Constraint Violation (CV) as Penalty

Another well-known way of handling constraints is removing the constraint and adding it as a penalty to objective(s). One easy way of achieving that is redefining the problem, as shown below using the ConstraintsAsPenalty class. Nevertheless, whenever two numbers are added, normalization can become an issue. Thus, commonly a penalty coefficient (here penalty) needs to be defined. It can be helpful to play with this parameter if the results are not satisfying.

[2]:
from pymoo.algorithms.soo.nonconvex.de import DE
from pymoo.constraints.as_penalty import ConstraintsAsPenalty
from pymoo.optimize import minimize
from pymoo.core.evaluator import Evaluator
from pymoo.core.individual import Individual

problem = ConstrainedProblem()

algorithm = DE()

res = minimize(ConstraintsAsPenalty(problem, penalty=100.0),
               algorithm,
               seed=1,
               verbose=False)

res = Evaluator().eval(problem, Individual(X=res.X))

print("Best solution found: \nX = %s\nF = %s\nCV = %s" % (res.X, res.F, res.CV))

Compiled modules for significant speedup can not be used!
https://pymoo.org/installation.html#installation

To disable this warning:
from pymoo.config import Config
Config.warnings['not_compiled'] = False

Best solution found:
X = [0.50058588 0.49942291]
F = [0.50000947]
CV = [0.]

Solution only almost feasible

Please note that this approach might not always find a feasible solution (because the algorithm does not know anything about whether a solution is feasible or not). For instance, see the example below:

[3]:
from pymoo.algorithms.soo.nonconvex.de import DE
from pymoo.constraints.as_penalty import ConstraintsAsPenalty
from pymoo.optimize import minimize
from pymoo.core.evaluator import Evaluator
from pymoo.core.individual import Individual

res = minimize(ConstraintsAsPenalty(problem, penalty=2.0),
               algorithm,
               seed=1,
               verbose=False)

res = Evaluator().eval(problem, Individual(X=res.X))

print("Best solution found: \nX = %s\nF = %s\nCV = %s" % (res.X, res.F, res.CV))
Best solution found:
X = [0.49918147 0.50081538]
F = [0.49999819]
CV = [3.14674782e-06]

In such cases, it can be helpful to perform another search for the solution found to the original problem to find a feasible one. This second search method can, for instance, be realized by a local search or by using again a population-based method injecting the solution found before. Here, we demonstrate the latter:

[4]:
from pymoo.operators.sampling.lhs import LHS

sampling = LHS().do(problem, 100)
sampling[0].X = res.X

algorithm = DE(sampling=sampling)

res = minimize(problem, algorithm)

print("Best solution found: \nX = %s\nF = %s\nCV = %s" % (res.X, res.F, res.CV))
Best solution found:
X = [0.52660211 0.47614088]
F = [0.50401992]
CV = [0.]