In this example, we'll be performing a simple optimization of single-objective functions using the global-best optimizer in pyswarms.single.GBestPSO
and the local-best optimizer in pyswarms.single.LBestPSO
. This aims to demonstrate the basic capabilities of the library when applied to benchmark problems.
In [1]:
# Import modules
import numpy as np
# Import PySwarms
import pyswarms as ps
from pyswarms.utils.functions import single_obj as fx
# Some more magic so that the notebook will reload external python modules;
# see http://stackoverflow.com/questions/1907993/autoreload-of-modules-in-ipython
%load_ext autoreload
%autoreload 2
For now let's just set some arbitrary parameters in our optimizers. There are, at minimum, three steps to perform optimization:
dict
.optimize()
method and have it store the optimal cost and position in a variable.The optimize()
method returns a tuple
of values, one of which includes the optimal cost and position after optimization. You can store it in a single variable and just index the values, or unpack it using several variables at once.
In [2]:
%%time
# Set-up hyperparameters
options = {'c1': 0.5, 'c2': 0.3, 'w':0.9}
# Call instance of PSO
optimizer = ps.single.GlobalBestPSO(n_particles=10, dimensions=2, options=options)
# Perform optimization
cost, pos = optimizer.optimize(fx.sphere, iters=1000)
We can see that the optimizer was able to find a good minima as shown above. You can control the verbosity of the output using the verbose
argument, and the number of steps to be printed out using the print_step
argument.
Now, let's try this one using local-best PSO:
In [3]:
%%time
# Set-up hyperparameters
options = {'c1': 0.5, 'c2': 0.3, 'w':0.9, 'k': 2, 'p': 2}
# Call instance of PSO
optimizer = ps.single.LocalBestPSO(n_particles=10, dimensions=2, options=options)
# Perform optimization
cost, pos = optimizer.optimize(fx.sphere, iters=1000)
Another thing that we can do is to set some bounds into our solution, so as to contain our candidate solutions within a specific range. We can do this simply by passing a bounds
parameter, of type tuple
, when creating an instance of our swarm. Let's try this using the global-best PSO with the Rastrigin function (rastrigin
in pyswarms.utils.functions.single_obj
).
Recall that the Rastrigin function is bounded within [-5.12, 5.12]
. If we pass an unbounded swarm into this function, then a ValueError
might be raised. So what we'll do is to create a bound within the specified range. There are some things to remember when specifying a bound:
numpy.ndarrays
so that we have a (min_bound, max_bound)
max_bound
should always be greater than the min_bound
. Their shapes should match the dimensions of the swarm.What we'll do now is to create a 10-particle, 2-dimensional swarm. This means that we have to set our maximum and minimum boundaries with the shape of 2. In case we want to initialize an n-dimensional swarm, we then have to set our bounds with the same shape n. A fast workaround for this would be to use the numpy.ones
function multiplied by a constant.
In [4]:
# Create bounds
max_bound = 5.12 * np.ones(2)
min_bound = - max_bound
bounds = (min_bound, max_bound)
In [5]:
%%time
# Initialize swarm
options = {'c1': 0.5, 'c2': 0.3, 'w':0.9}
# Call instance of PSO with bounds argument
optimizer = ps.single.GlobalBestPSO(n_particles=10, dimensions=2, options=options, bounds=bounds)
# Perform optimization
cost, pos = optimizer.optimize(fx.rastrigin, iters=1000)
Here, we will run a basic optimization using an objective function that needs parameterization. We will use the single.GBestPSO
and a version of the rosenbrock function to demonstrate
In [6]:
# import modules
import numpy as np
# create a parameterized version of the classic Rosenbrock unconstrained optimzation function
def rosenbrock_with_args(x, a, b, c=0):
f = (a - x[:, 0]) ** 2 + b * (x[:, 1] - x[:, 0] ** 2) ** 2 + c
return f
Arguments can either be passed in using a tuple or a dictionary, using the kwargs={}
paradigm. First lets optimize the Rosenbrock function using keyword arguments. Note in the definition of the Rosenbrock function above, there were two arguments that need to be passed other than the design variables, and one optional keyword argument, a
, b
, and c
, respectively
In [7]:
from pyswarms.single.global_best import GlobalBestPSO
# instatiate the optimizer
x_max = 10 * np.ones(2)
x_min = -1 * x_max
bounds = (x_min, x_max)
options = {'c1': 0.5, 'c2': 0.3, 'w': 0.9}
optimizer = GlobalBestPSO(n_particles=10, dimensions=2, options=options, bounds=bounds)
# now run the optimization, pass a=1 and b=100 as a tuple assigned to args
cost, pos = optimizer.optimize(rosenbrock_with_args, 1000, a=1, b=100, c=0)
It is also possible to pass a dictionary of key word arguments by using **
decorator when passing the dict
In [8]:
kwargs={"a": 1.0, "b": 100.0, 'c':0}
cost, pos = optimizer.optimize(rosenbrock_with_args, 1000, **kwargs)
Any key word arguments in the objective function can be left out as they will be passed the default as defined in the prototype. Note here, c
is not passed into the function.
In [9]:
cost, pos = optimizer.optimize(rosenbrock_with_args, 1000, a=1, b=100)