A flexible genetic algorithm
[pop_opt,fobj_pop_opt,pop_init,fobj_pop_init] = optim_ga(ga_f,pop_size,nb_generation,p_mut,p_cross,Log,param)
: :p_mut the mutation probability (default value: 0.1). : :p_cross the crossover probability (default value: 0.7). : :Log if %T, we will display to information message during the run of
the genetic algorithm.
: :param a list of parameters.
- “codage_func”: the function which will perform the coding and decoding of individuals (default function: coding_ga_identity).
- “init_func”: the function which will perform the initialization of the population (default function: init_ga_default).
- “crossover_func”: the function which will perform the crossover between two individuals (default function: crossover_ga_default).
- “mutation_func”: the function which will perform the mutation of one individual (default function: mutation_ga_default).
- “selection_func”: the function whcih will perform the selection of individuals at the end of a generation (default function: selection_ga_elitist).
- “nb_couples”: the number of couples which will be selected so as to perform the crossover and mutation (default value: 100).
- “pressure”: the value the efficiency of the worst individual (default value: 0.05).
: :pop_opt the population of optimal individuals. : :fobj_pop_opt the set of objective function values associated to
pop_opt (optional).
: :pop_init the initial population of individuals (optional). : :fobj_pop_init the set of objective function values associated to
pop_init (optional).
:
This function implements the classical genetic algorithm.
A great flexibility is authorized in customizing the behaviour of the optim_ga function. This flexibility is provided by the various functions which can be set in the param variable. In order to analyze the header of these functions (i.e. the input and output arguments), we may read the help page corresponding to the default function. For example, in order to understand what are the input and output arguments of the “codage_func” function, we may read the page of the coding_identity function.
See in the Demonstrations for more examples for this function.
The following session presents the simplest possible example. We minimize a quadratic function in dimension 3. By default, all the parameters are taken in the interval [0,1]^3. The “dimension” field is passed to the function which computes the initial population, which is init_ga_default function by default. In the case where the “dimension” field is not customized, the default value is used, which is equal to 2.
function y=f(x)
y = `sum`_(x.^2)
endfunction
PopSize = 100;
Proba_cross = 0.7;
Proba_mut = 0.1;
NbGen = 10;
Log = %T;
ga_params = `init_param`_();
// Parameters to control the initial population.
ga_params = `add_param`_(ga_params,"dimension",3);
[pop_opt, fobj_pop_opt] = ..
optim_ga(f, PopSize, NbGen, Proba_mut, Proba_cross, Log, ga_params);
Once the algorithm done, we can analyze the results. In the following script, we compute some basic statistics about the optimum population and get the best and the worst points.
// Display basic statistics
// min, mean and max function values of the population.
`disp`_([`min`_(fobj_pop_opt) `mean`_(fobj_pop_opt) `max`_(fobj_pop_opt)])
// Get the best x (i.e. the one which achieves the minimum function value)
[fmin ,k] = `min`_(fobj_pop_opt)
xmin = pop_opt(k)
// Get the worst x
[fmax ,k] = `max`_(fobj_pop_opt)
xmax = pop_opt(k)
In the following example, we customize all the options in order to show all the features of the algorithm.
function y=f(x)
y = `sum`_(x.^2)
endfunction
PopSize = 100;
Proba_cross = 0.7;
Proba_mut = 0.1;
NbGen = 10;
NbCouples = 110;
Log = %T;
pressure = 0.05;
ga_params = `init_param`_();
// Parameters to adapt to the shape of the optimization problem
ga_params = `add_param`_(ga_params,"minbound",[-2; -2]);
ga_params = `add_param`_(ga_params,"maxbound",[2; 2]);
ga_params = `add_param`_(ga_params,"dimension",2);
ga_params = `add_param`_(ga_params,"beta",0);
ga_params = `add_param`_(ga_params,"delta",0.1);
// Parameters to fine tune the Genetic algorithm.
// All these parameters are optional for continuous optimization
// If you need to adapt the GA to a special problem, you
ga_params = `add_param`_(ga_params,"init_func",`init_ga_default`_);
ga_params = `add_param`_(ga_params,"crossover_func",`crossover_ga_default`_);
ga_params = `add_param`_(ga_params,"mutation_func",`mutation_ga_default`_);
ga_params = `add_param`_(ga_params,"codage_func",`coding_ga_identity`_);
ga_params = `add_param`_(ga_params,"selection_func",`selection_ga_elitist`_);
//ga_params = add_param(ga_params,"selection_func",selection_ga_random);
ga_params = `add_param`_(ga_params,"nb_couples",NbCouples);
ga_params = `add_param`_(ga_params,"pressure",pressure);
[pop_opt, fobj_pop_opt, pop_init, fobj_pop_init] = ..
optim_ga(f, PopSize, NbGen, Proba_mut, Proba_cross, Log, ga_params);
In the following example, we customize the init function, which computes the initial population. In the myinitga function, we use the grand function (instead of the default rand used in init_ga_default). We could use any other type of population generator, including, for example, a low discrepancy sequence such as the Halton or Sobol sequence.
function y=f(x)
y = `sum`_(x.^2)
endfunction
function Pop_init=myinitga(popsize, param)
// This message is to be displayed in the console
// for demonstration purpose only :
// remove it in a real application!
`disp`_("Initializing the Population with grand")
// We deal with some parameters to take into account
// the boundary of the domain and the neighborhood size
[Dim,err] = `get_param`_(param,"dimension",2)
[MinBounds,err] = `get_param`_(param,"minbound",-2*`ones`_(1,Dim))
[MaxBounds,err] = `get_param`_(param,"maxbound",2*`ones`_(1,Dim))
// Pop_init must be a list()
Pop_init = `list`_()
nr = `size`_(MaxBounds,1)
nc = `size`_(MaxBounds,2)
for i=1:popsize
u = `grand`_(nr,nc,"def")
Pop_init(i) = (MaxBounds - MinBounds).*u + MinBounds
end
endfunction
PopSize = 100;
Proba_cross = 0.7;
Proba_mut = 0.1;
NbGen = 10;
NbCouples = 110;
Log = %T;
ga_params = `init_param`_();
// Parameters to adapt to the shape of the optimization problem
ga_params = `add_param`_(ga_params,"minbound",[-2; -2]);
ga_params = `add_param`_(ga_params,"maxbound",[2; 2]);
ga_params = `add_param`_(ga_params,"dimension",2);
ga_params = `add_param`_(ga_params,"init_func",myinitga);
[pop_opt, fobj_pop_opt, pop_init, fobj_pop_init] = ..
optim_ga(f, PopSize, NbGen, Proba_mut, Proba_cross, Log, ga_params);
In some cases, the objective function needs additionnal parameters in order to be evaluated. In this case, we can pass a list to the optim_ga function, where the first element of the list is the function and the remaining elements are the extra parameters.
This is done in the following script, where the function f needs the two extra parameters a1 and a2. This is why we define the list myobjfun and pass it to the optim_ga solver.
function y=f(x, a1, a2)
y = a1*`sum`_(x.^2) + a2
endfunction
PopSize = 100;
Proba_cross = 0.7;
Proba_mut = 0.1;
NbGen = 10;
NbCouples = 110;
Log = %T;
ga_params = `init_param`_();
// Parameters to control the initial population.
ga_params = `add_param`_(ga_params,"dimension",3);
// Pass the extra parameters to the objective function
a1 = 12;
a2 = 7;
myobjfun = `list`_(f,a1,a2);
// Optimize !
[pop_opt, fobj_pop_opt] = ..
optim_ga(myobjfun, PopSize, NbGen, Proba_mut, Proba_cross, Log, ga_params);