00:00, 3139.45 it/sec, obj=-1.15]
INFO - 16:58:58: ...  30%|███       | 3/10 [00:00<00:00, 3052.62 it/sec, obj=-1.24]
INFO - 16:58:58: ...  40%|████      | 4/10 [00:00<00:00, 2849.88 it/sec, obj=-1.13]
INFO - 16:58:58: ...  50%|█████     | 5/10 [00:00<00:00, 2747.84 it/sec, obj=-2.91]
INFO - 16:58:58: ...  60%|██████    | 6/10 [00:00<00:00, 2719.75 it/sec, obj=-1.75]
INFO - 16:58:58: ...  70%|███████   | 7/10 [00:00<00:00, 2697.05 it/sec, obj=-1.14]
INFO - 16:58:58: ...  80%|████████  | 8/10 [00:00<00:00, 2732.89 it/sec, obj=-1.05]
INFO - 16:58:58: ...  90%|█████████ | 9/10 [00:00<00:00, 2761.23 it/sec, obj=-1.23]
INFO - 16:58:58: ... 100%|██████████| 10/10 [00:00<00:00, 2786.54 it/sec, obj=-1]
INFO - 16:58:58: Optimization result:
INFO - 16:58:58:    Optimizer info:
INFO - 16:58:58:       Status: None
INFO - 16:58:58:       Message: None
INFO - 16:58:58:       Number of calls to the objective function by the optimizer: 18
INFO - 16:58:58:    Solution:
INFO - 16:58:58:       Objective: -5.174108803965848
INFO - 16:58:58:       Design space:
INFO - 16:58:58:       +------+-------------+-------------------+-------------+-------+
INFO - 16:58:58:       | name | lower_bound |       value       | upper_bound | type  |
INFO - 16:58:58:       +------+-------------+-------------------+-------------+-------+
INFO - 16:58:58:       | x    |      -2     | 1.815526693601343 |      2      | float |
INFO - 16:58:58:       +------+-------------+-------------------+-------------+-------+
Optimum =  Optimization result:
Optimizer info:
Status: None
Message: None
Number of calls to the objective function by the optimizer: 18
Solution:
Objective: -5.174108803965848


Total running time of the script: ( 0 minutes 0.967 seconds)

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