# Co-evolution penalty method¶

Co-evolution mimics the biological behaviour of two distinct populations whose evolutions are interdependent. This technique has been used by some authors to deal with constraints in optimization and has been implemented in PaGMO/PyGMO. In this tutorial, we will learn how to solve a constrained problem with this technique.

## Method¶

The co-evolution method implemented in PaGMO/PyGMO has two populations evolving at the same time. The first one is a population associated to the initial problem from which the constraints are removed and where a penalized fitness is assigned to individuals that do not satisfy the constraints. To apply such a penalization the penalty coefficients need to be assessed, according to the level of infeasibility of the individuals in the population, in order to drive it towards feasible region of the search space. As the coefficients are cumbersome to determine, the co-evolution technique updates them by mean of a second population that encodes them. The fitness of the second population depends on the constraints satisfaction of the individuals in the first population. Hence the co-evolution happens sequentially: a set of penalty coefficents are firstly assigned ramdomly (the individuals in the second population). For each set of coefficient (for each individual in the second population) the first population is evolved for a certain number of iterations. Then the second population is evolved, with a fitness definition that depends on the results of the evolution of the first population and process then iterates with the new set of penalty coefficients.

In PaGMO/PyGMO this method is implemented within a meta-algorithm. The meta-algorithm takes the initial population to be optimized and two algorithms to evolve respectively the first and second populations. In the following we are going to see how to effectively solve a constrained problem.

## Application¶

The problem considered here is the problem g05 from the Congress on Evolutionary Computation 2006 (CEC2006). This problem has a cubic objective function with two linear inequality constraints and 3 non linear equality constraints. To solve this problem the Differential Evolution (DE) is chosen for both the first and second population. The first population size is set to 60 and the second to 30. It is good practice to set the second population size smaller than the first population size. Half the size of it seems to be experimentally a good compromize. The first algorithm evolves for 25 generations while the second algorithm number of generations must be set to 1. This is important as the second population uses the information from the first population and can not be evolved more than one time. This is repeated for 50 generations until convergence occurs.

First import the PyGMO library and choose the populations size and the number of generation for the meta-algorithm.

```
In [1]: from PyGMO import *
In [2]: pop_1_size = 60
In [3]: pop_2_size = 30
In [4]: n_gen = 50
```

Then creates the algorithms you wish to use for both populations. Here we have decided to use the Differential Evolution for both populations.

```
In [5]: algo_1 = algorithm.de(gen = 25, xtol=1e-30, ftol=1e-30)
In [6]: algo_2 = algorithm.de(gen = 1, xtol=1e-30, ftol=1e-30)
```

Select the problem and associate a population to this problem.

```
In [7]: prob = problem.cec2006(5)
In [8]: pop = population(prob,pop_1_size)
```

Creates the meta-algorithm with these informations.

```
In [9]: algo_coevo = algorithm.cstrs_co_evolution(original_algo = algo_1, original_algo_penalties = algo_2, pop_penalties_size = pop_2_size, gen = n_gen)
```

Evolve the algorithm.

```
In [10]: pop = algo_coevo.evolve(pop)
```

And finally, print the solutions.

```
In [11]: print(pop.champion.x)
In [12]: print(pop.champion.f)
In [13]: print(pop.champion.c)
Out [1]:
(679.9451523687442, 1026.0669716493055, 0.11887636619084481, -0.3962334865933514)
(5126.4967140070985,)
(9.999999997489795e-05, 9.999999997489795e-05, 9.999999997489795e-05, -0.03489014721580386, -1.0651098527841962)
```

The solution found by this method is the global optimum of the constrainted problem. Due to the stochastic behavior of the algorithm performing multiple runs is always raccomanded.