Note

Go to the end to download the full example code.

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 sin

from gemseo import configure_logger
from gemseo import execute_algo
from gemseo import execute_post
from gemseo import get_available_opt_algorithms
from gemseo.algos.design_space import DesignSpace
from gemseo.algos.optimization_problem import OptimizationProblem
from gemseo.core.mdo_functions.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", lower_bound=-2.0, upper_bound=2.0, value=-0.5)

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#

optimization_result = execute_algo(problem, algo_name="L-BFGS-B")
optimization_result

INFO - 08:35:46: Optimization problem:
INFO - 08:35:46:    minimize [f_1-f_2] = sin(x)-exp(x)
INFO - 08:35:46:    with respect to x
INFO - 08:35:46:    over the design space:
INFO - 08:35:46:       +------+-------------+-------+-------------+-------+
INFO - 08:35:46:       | Name | Lower bound | Value | Upper bound | Type  |
INFO - 08:35:46:       +------+-------------+-------+-------------+-------+
INFO - 08:35:46:       | x    |      -2     |  -0.5 |      2      | float |
INFO - 08:35:46:       +------+-------------+-------+-------------+-------+
INFO - 08:35:46: Solving optimization problem with algorithm L-BFGS-B:
INFO - 08:35:46:      1%|          | 6/1000 [00:00<00:00, 1270.30 it/sec, obj=-1.24]
INFO - 08:35:46:      1%|          | 7/1000 [00:00<00:00, 1211.83 it/sec, obj=-1.24]
INFO - 08:35:46: Optimization result:
INFO - 08:35:46:    Optimizer info:
INFO - 08:35:46:       Status: 0
INFO - 08:35:46:       Message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
INFO - 08:35:46:       Number of calls to the objective function by the optimizer: 8
INFO - 08:35:46:    Solution:
INFO - 08:35:46:       Objective: -1.2361083418592416
INFO - 08:35:46:       Design space:
INFO - 08:35:46:          +------+-------------+--------------------+-------------+-------+
INFO - 08:35:46:          | Name | Lower bound |       Value        | Upper bound | Type  |
INFO - 08:35:46:          +------+-------------+--------------------+-------------+-------+
INFO - 08:35:46:          | x    |      -2     | -1.292695718944152 |      2      | float |
INFO - 08:35:46:          +------+-------------+--------------------+-------------+-------+
Optimization result:
  • Design variables: [-1.29269572]
  • Objective function: -1.2361083418592416
  • Feasible solution: True


Note that you can get all the optimization algorithms names:

get_available_opt_algorithms()

['Augmented_Lagrangian_order_0', 'Augmented_Lagrangian_order_1', 'MNBI', 'MultiStart', 'NLOPT_MMA', 'NLOPT_COBYLA', 'NLOPT_SLSQP', 'NLOPT_BOBYQA', 'NLOPT_BFGS', 'NLOPT_NEWUOA', 'DUAL_ANNEALING', 'SHGO', 'DIFFERENTIAL_EVOLUTION', 'INTERIOR_POINT', 'DUAL_SIMPLEX', '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 - 08:35:46: Exporting the optimization problem to the file my_optim.hdf5 at node

Post-process the results#

execute_post(problem, post_name="OptHistoryView", save=False, show=True)
  • Evolution of the optimization variables
  • Evolution of the objective value
  • Evolution of the distance to the optimum
<gemseo.post.opt_history_view.OptHistoryView object at 0x7f2522c40c10>

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 problem and solve it by means of a design of experiments (DOE).

problem.reset()
optimization_result = execute_algo(
    problem, algo_name="PYDOE_LHS", n_samples=10, algo_type="doe"
)
optimization_result

INFO - 08:35:47: Optimization problem:
INFO - 08:35:47:    minimize [f_1-f_2] = sin(x)-exp(x)
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    |      -2     |  -0.5 |      2      | float |
INFO - 08:35:47:       +------+-------------+-------+-------------+-------+
INFO - 08:35:47: Solving optimization problem with algorithm PYDOE_LHS:
INFO - 08:35:47:     10%|█         | 1/10 [00:00<00:00, 5090.17 it/sec, obj=-5.17]
INFO - 08:35:47:     20%|██        | 2/10 [00:00<00:00, 3951.30 it/sec, obj=-1.15]
INFO - 08:35:47:     30%|███       | 3/10 [00:00<00:00, 3960.63 it/sec, obj=-1.24]
INFO - 08:35:47:     40%|████      | 4/10 [00:00<00:00, 3963.43 it/sec, obj=-1.13]
INFO - 08:35:47:     50%|█████     | 5/10 [00:00<00:00, 3973.38 it/sec, obj=-2.91]
INFO - 08:35:47:     60%|██████    | 6/10 [00:00<00:00, 3992.04 it/sec, obj=-1.75]
INFO - 08:35:47:     70%|███████   | 7/10 [00:00<00:00, 4020.83 it/sec, obj=-1.14]
INFO - 08:35:47:     80%|████████  | 8/10 [00:00<00:00, 4043.19 it/sec, obj=-1.05]
INFO - 08:35:47:     90%|█████████ | 9/10 [00:00<00:00, 4019.24 it/sec, obj=-1.23]
INFO - 08:35:47:    100%|██████████| 10/10 [00:00<00:00, 4035.31 it/sec, obj=-1]
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: 10
INFO - 08:35:47:    Solution:
INFO - 08:35:47:       Objective: -5.174108803965849
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    |      -2     | 1.815526693601343 |      2      | float |
INFO - 08:35:47:          +------+-------------+-------------------+-------------+-------+
Optimization result:
  • Design variables: [1.81552669]
  • Objective function: -5.174108803965849
  • Feasible solution: True


Total running time of the script: (0 minutes 0.801 seconds)

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