The self-adaptive penalty is a method to handle constraints in optimization problems. It is an adaptive penalty method where the penalization factor is computed dynamically according to the individuals in the population at the current generation. In this tutorial, it is explained how to solve a single objective constrained problem using the self-adaptive technique.

## Method¶

The self-adaptive method is a part of what is called adaptive penalty methods for constrained optimization. The idea is to penalize in term of cost function the infeasible individuals. Minimization is assumed in the design of the algorithm.

The infeasibility measure is computed through a two-stage penalty approach. The first stage applies just if there is one or more infeasible solution in the population that has a lower, hence potentially better, objective function value than the best feasible solution in the population. The penalty is hence applied to all the infeasible solutions in the population according to the relative order of infeasibility, in such a way that the solution with a high infeasibility rate but a low objective function has a fitness value close to the one of the best individual in the population. This is to avoid the elimination of infeasible solutions potentially close to the real optimum. The second stage penalizes exponentially all the infeasible individual according to their infeasibility rate and a scaling factor that is a measure, in the fitness space, of the distance between the infeasible solution with the lower objective and the individual in the population with the highest objective function.

The aim of the technique is not to reject completely individuals that are not totally feasible, or in other words that slightly violate some of the constraints. The actual implementation of the self-adaptive method in PaGMO/PyGMO is based on population fitness evaluation modification that adapts at each algorithm generation. The main advantages of this techniques are that it does not require parameter tuning and can be used also without a feasible solution in the initial population.

## Application¶

The problem considered here is the problem g01 from the Congress on Evolutionary Computation 2006 (CEC2006). This problem has a quadratic objective function with nine linear inequality constraints. In the optimum six constraints are active.

To solve it, a simple genetic algorithm with Gray encoding has been used in the meta-algorithm defined by the constraints handling technique described above. The meta-algorithm wraps the original algorithm that was only dedicated to solve unconstrained problems. It assignes to the population a modified problem that performes the adaptive penalization. In the example 70 individuals are considered into the population.

Step by step, the PyGMO library is imported and the populations size and the number of generation for the algorithm are initialized.

In [1]: from PyGMO import *
In [2]: pop_size = 70
In [3]: n_gen = 20000


Then the problem is created, a corresponding population randomly initialized and the genetic algorithm instantiated.

In [4]: prob = problem.cec2006(1)
In [5]: pop = population(prob,pop_size)
In [6]: algo = algorithm.sga_gray(gen=1,cr=0.9,m=0.004,elitism=0,mutation=algorithm.sga_gray.mutation.UNIFORM,selection=algorithm.sga_gray.selection.ROULETTE,crossover=algorithm.sga_gray.crossover.SINGLE_POINT)


In this example the genetic algorithm has a crossover probability of 90%, a mutation probability of 0.4%. Furthermore, it uses a uniform mutation, a roulette selection and a single point crossover. For more details on this algorithm, please refer to its documentation.

The important point here is the number of generation of the algorithm. In fact, as the number of generation is governed by the meta-algorithm and that the meta-algorithm modifies the fitness of the population at each iterations, the number of generation of the algorithm must be set to 1 to work properly. The same comment holds for the elitism.

The next step consists in creating the meta-algorithm that implements the self-adaptive constraints handling technique presented.

In [7]: algo_self_adaptive = algorithm.cstrs_self_adaptive(algo, n_gen)


Then we evolve it and print the solutions.

In [8]: pop = algo_self_adaptive.evolve(pop)
In [9]: print(pop.champion.x)
In [10]: print(pop.champion.f)
In [11]: print(pop.champion.c)
Out [1]:
(0.9999999701976767, 0.9999997615814138, 0.9999998509883836, 1.0, 1.0, 0.9999980032443405, 0.9999997615814138, 0.9999997615814138, 1.0, 2.9999972283839353, 2.99999126791928, 2.99997934698997, 0.9999922513959483)
(-14.999955534934077,)
(-0.008994877606477658, -0.0184053188087141, -0.01848650033731758, -5.004269987471997, -5.004496604338187, -4.982709675512006, -0.0023840070481302433, -0.004443645609725877, -0.0037382544201092216)


The solution found by this algorithm is close to the optimum given with the following:

In [12]: print(prob.best_x)
In [13]: print(prob.best_f)
In [14]: print(prob.best_c)
Out [2]:
((1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 3.0, 3.0, 3.0, 1.0),)
((-15.0,),)
((0.0, 0.0, 0.0, -5.0, -5.0, -5.0, 0.0, 0.0, 0.0),)


Even if the solutions are really close to the optimum, the exact same performances of this algorithm as described by R. Farmani and J. A. Wright in their paper could not be retrieved with our configuration. This might be due to a different implementation of the heuristic technique used for the optimization.