# 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 numpy import cos
from numpy import exp
from numpy import ones
from numpy import sin

from gemseo import configure_logger
from gemseo import execute_post
from gemseo.algos.design_space import DesignSpace
from gemseo.algos.doe.factory import DOELibraryFactory
from gemseo.algos.opt.factory import OptimizationLibraryFactory
from gemseo.algos.optimization_problem import OptimizationProblem
from gemseo.core.mdofunctions.mdo_function import MDOFunction

configure_logger()

<RootLogger root (INFO)>


## Define the objective function¶

We define the objective function $$f(x)=\sin(x)-\exp(x)$$ using an 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


See also

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

INFO - 01:01:24: Optimization problem:
INFO - 01:01:24:    minimize [f_1-f_2] = sin(x)-exp(x)
INFO - 01:01:24:    with respect to x
INFO - 01:01:24:    over the design space:
INFO - 01:01:24:       +------+-------------+-------+-------------+-------+
INFO - 01:01:24:       | Name | Lower bound | Value | Upper bound | Type  |
INFO - 01:01:24:       +------+-------------+-------+-------------+-------+
INFO - 01:01:24:       | x    |      -2     |  -0.5 |      2      | float |
INFO - 01:01:24:       +------+-------------+-------+-------------+-------+
INFO - 01:01:24: Solving optimization problem with algorithm L-BFGS-B:
INFO - 01:01:24:      1%|          | 5/999 [00:00<00:00, 1597.59 it/sec, obj=-1.24]
INFO - 01:01:24:      1%|          | 6/999 [00:00<00:00, 1499.66 it/sec, obj=-1.24]
INFO - 01:01:24:      1%|          | 7/999 [00:00<00:00, 1441.41 it/sec, obj=-1.24]
INFO - 01:01:24: Optimization result:
INFO - 01:01:24:    Optimizer info:
INFO - 01:01:24:       Status: 0
INFO - 01:01:24:       Message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
INFO - 01:01:24:       Number of calls to the objective function by the optimizer: 8
INFO - 01:01:24:    Solution:
INFO - 01:01:24:       Objective: -1.2361083418592416
INFO - 01:01:24:       Design space:
INFO - 01:01:24:          +------+-------------+--------------------+-------------+-------+
INFO - 01:01:24:          | Name | Lower bound |       Value        | Upper bound | Type  |
INFO - 01:01:24:          +------+-------------+--------------------+-------------+-------+
INFO - 01:01:24:          | x    |      -2     | -1.292695718944152 |      2      | float |
INFO - 01:01:24:          +------+-------------+--------------------+-------------+-------+

Optimization result:
• Design variables: [-1.29269572]
• Objective function: -1.2361083418592416
• Feasible solution: True

Note that you can get all the optimization algorithms names:

OptimizationLibraryFactory().algorithms

['Augmented_Lagrangian_order_0', 'Augmented_Lagrangian_order_1', 'MMA', 'MNBI', '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', 'HIGHS_INTERIOR_POINT', 'HIGHS_DUAL_SIMPLEX', 'HIGHS', 'Scipy_MILP', 'SLSQP', 'L-BFGS-B', 'TNC', 'NELDER-MEAD']


### Save the optimization results¶

We can serialize the results for further exploitation.

problem.to_hdf("my_optim.hdf5")

INFO - 01:01:24: Exporting the optimization problem to the file my_optim.hdf5 at node


### Post-process the results¶

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

<gemseo.post.opt_history_view.OptHistoryView object at 0x7f1191bbf220>


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 = DOELibraryFactory().execute(
problem, "lhs", n_samples=10, normalize_design_space=True
)
opt

INFO - 01:01:24: Optimization problem:
INFO - 01:01:24:    minimize [f_1-f_2] = sin(x)-exp(x)
INFO - 01:01:24:    with respect to x
INFO - 01:01:24:    over the design space:
INFO - 01:01:24:       +------+-------------+--------------------+-------------+-------+
INFO - 01:01:24:       | Name | Lower bound |       Value        | Upper bound | Type  |
INFO - 01:01:24:       +------+-------------+--------------------+-------------+-------+
INFO - 01:01:24:       | x    |      -2     | -1.292695718944152 |      2      | float |
INFO - 01:01:24:       +------+-------------+--------------------+-------------+-------+
INFO - 01:01:24: Solving optimization problem with algorithm lhs:
INFO - 01:01:24:     10%|█         | 1/10 [00:00<00:00, 4076.10 it/sec, obj=-5.17]
INFO - 01:01:24:     20%|██        | 2/10 [00:00<00:00, 3318.28 it/sec, obj=-1.15]
INFO - 01:01:24:     30%|███       | 3/10 [00:00<00:00, 3280.22 it/sec, obj=-1.24]
INFO - 01:01:24:     40%|████      | 4/10 [00:00<00:00, 3277.44 it/sec, obj=-1.13]
INFO - 01:01:24:     50%|█████     | 5/10 [00:00<00:00, 3282.96 it/sec, obj=-2.91]
INFO - 01:01:24:     60%|██████    | 6/10 [00:00<00:00, 3286.21 it/sec, obj=-1.75]
INFO - 01:01:24:     70%|███████   | 7/10 [00:00<00:00, 3291.86 it/sec, obj=-1.14]
INFO - 01:01:24:     80%|████████  | 8/10 [00:00<00:00, 3298.06 it/sec, obj=-1.05]
INFO - 01:01:24:     90%|█████████ | 9/10 [00:00<00:00, 3299.43 it/sec, obj=-1.23]
INFO - 01:01:24:    100%|██████████| 10/10 [00:00<00:00, 3299.22 it/sec, obj=-1]
INFO - 01:01:24: Optimization result:
INFO - 01:01:24:    Optimizer info:
INFO - 01:01:24:       Status: None
INFO - 01:01:24:       Message: None
INFO - 01:01:24:       Number of calls to the objective function by the optimizer: 18
INFO - 01:01:24:    Solution:
INFO - 01:01:24:       Objective: -5.174108803965849
INFO - 01:01:24:       Design space:
INFO - 01:01:24:          +------+-------------+-------------------+-------------+-------+
INFO - 01:01:24:          | Name | Lower bound |       Value       | Upper bound | Type  |
INFO - 01:01:24:          +------+-------------+-------------------+-------------+-------+
INFO - 01:01:24:          | x    |      -2     | 1.815526693601343 |      2      | float |
INFO - 01:01:24:          +------+-------------+-------------------+-------------+-------+

Optimization result:
• Design variables: [1.81552669]
• Objective function: -5.174108803965849
• Feasible solution: True

Total running time of the script: (0 minutes 0.660 seconds)

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