Note
Click here 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 division, unicode_literals
from numpy import cos, exp, ones, sin
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, execute_post
from gemseo.core.mdofunctions.mdo_function import MDOFunction
configure_logger()
Out:
<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
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", 1, 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)
Out:
INFO - 21:50:44: Optimization problem:
INFO - 21:50:44: Minimize: f_1-f_2 = sin(x)-exp(x)
INFO - 21:50:44: With respect to: x
INFO - 21:50:44: Design space:
INFO - 21:50:44: +------+-------------+-------+-------------+-------+
INFO - 21:50:44: | name | lower_bound | value | upper_bound | type |
INFO - 21:50:44: +------+-------------+-------+-------------+-------+
INFO - 21:50:44: | x | -2 | -0.5 | 2 | float |
INFO - 21:50:44: +------+-------------+-------+-------------+-------+
INFO - 21:50:44: Optimization: 0%| | 0/999 [00:00<?, ?it]
INFO - 21:50:44: Optimization: 1%| | 7/999 [00:00<00:00, 150485.19 it/sec, obj=[-1.23610834]]
INFO - 21:50:44: Optimization result:
INFO - 21:50:44: Objective value = [-1.23610834]
INFO - 21:50:44: The result is feasible.
INFO - 21:50:44: Status: 0
INFO - 21:50:44: Optimizer message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
INFO - 21:50:44: Number of calls to the objective function by the optimizer: 8
INFO - 21:50:44: Design space:
INFO - 21:50:44: +------+-------------+--------------------+-------------+-------+
INFO - 21:50:44: | name | lower_bound | value | upper_bound | type |
INFO - 21:50:44: +------+-------------+--------------------+-------------+-------+
INFO - 21:50:44: | x | -2 | -1.292695718944152 | 2 | float |
INFO - 21:50:44: +------+-------------+--------------------+-------------+-------+
Optimum = Optimization result:
Objective value = [-1.23610834]
The result is feasible.
Status: 0
Optimizer message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
Number of calls to the objective function by the optimizer: 8
Note that you can get all the optimization algorithms names:
algo_list = OptimizersFactory().algorithms
print("Available algorithms ", algo_list)
Out:
Available algorithms ['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', '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")
Out:
INFO - 21:50:44: Export optimization problem to file: my_optim.hdf5
Post-process the results¶
execute_post(problem, "OptHistoryView", show=True, save=False)
Out:
WARNING - 21:50:44: Failed to create Hessian approximation.
Traceback (most recent call last):
File "/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.2.0/lib/python3.8/site-packages/gemseo/post/opt_history_view.py", line 610, in _create_hessian_approx_plot
_, diag, _, _ = approximator.build_approximation(
File "/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.2.0/lib/python3.8/site-packages/gemseo/post/core/hessians.py", line 368, in build_approximation
x_hist, grad_hist, _, _ = self.get_x_grad_history(
File "/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.2.0/lib/python3.8/site-packages/gemseo/post/core/hessians.py", line 172, in get_x_grad_history
raise ValueError(
ValueError: Inconsistent gradient and design variables optimization history.
<gemseo.post.opt_history_view.OptHistoryView object at 0x7f618fea2340>
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)
Out:
WARNING - 21:50:44: Driver lhs has no option normalize_design_space, option is ignored.
INFO - 21:50:44: Optimization problem:
INFO - 21:50:44: Minimize: f_1-f_2 = sin(x)-exp(x)
INFO - 21:50:44: With respect to: x
INFO - 21:50:44: DOE sampling: 0%| | 0/10 [00:00<?, ?it]
INFO - 21:50:44: DOE sampling: 100%|██████████| 10/10 [00:00<00:00, 1444.87 it/sec, obj=[-1.00069899]]
INFO - 21:50:44: Optimization result:
INFO - 21:50:44: Objective value = [-5.1741088]
INFO - 21:50:44: The result is feasible.
INFO - 21:50:44: Status: None
INFO - 21:50:44: Optimizer message: None
INFO - 21:50:44: Number of calls to the objective function by the optimizer: 18
INFO - 21:50:44: Design space:
INFO - 21:50:44: +------+-------------+-------------------+-------------+-------+
INFO - 21:50:44: | name | lower_bound | value | upper_bound | type |
INFO - 21:50:44: +------+-------------+-------------------+-------------+-------+
INFO - 21:50:44: | x | -2 | 1.815526693601343 | 2 | float |
INFO - 21:50:44: +------+-------------+-------------------+-------------+-------+
Optimum = Optimization result:
Objective value = [-5.1741088]
The result is feasible.
Status: None
Optimizer message: None
Number of calls to the objective function by the optimizer: 18
Total running time of the script: ( 0 minutes 0.125 seconds)