Note
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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 import configure_logger
from gemseo import execute_post
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.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
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 = OptimizersFactory().execute(problem, "L-BFGS-B", normalize_design_space=True)
print("Optimum = ", opt)
INFO - 16:27:02: Optimization problem:
INFO - 16:27:02: minimize f_1-f_2 = sin(x)-exp(x)
INFO - 16:27:02: with respect to x
INFO - 16:27:02: over the design space:
INFO - 16:27:02: +------+-------------+-------+-------------+-------+
INFO - 16:27:02: | name | lower_bound | value | upper_bound | type |
INFO - 16:27:02: +------+-------------+-------+-------------+-------+
INFO - 16:27:02: | x | -2 | -0.5 | 2 | float |
INFO - 16:27:02: +------+-------------+-------+-------------+-------+
INFO - 16:27:02: Solving optimization problem with algorithm L-BFGS-B:
INFO - 16:27:02: ... 0%| | 0/999 [00:00<?, ?it]
INFO - 16:27:02: ... 0%| | 1/999 [00:00<00:00, 2219.21 it/sec, obj=-1.09]
INFO - 16:27:02: ... 0%| | 2/999 [00:00<00:00, 1267.35 it/sec, obj=-1.04]
INFO - 16:27:02: ... 0%| | 3/999 [00:00<00:00, 1387.92 it/sec, obj=-1.24]
INFO - 16:27:02: ... 0%| | 4/999 [00:00<00:00, 1201.98 it/sec, obj=-1.23]
INFO - 16:27:02: ... 1%| | 5/999 [00:00<00:00, 1134.39 it/sec, obj=-1.24]
INFO - 16:27:02: ... 1%| | 6/999 [00:00<00:00, 1093.79 it/sec, obj=-1.24]
INFO - 16:27:02: ... 1%| | 7/999 [00:00<00:00, 1064.58 it/sec, obj=-1.24]
INFO - 16:27:02: Optimization result:
INFO - 16:27:02: Optimizer info:
INFO - 16:27:02: Status: 0
INFO - 16:27:02: Message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
INFO - 16:27:02: Number of calls to the objective function by the optimizer: 8
INFO - 16:27:02: Solution:
INFO - 16:27:02: Objective: -1.236108341859242
INFO - 16:27:02: Design space:
INFO - 16:27:02: +------+-------------+--------------------+-------------+-------+
INFO - 16:27:02: | name | lower_bound | value | upper_bound | type |
INFO - 16:27:02: +------+-------------+--------------------+-------------+-------+
INFO - 16:27:02: | x | -2 | -1.292695718944152 | 2 | float |
INFO - 16:27:02: +------+-------------+--------------------+-------------+-------+
Optimum = Optimization result:
Optimizer info:
Status: 0
Message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
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', 'SBO']
Save the optimization results¶
We can serialize the results for further exploitation.
problem.to_hdf("my_optim.hdf5")
INFO - 16:27:02: Export optimization problem to file: my_optim.hdf5
Post-process the results¶
execute_post(problem, "OptHistoryView", show=True, save=False)
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/stable/lib/python3.9/site-packages/genson/schema/strategies/base.py:42: UserWarning: Schema incompatible. Keyword 'description' has conflicting values ('The width and height of the figure in inches, e.g. ``(w, h)``.\nIf ``None``, use the :attr:`.OptPostProcessor.DEFAULT_FIG_SIZE`\nof the post-processor.' vs. 'The width and height of the figure in inches, e.g. `(w, h)`.\nIf ``None``, use the :attr:`.OptPostProcessor.DEFAULT_FIG_SIZE`\nof the post-processor.'). Using 'The width and height of the figure in inches, e.g. ``(w, h)``.\nIf ``None``, use the :attr:`.OptPostProcessor.DEFAULT_FIG_SIZE`\nof the post-processor.'
warn(('Schema incompatible. Keyword {0!r} has conflicting '
<gemseo.post.opt_history_view.OptHistoryView object at 0x7ffae48155b0>
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 - 16:27:03: Optimization problem:
INFO - 16:27:03: minimize f_1-f_2 = sin(x)-exp(x)
INFO - 16:27:03: with respect to x
INFO - 16:27:03: over the design space:
INFO - 16:27:03: +------+-------------+--------------------+-------------+-------+
INFO - 16:27:03: | name | lower_bound | value | upper_bound | type |
INFO - 16:27:03: +------+-------------+--------------------+-------------+-------+
INFO - 16:27:03: | x | -2 | -1.292695718944152 | 2 | float |
INFO - 16:27:03: +------+-------------+--------------------+-------------+-------+
INFO - 16:27:03: Solving optimization problem with algorithm lhs:
INFO - 16:27:03: ... 0%| | 0/10 [00:00<?, ?it]
INFO - 16:27:03: ... 10%|█ | 1/10 [00:00<00:00, 2985.27 it/sec, obj=-5.17]
INFO - 16:27:03: ... 20%|██ | 2/10 [00:00<00:00, 2581.11 it/sec, obj=-1.15]
INFO - 16:27:03: ... 30%|███ | 3/10 [00:00<00:00, 2566.89 it/sec, obj=-1.24]
INFO - 16:27:03: ... 40%|████ | 4/10 [00:00<00:00, 2559.45 it/sec, obj=-1.13]
INFO - 16:27:03: ... 50%|█████ | 5/10 [00:00<00:00, 2540.16 it/sec, obj=-2.91]
INFO - 16:27:03: ... 60%|██████ | 6/10 [00:00<00:00, 2537.64 it/sec, obj=-1.75]
INFO - 16:27:03: ... 70%|███████ | 7/10 [00:00<00:00, 2498.10 it/sec, obj=-1.14]
INFO - 16:27:03: ... 80%|████████ | 8/10 [00:00<00:00, 2460.72 it/sec, obj=-1.05]
INFO - 16:27:03: ... 90%|█████████ | 9/10 [00:00<00:00, 2466.43 it/sec, obj=-1.23]
INFO - 16:27:03: ... 100%|██████████| 10/10 [00:00<00:00, 2476.71 it/sec, obj=-1]
INFO - 16:27:03: Optimization result:
INFO - 16:27:03: Optimizer info:
INFO - 16:27:03: Status: None
INFO - 16:27:03: Message: None
INFO - 16:27:03: Number of calls to the objective function by the optimizer: 18
INFO - 16:27:03: Solution:
INFO - 16:27:03: Objective: -5.174108803965848
INFO - 16:27:03: Design space:
INFO - 16:27:03: +------+-------------+-------------------+-------------+-------+
INFO - 16:27:03: | name | lower_bound | value | upper_bound | type |
INFO - 16:27:03: +------+-------------+-------------------+-------------+-------+
INFO - 16:27:03: | x | -2 | 1.815526693601343 | 2 | float |
INFO - 16:27:03: +------+-------------+-------------------+-------------+-------+
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 1.079 seconds)