# Moving Peaks Benchmark with Multiswarm PSO¶

In this example we show how to use the `MovingPeaks`

benchmark. A popular algorithm on this benchmark is the Multiswarm PSO (MPSO) [Blackwell2004] which achieve a great offline error and is able to follow multiple peaks at the same time.

## Choosing the Scenario¶

The moving peak benchmark allows to choose from the 3 original scenarios proposed in the original studies. This is done by retreiving one of the constants defined in the `movingpeaks`

module. Here we will use Scenario 2.

```
from deap.benchmarks import movingpeaks
scenario = movingpeaks.SCENARIO_2
```

Once the scenario is retrieved, we need to set a few more constants and instantiate the benchmark, here the number of dimensions and the bounds of the problem.

For a list of all the variables defined in the `SENARIO_X`

dictionaries see `MovingPeaks`

class documentation.

## Initialization¶

As in every DEAP example we are required to create the objects. The moving peak benchmark is a max problem, thus we need a maximizing fitness. And, we associate that fitness to a particle as in the Particle Swarm Optimization Basics example.

```
creator.create("FitnessMax", base.Fitness, weights=(1.0,))
creator.create("Particle", list, fitness=creator.FitnessMax, speed=list,
best=None, bestfit=creator.FitnessMax)
creator.create("Swarm", list, best=None, bestfit=creator.FitnessMax)
```

Then, the particle generator is defined. It takes the particle class object `pclass`

into which to put the data. Remeber that `creator.Particle`

, which is gonna be give to this argument in the toolbox, inherits from `list`

and can be initialized with an iterable. The position (elements of the list) and the speed (attribute) of the particle is set to randomly generated numbers between the given bounds.

```
def generate(pclass, dim, pmin, pmax, smin, smax):
part = pclass(random.uniform(pmin, pmax) for _ in range(dim))
part.speed = [random.uniform(smin, smax) for _ in range(dim)]
return part
```

The next function update the particle position and speed.

```
def updateParticle(part, best, chi, c):
ce1 = (c * random.uniform(0, 1) for _ in range(len(part)))
ce2 = (c * random.uniform(0, 1) for _ in range(len(part)))
ce1_p = map(operator.mul, ce1, map(operator.sub, best, part))
ce2_g = map(operator.mul, ce2, map(operator.sub, part.best, part))
a = map(operator.sub,
map(operator.mul,
itertools.repeat(chi),
map(operator.add, ce1_p, ce2_g)),
map(operator.mul,
itertools.repeat(1 - chi),
part.speed))
part.speed = list(map(operator.add, part.speed, a))
part[:] = list(map(operator.add, part, part.speed))
```

Thereafter, a function “converting” a particle to a quantum particle with different possible distributions is defined.

```
def convertQuantum(swarm, rcloud, centre, dist):
dim = len(swarm[0])
for part in swarm:
position = [random.gauss(0, 1) for _ in range(dim)]
dist = math.sqrt(sum(x**2 for x in position))
if dist == "gaussian":
u = abs(random.gauss(0, 1.0/3.0))
part[:] = [(rcloud * x * u**(1.0/dim) / dist) + c for x, c in zip(position, centre)]
elif dist == "uvd":
u = random.random()
part[:] = [(rcloud * x * u**(1.0/dim) / dist) + c for x, c in zip(position, centre)]
elif dist == "nuvd":
u = abs(random.gauss(0, 1.0/3.0))
part[:] = [(rcloud * x * u / dist) + c for x, c in zip(position, centre)]
del part.fitness.values
del part.bestfit.values
part.best = None
return swarm
```

Finally, all the functions are registered in the toolbox for further use in the algorithm.

```
toolbox = base.Toolbox()
toolbox.register("particle", generate, creator.Particle, dim=NDIM,
pmin=BOUNDS[0], pmax=BOUNDS[1], smin=-(BOUNDS[1] - BOUNDS[0])/2.0,
smax=(BOUNDS[1] - BOUNDS[0])/2.0)
toolbox.register("swarm", tools.initRepeat, creator.Swarm, toolbox.particle)
toolbox.register("update", updateParticle, chi=0.729843788, c=2.05)
toolbox.register("convert", convertQuantum, dist="nuvd")
toolbox.register("evaluate", mpb)
```

## Moving Peaks¶

The registered evaluation function in the toolbox refers directly to the instance of the `MovingPeaks`

benchmark object `mpb`

. The call to `mpb()`

evaluates the given individuals as any other evaluation function.

## Algorithm¶

The algorithm is fully detailed in the file examples/pso/multiswarm, it reflects what is described in [Blackwell2004].

[Blackwell2004] | (1, 2) Blackwell, T., & Branke, J. (2004). Multi-swarm optimization in dynamic environments. In Applications of Evolutionary Computing (pp. 489-500). Springer Berlin Heidelberg. |