Multi-objective Fonseca-Fleming example with the mNBI algorithm

In this example, the modified Normal Boundary Intersection algorithm (mNBI) is used to solve the FonsecaFleming optimization problem [FF95]:

\[\begin{split}\begin{aligned} \text{minimize the objective function } & f_1(x) = 1 - exp(-\sum_{i=1}^{d}((x_i - 1 / sqrt(d)) ^ 2)) \\ & f_2(x) = 1 + exp(-\sum_{i=1}^{d}((x_i + 1 / sqrt(d)) ^ 2)) \\ \text{with respect to the design variables }&x\\ \text{subject to the bound constraint} & x\in[-4,4]^d \end{aligned}\end{split}\]

We also show how the Pareto front can be refined.

from __future__ import annotations

from gemseo import configure_logger
from gemseo import execute_algo
from gemseo import execute_post
from gemseo.problems.optimization.fonseca_fleming import FonsecaFleming

configure_logger()

Traceback (most recent call last):
  File "/home/docs/checkouts/readthedocs.org/user_builds/gemseo/checkouts/develop/doc_src/_examples/multi_objective/plot_mnbi_fonseca.py", line 46, in <module>
    from gemseo.problems.optimization.fonseca_fleming import FonsecaFleming
ModuleNotFoundError: No module named 'gemseo.problems.optimization.fonseca_fleming'

Solve the Fonseca-Fleming optimization problem

The 3 sub-optimization problems of mNBI are solved with SLSQP, a gradient-based optimization algorithm from the NLOPT library, with a maximum of 100 iterations. The analytic gradients are provided.

opt_problem = FonsecaFleming()
result = execute_algo(
    opt_problem,
    "MNBI",
    max_iter=1000,
    sub_optim_max_iter=100,
    n_sub_optim=3,
    sub_optim_algo="NLOPT_SLSQP",
)

Display the Pareto front

GEMSEO detects the Pareto optimal points and the dominated ones. The Fonseca-Fleming problem is interesting because its Pareto front is not convex. The mNBI algorithm successfully computes it.

execute_post(opt_problem, "ParetoFront", save=False, show=True)

Solve the Fonseca-Fleming optimization problem more finely

The Pareto front is then refined with 10 sub-optimizations instead of 3.

opt_problem = FonsecaFleming()
result = execute_algo(
    opt_problem,
    "MNBI",
    max_iter=1000,
    sub_optim_max_iter=100,
    n_sub_optim=10,
    sub_optim_algo="NLOPT_SLSQP",
)

Display the Pareto front

We can clearly see the effect of the refinement.

execute_post(opt_problem, "ParetoFront", save=False, show=True)