There are two ways to specify options for the genetic algorithm,
depending on whether you are using the Optimization app or calling
the functions ga
or gamultiobj
at
the command line:
If you are using the Optimization app (optimtool
),
select an option from a drop-down list or enter the value of the option
in a text field.
If you are calling ga
or gamultiobj
at
the command line, create options
using the function optimoptions
,
as follows:
options = optimoptions('ga','Param1', value1, 'Param2', value2, ...); % or options = optimoptions('gamultiobj','Param1', value1, 'Param2', value2, ...);
See Setting Options at the Command Line for examples.
In this section, each option is listed in two ways:
By its label, as it appears in the Optimization app
By its field name in the options
structure
For example:
Population type is the label of the option in the Optimization app.
PopulationType
is the corresponding
field of the options
structure.
Plot options let you plot data from the genetic algorithm while it is running. You can stop the algorithm at any time by clicking the Stop button on the plot window.
Plot interval (PlotInterval
)
specifies the number of generations between consecutive calls to the
plot function.
You can select any of the following plot functions in the Plot
functions pane for both ga
and gamultiobj
:
Score diversity (@gaplotscorediversity
)
plots a histogram of the scores at each generation.
Stopping (@gaplotstopping
)
plots stopping criteria levels.
Genealogy (@gaplotgenealogy
)
plots the genealogy of individuals. Lines from one generation to the
next are color-coded as follows:
Red lines indicate mutation children.
Blue lines indicate crossover children.
Black lines indicate elite individuals.
Scores (@gaplotscores
)
plots the scores of the individuals at each generation.
Distance (@gaplotdistance
)
plots the average distance between individuals at each generation.
Selection (@gaplotselection
)
plots a histogram of the parents.
Max constraint (@gaplotmaxconstr
)
plots the maximum nonlinear constraint violation at each generation.
For ga
, available only for the Augmented
Lagrangian
('auglag'
) Nonlinear
constraint algorithm (NonlinearConstraintAlgorithm
)
option. Therefore, not available for integer-constrained problems,
as they use the Penalty
('penalty'
)
nonlinear constraint algorithm.
Custom function lets you use plot functions of your own. To specify the plot function if you are using the Optimization app,
Select Custom function.
Enter @myfun
in the text box, where myfun
is
the name of your function.
The following plot functions are available for ga
only:
Best fitness (@gaplotbestf
)
plots the best score value and mean score versus generation.
Best individual (@gaplotbestindiv
)
plots the vector entries of the individual with the best fitness function
value in each generation.
Expectation (@gaplotexpectation
)
plots the expected number of children versus the raw scores at each
generation.
Range (@gaplotrange
)
plots the minimum, maximum, and mean score values in each generation.
The following plot functions are available for gamultiobj
only:
Pareto front (@gaplotpareto
)
plots the Pareto front for the first two objective functions.
Average Pareto distance (@gaplotparetodistance
)
plots a bar chart of the distance of each individual from its neighbors.
Rank histogram (@gaplotrankhist
)
plots a histogram of the ranks of the individuals. Individuals of
rank 1 are on the Pareto frontier. Individuals of rank 2 are lower
than at least one rank 1 individual, but are not lower than any individuals
from other ranks, etc.
Average Pareto spread (@gaplotspread
)
plots the average spread as a function of iteration number.
To display a plot when calling ga
or gamultiobj
from
the command line, set the PlotFcn
option to be
a function handle to the plot function. For example, to display the
best fitness plot, set options
as follows:
options = optimoptions('ga','PlotFcn', @gaplotbestf);
To display multiple plots, use the syntax
options = optimoptions('ga','PlotFcn', {@plotfun1, @plotfun2, ...});
where @plotfun1
, @plotfun2
,
and so on are function handles to the plot functions.
If you specify multiple plot functions, all plots appear as subplots in the same window. Right-click any subplot to obtain a larger version in a separate figure window.
The first line of a plot function has this form:
function state = plotfun(options,state,flag)
The input arguments to the function are
options
— Structure containing
all the current options settings.
state
— Structure containing
information about the current generation. The State Structure describes the fields of state
.
flag
— Description of the
stage the algorithm is currently in.
Passing Extra Parameters (Optimization Toolbox) explains how to provide additional parameters to the function.
The output argument state
is a state structure
as well. Pass the input argument, modified if you like; see Changing the State Structure. To stop
the iterations, set state.StopFlag
to a nonempty
character vector, such as 'y'
.
ga. The state structure for ga
, which is an
input argument to plot, mutation, and output functions, contains the
following fields:
Generation
— Current generation
number
StartTime
— Time when genetic
algorithm started, returned by tic
StopFlag
— Reason for stopping,
a character vector
LastImprovement
— Generation
at which the last improvement in fitness value occurred
LastImprovementTime
— Time
at which last improvement occurred
Best
— Vector containing
the best score in each generation
how
— The 'augLag'
nonlinear
constraint algorithm reports one of the following actions: 'Infeasible
point'
, 'Update multipliers'
, or 'Increase
penalty'
; see Augmented Lagrangian Genetic Algorithm
FunEval
— Cumulative number
of function evaluations
Expectation
— Expectation
for selection of individuals
Selection
— Indices of individuals
selected for elite, crossover, and mutation
Population
— Population
in the current generation
Score
— Scores of the current
population
NonlinIneq
— Nonlinear inequality
constraints at current point, present only when a nonlinear constraint
function is specified and there are no integer variables
NonlinEq
— Nonlinear equality
constraints at current point, present only when a nonlinear constraint
function is specified and there are no integer variables
gamultiobj. The state structure for gamultiobj
, which
is an input argument to plot, mutation, and output functions, contains
the following fields:
Population
— Population
in the current generation
Score
— Scores of the current
population, a Population
-by-nObjectives
matrix,
where nObjectives
is the number of objectives
Generation
— Current generation
number
StartTime
— Time when genetic
algorithm started, returned by tic
StopFlag
— Reason for stopping,
a character vector
FunEval
— Cumulative number
of function evaluations
Selection
— Indices of individuals
selected for elite, crossover, and mutation
Rank
— Vector of the ranks
of members in the population
Distance
— Vector of distances
of each member of the population to the nearest neighboring member
AverageDistance
— The average
of Distance
Spread
— Vector where the
entries are the spread in each generation
mIneq
— Number of nonlinear
inequality constraints
mEq
— Number of nonlinear
equality constraints
mAll
— Total number of nonlinear
constraints, mAll
= mIneq
+ mEq
C
— Nonlinear inequality
constraints at current point, a PopulationSize
-by-mIneq
matrix
Ceq
— Nonlinear equality
constraints at current point, a PopulationSize
-by-mEq
matrix
isFeas
— Feasibility of
population, a logical vector with PopulationSize
elements
maxLinInfeas
— Maximum infeasibility
with respect to linear constraints for the population
Population options let you specify the parameters of the population that the genetic algorithm uses.
Population type (PopulationType
)
specifies the type of input to the fitness function. Types and their
restrictions are:
Double vector
('doubleVector'
)
— Use this option if the individuals in the population have
type double
. Use this option for mixed integer
programming. This is the default.
Bit string
('bitstring'
)
— Use this option if the individuals in the population have
components that are 0
or 1
.
Caution:
The individuals in a |
For Creation function (CreationFcn
)
and Mutation function (MutationFcn
),
use Uniform
(@gacreationuniform
and @mutationuniform
)
or Custom
. For Crossover function (CrossoverFcn
),
use Scattered
(@crossoverscattered
), Single
point
(@crossoversinglepoint
), Two
point
(@crossovertwopoint
), or Custom
.
You cannot use a Hybrid function, and ga
ignores
all constraints, including bounds, linear constraints, and nonlinear
constraints.
Custom
— For Crossover
function and Mutation function, use Custom
.
For Creation function, either use Custom
,
or provide an Initial population. You cannot
use a Hybrid function, and ga
ignores
all constraints, including bounds, linear constraints, and nonlinear
constraints.
Population size (PopulationSize
)
specifies how many individuals there are in each generation. With
a large population size, the genetic algorithm searches the solution
space more thoroughly, thereby reducing the chance that the algorithm
returns a local minimum that is not a global minimum. However, a large
population size also causes the algorithm to run more slowly.
If you set Population size to a vector, the genetic algorithm creates multiple subpopulations, the number of which is the length of the vector. The size of each subpopulation is the corresponding entry of the vector. See Migration Options.
Creation
function (CreationFcn
) specifies the
function that creates the initial population for ga
.
Do not specify a creation function with integer problems because ga
overrides
any choice you make. Choose from:
[]
uses the default creation function
for your problem.
Uniform
(@gacreationuniform
)
creates a random initial population with a uniform distribution. This
is the default when there are no linear constraints, or when there
are integer constraints. The uniform distribution is in the initial
population range (InitialPopulationRange
). The
default values for InitialPopulationRange
are [-10;10]
for
every component, or [-9999;10001]
when there are
integer constraints. These bounds are shifted and scaled to match
any existing bounds lb
and ub
.
Caution:
Do not use |
Feasible population
(@gacreationlinearfeasible
),
the default when there are linear constraints and no integer constraints,
creates a random initial population that satisfies all bounds and
linear constraints. If there are linear constraints, Feasible
population
creates many individuals on the boundaries
of the constraint region, and creates a well-dispersed population. Feasible
population
ignores Initial range (InitialPopulationRange
).
gacreationlinearfeasible
calls linprog
to
create a feasible population with respect to bounds and linear constraints.
For an example showing its behavior, see Custom Plot Function and Linear Constraints in ga
.
Nonlinear Feasible population
(@gacreationnonlinearfeasible
)
is the default creation function for the 'penalty'
nonlinear
constraint algorithm. For details, see Constraint Parameters.
Custom
lets
you write your own creation function, which must generate data of
the type that you specify in Population type.
To specify the creation function if you are using the Optimization
app,
Set Creation function to Custom
.
Set Function name to @myfun
,
where myfun
is the name of your function.
If you are using ga
, set
options = optimoptions('ga','CreationFcn', @myfun);
Your creation function must have the following calling syntax.
function Population = myfun(GenomeLength, FitnessFcn, options)
The input arguments to the function are:
Genomelength
— Number of
independent variables for the fitness function
FitnessFcn
— Fitness function
options
— Options structure
The function returns Population
, the initial
population for the genetic algorithm.
Passing Extra Parameters (Optimization Toolbox) explains how to provide additional parameters to the function.
Caution: When you have bounds or linear constraints, ensure that your creation function creates individuals that satisfy these constraints. Otherwise, your population might not satisfy the constraints. |
Initial population (InitialPopulationMatrix
)
specifies an initial population for the genetic algorithm. The default
value is []
, in which case ga
uses
the default Creation function to create an initial
population. If you enter a nonempty array in the Initial
population field, the array must have no more than Population
size rows, and exactly Number of variables columns.
If you have a partial initial population, meaning
fewer than Population size rows, then the genetic
algorithm calls a Creation function to generate
the remaining individuals.
Initial scores (InitialScoreMatrix
)
specifies initial scores for the initial population. The initial scores
can also be partial. Do not specify initial scores with integer problems
because ga
overrides any choice you make.
Initial range (InitialPopulationRange
)
specifies the range of the vectors in the initial population that
is generated by the gacreationuniform
creation
function. You can set Initial range to be a matrix
with two rows and Number of variables columns,
each column of which has the form [lb;ub]
, where lb
is
the lower bound and ub
is the upper bound for the
entries in that coordinate. If you specify Initial range to
be a 2-by-1 vector, each entry is expanded to a constant row of length Number
of variables. If you do not specify an Initial
range, the default is [-10;10]
([-1e4+1;1e4+1]
for
integer-constrained problems), modified to match any existing bounds.
See Setting the Initial Range for an example.
Fitness scaling converts the raw fitness scores that are returned by the fitness function to values in a range that is suitable for the selection function. You can specify options for fitness scaling in the Fitness scaling pane.
Scaling function (FitnessScalingFcn
)
specifies the function that performs the scaling. The options are
Rank
(@fitscalingrank
)
— The default fitness scaling function, Rank
,
scales the raw scores based on the rank of each individual instead
of its score. The rank of an individual is its position in the sorted
scores. An individual with rank r has scaled score
proportional to $$1/\sqrt{r}$$. So the scaled
score of the most fit individual is proportional to 1, the scaled
score of the next most fit is proportional to $$1/\sqrt{2}$$, and so on. Rank fitness scaling
removes the effect of the spread of the raw scores. The square root
makes poorly ranked individuals more nearly equal in score, compared
to rank scoring. For more information, see Fitness Scaling.
Proportional
(@fitscalingprop
)
— Proportional scaling makes the scaled value of an individual
proportional to its raw fitness score.
Top
(@fitscalingtop
)
— Top scaling scales the top individuals equally. Selecting Top
displays
an additional field, Quantity, which specifies
the number of individuals that are assigned positive scaled values. Quantity can
be an integer from 1 through the population size or a fraction from
0 through 1 specifying a fraction of the population size. The default
value is 0.4
. Each of the individuals that produce
offspring is assigned an equal scaled value, while the rest are assigned
the value 0. The scaled values have the form [01/n 1/n 0 0 1/n 0 0
1/n ...].
To change the default value for Quantity at the command line, use the following syntax:
options = optimoptions('ga','FitnessScalingFcn', {@fitscalingtop,quantity})
where quantity
is the value of Quantity.
Shift linear
(@fitscalingshiftlinear
)
— Shift linear scaling scales the raw scores so that the expectation
of the fittest individual is equal to a constant multiplied by the
average score. You specify the constant in the Max survival
rate field, which is displayed when you select Shift
linear
. The default value is 2
.
To change the default value of Max survival rate at the command line, use the following syntax
options = optimoptions('ga','FitnessScalingFcn', {@fitscalingshiftlinear, rate})
where rate
is the value of Max
survival rate.
Custom
lets
you write your own scaling function. To specify the scaling function
using the Optimization app,
Set Scaling function to Custom
.
Set Function name to @myfun
,
where myfun
is the name of your function.
If you are using ga
at the command line,
set
options = optimoptions('ga','FitnessScalingFcn', @myfun);
Your scaling function must have the following calling syntax:
function expectation = myfun(scores, nParents)
The input arguments to the function are:
scores
— A vector of scalars,
one for each member of the population
nParents
— The number of
parents needed from this population
The function returns expectation
, a column
vector of scalars of the same length as scores
,
giving the scaled values of each member of the population. The sum
of the entries of expectation
must equal nParents
.
Passing Extra Parameters (Optimization Toolbox) explains how to provide additional parameters to the function.
See Fitness Scaling for more information.
Selection options specify how the genetic algorithm chooses
parents for the next generation. You can specify the function the
algorithm uses in the Selection function (SelectionFcn
)
field in the Selection options pane. Do
not use with integer problems.
gamultiobj
uses only the Tournament
(@selectiontournament
)
selection function.
For ga
the options are:
Stochastic uniform
(@selectionstochunif
)
— The ga
default selection function, Stochastic
uniform
, lays out a line in which each parent corresponds
to a section of the line of length proportional to its scaled value.
The algorithm moves along the line in steps of equal size. At each
step, the algorithm allocates a parent from the section it lands on.
The first step is a uniform random number less than the step size.
Remainder
(@selectionremainder
)
— Remainder selection assigns parents deterministically from
the integer part of each individual's scaled value and then uses roulette
selection on the remaining fractional part. For example, if the scaled
value of an individual is 2.3, that individual is listed twice as
a parent because the integer part is 2. After parents have been assigned
according to the integer parts of the scaled values, the rest of the
parents are chosen stochastically. The probability that a parent is
chosen in this step is proportional to the fractional part of its
scaled value.
Uniform
(@selectionuniform
)
— Uniform selection chooses parents using the expectations
and number of parents. Uniform selection is useful for debugging and
testing, but is not a very effective search strategy.
Roulette
(@selectionroulette
)
— Roulette selection chooses parents by simulating a roulette
wheel, in which the area of the section of the wheel corresponding
to an individual is proportional to the individual's expectation.
The algorithm uses a random number to select one of the sections with
a probability equal to its area.
Tournament
(@selectiontournament
)
— Tournament selection chooses each parent by choosing Tournament
size players at random and then choosing the best individual
out of that set to be a parent. Tournament size must
be at least 2. The default value of Tournament size is 4
.
To change the default value of Tournament size at the command line, use the syntax
options = optimoptions('ga','SelectionFcn',... {@selectiontournament,size})
where size
is the value of Tournament
size.
When Constraint parameters > Nonlinear constraint
algorithm is Penalty
, ga
uses Tournament
with
size 2
.
Custom
enables
you to write your own selection function. To specify the selection
function using the Optimization app,
Set Selection function to Custom
.
Set Function name to @myfun
,
where myfun
is the name of your function.
If you are using ga
at the command line,
set
options = optimoptions('ga','SelectionFcn', @myfun);
Your selection function must have the following calling syntax:
function parents = myfun(expectation, nParents, options)
ga
provides the input arguments expectation
, nParents
,
and options
. Your function returns the indices
of the parents.
The input arguments to the function are:
expectation
For ga
, expectation
is
a column vector of the scaled fitness of each member of the population.
The scaling comes from the Fitness Scaling Options.
Tip:
You can ensure that you have a column vector by using |
For gamultiobj
, expectation
is
a matrix whose first column is the rank of the individuals, and whose
second column is the distance measure of the individuals. See Multiobjective Options.
nParents
— Number of parents
to select.
options
— Genetic algorithm options
structure.
The function returns parents
, a row vector
of length nParents
containing the indices of the
parents that you select.
Passing Extra Parameters (Optimization Toolbox) explains how to provide additional parameters to the function.
See Selection for more information.
Reproduction options specify how the genetic algorithm creates children for the next generation.
Elite count (EliteCount
)
specifies the number of individuals that are guaranteed to survive
to the next generation. Set Elite count to be
a positive integer less than or equal to the population size. The
default value is ceil(0.05*PopulationSize)
for
continuous problems, and 0.05*(default PopulationSize)
for
mixed-integer problems.
Crossover fraction (CrossoverFraction
)
specifies the fraction of the next generation, other than elite children,
that are produced by crossover. Set Crossover fraction to
be a fraction between 0
and 1
,
either by entering the fraction in the text box or moving the slider.
The default value is 0.8
.
See Setting the Crossover Fraction for an example.
Mutation options specify how the genetic algorithm makes small
random changes in the individuals in the population to create mutation
children. Mutation provides genetic diversity and enables the genetic
algorithm to search a broader space. You can specify the mutation
function in the Mutation function (MutationFcn
)
field in the Mutation options pane. Do not use
with integer problems. You can choose from the following functions:
Gaussian
(mutationgaussian
)
— The default mutation function for unconstrained problems, Gaussian
,
adds a random number taken from a Gaussian distribution with mean
0 to each entry of the parent vector. The standard deviation of this
distribution is determined by the parameters Scale and Shrink,
which are displayed when you select Gaussian
, and
by the Initial range setting in the Population options.
The Scale parameter determines
the standard deviation at the first generation. If you set Initial
range to be a 2-by-1 vector v
, the initial
standard deviation is the same at all coordinates of the parent vector,
and is given by Scale*(v(2)-v(1))
.
If you set Initial range to be a vector v
with
two rows and Number of variables columns, the
initial standard deviation at coordinate i
of the
parent vector is given by Scale*(v(i,2)
- v(i,1))
.
The Shrink parameter controls how the standard deviation shrinks as generations go by. If you set Initial range to be a 2-by-1 vector, the standard deviation at the kth generation, σ_{k}, is the same at all coordinates of the parent vector, and is given by the recursive formula
$${\sigma}_{k}={\sigma}_{k-1}\left(1-\text{Shrink}\frac{k}{\text{Generations}}\right).$$
If you set Initial range to be a vector with two rows and Number of variables columns, the standard deviation at coordinate i of the parent vector at the kth generation, σ_{i,k}, is given by the recursive formula
$${\sigma}_{i,k}={\sigma}_{i,k-1}\left(1-\text{Shrink}\frac{k}{\text{Generations}}\right).$$
If you set Shrink to 1
,
the algorithm shrinks the standard deviation in each coordinate linearly
until it reaches 0 at the last generation is reached. A negative value
of Shrink causes the standard deviation to grow.
The default value of both Scale and Shrink is 1. To change the default values at the command line, use the syntax
options = optimoptions('ga','MutationFcn', ... {@mutationgaussian, scale, shrink})
where scale
and shrink
are
the values of Scale and Shrink,
respectively.
Caution:
Do not use |
Uniform
(mutationuniform
)
— Uniform mutation is a two-step process. First, the algorithm
selects a fraction of the vector entries of an individual for mutation,
where each entry has a probability Rate of being
mutated. The default value of Rate is 0.01
.
In the second step, the algorithm replaces each selected entry by
a random number selected uniformly from the range for that entry.
To change the default value of Rate at the command line, use the syntax
options = optimoptions('ga','MutationFcn', {@mutationuniform, rate})
where rate
is the value of Rate.
Caution:
Do not use |
Adaptive Feasible
(mutationadaptfeasible
),
the default mutation function when there are constraints, randomly
generates directions that are adaptive with respect to the last successful
or unsuccessful generation. The mutation chooses a direction and step
length that satisfies bounds and linear constraints.
Custom
enables
you to write your own mutation function. To specify the mutation function
using the Optimization app,
Set Mutation function to Custom
.
Set Function name to @myfun
,
where myfun
is the name of your function.
If you are using ga
, set
options = optimoptions('ga','MutationFcn', @myfun);
Your mutation function must have this calling syntax:
function mutationChildren = myfun(parents, options, nvars, FitnessFcn, state, thisScore, thisPopulation)
The arguments to the function are
parents
— Row vector of
parents chosen by the selection function
options
— Options structure
nvars
— Number of variables
FitnessFcn
— Fitness function
state
— Structure containing
information about the current generation. The State Structure describes the fields of state
.
thisScore
— Vector of scores
of the current population
thisPopulation
— Matrix
of individuals in the current population
The function returns mutationChildren
—the
mutated offspring—as a matrix where rows correspond to the
children. The number of columns of the matrix is Number
of variables.
Passing Extra Parameters (Optimization Toolbox) explains how to provide additional parameters to the function.
Caution: When you have bounds or linear constraints, ensure that your mutation function creates individuals that satisfy these constraints. Otherwise, your population will not necessarily satisfy the constraints. |
Crossover options specify how the genetic algorithm combines two individuals, or parents, to form a crossover child for the next generation.
Crossover function (CrossoverFcn
)
specifies the function that performs the crossover. Do not use with
integer problems. You can choose from the following functions:
Scattered
(@crossoverscattered
),
the default crossover function for problems without linear constraints,
creates a random binary vector and selects the genes where the vector
is a 1 from the first parent, and the genes where the vector is a
0 from the second parent, and combines the genes to form the child.
For example, if p1
and p2
are
the parents
p1 = [a b c d e f g h] p2 = [1 2 3 4 5 6 7 8]
and the binary vector is [1 1 0 0 1 0 0 0], the function returns the following child:
child1 = [a b 3 4 e 6 7 8]
Caution:
Do not use |
Single point
(@crossoversinglepoint
)
chooses a random integer n between 1 and Number of variables and
then
Selects vector entries numbered less than or equal to n from the first parent.
Selects vector entries numbered greater than n from the second parent.
Concatenates these entries to form a child vector.
For example, if p1
and p2
are
the parents
p1 = [a b c d e f g h] p2 = [1 2 3 4 5 6 7 8]
and the crossover point is 3, the function returns the following child.
child = [a b c 4 5 6 7 8]
Caution:
Do not use |
Two point
(@crossovertwopoint
)
selects two random integers m
and n
between 1
and Number
of variables. The function selects
Vector entries numbered less than or equal to m
from
the first parent
Vector entries numbered from m+1
to n
,
inclusive, from the second parent
Vector entries numbered greater than n
from
the first parent.
The algorithm then concatenates these genes to form a single
gene. For example, if p1
and p2
are
the parents
p1 = [a b c d e f g h] p2 = [1 2 3 4 5 6 7 8]
and the crossover points are 3 and 6, the function returns the following child.
child = [a b c 4 5 6 g h]
Caution:
Do not use |
Intermediate
(@crossoverintermediate
),
the default crossover function when there are linear constraints,
creates children by taking a weighted average of the parents. You
can specify the weights by a single parameter, Ratio,
which can be a scalar or a row vector of length Number of
variables. The default is a vector of all 1's. The function
creates the child from parent1
and parent2
using
the following formula.
child = parent1 + rand * Ratio * ( parent2 - parent1)
If all the entries of Ratio lie in the range [0, 1], the children produced are within the hypercube defined by placing the parents at opposite vertices. If Ratio is not in that range, the children might lie outside the hypercube. If Ratio is a scalar, then all the children lie on the line between the parents.
To change the default value of Ratio at the command line, use the syntax
options = optimoptions('ga','CrossoverFcn', ... {@crossoverintermediate, ratio});
where ratio
is the value of Ratio.
Heuristic
(@crossoverheuristic
)
returns a child that lies on the line containing the two parents,
a small distance away from the parent with the better fitness value
in the direction away from the parent with the worse fitness value.
You can specify how far the child is from the better parent by the
parameter Ratio, which appears when you select Heuristic
.
The default value of Ratio is 1.2. If parent1
and parent2
are
the parents, and parent1
has the better fitness
value, the function returns the child
child = parent2 + R * (parent1 - parent2);
To change the default value of Ratio at the command line, use the syntax
options = optimoptions('ga','CrossoverFcn',... {@crossoverheuristic,ratio});
where ratio
is the value of Ratio.
Arithmetic
(@crossoverarithmetic
)
creates children that are the weighted arithmetic mean of two parents.
Children are always feasible with respect to linear constraints and
bounds.
Custom
enables
you to write your own crossover function. To specify the crossover
function using the Optimization app,
Set Crossover function to Custom
.
Set Function name to @myfun
,
where myfun
is the name of your function.
If you are using ga
, set
options = optimoptions('ga','CrossoverFcn',@myfun);
Your crossover function must have the following calling syntax.
xoverKids = myfun(parents, options, nvars, FitnessFcn, ... unused,thisPopulation)
The arguments to the function are
parents
— Row vector of
parents chosen by the selection function
options
— options
structure
nvars
— Number of variables
FitnessFcn
— Fitness function
unused
— Placeholder not
used
thisPopulation
— Matrix
representing the current population. The number of rows of the matrix
is Population size and the number of columns
is Number of variables.
The function returns xoverKids
—the
crossover offspring—as a matrix where rows correspond to the
children. The number of columns of the matrix is Number
of variables.
Passing Extra Parameters (Optimization Toolbox) explains how to provide additional parameters to the function.
Caution: When you have bounds or linear constraints, ensure that your crossover function creates individuals that satisfy these constraints. Otherwise, your population will not necessarily satisfy the constraints. |
Note:
Subpopulations refer to a form of parallel
processing for the genetic algorithm. Because |
Migration options specify how individuals move between subpopulations. Migration occurs if you set Population size to be a vector of length greater than 1. When migration occurs, the best individuals from one subpopulation replace the worst individuals in another subpopulation. Individuals that migrate from one subpopulation to another are copied. They are not removed from the source subpopulation.
You can control how migration occurs by the following three fields in the Migration options pane:
Direction (MigrationDirection
)
— Migration can take place in one or both directions.
If you set Direction to Forward
('forward'
),
migration takes place toward the last subpopulation. That is, the nth
subpopulation migrates into the (n+1)th subpopulation.
If you set Direction to Both
('both'
),
the n^{th} subpopulation
migrates into both the (n–1)th and the (n+1)th
subpopulation.
Migration wraps at the ends of the subpopulations. That is, the last subpopulation migrates into the first, and the first may migrate into the last.
Interval (MigrationInterval
)
— Specifies how many generation pass between migrations. For
example, if you set Interval to 20
,
migration takes place every 20 generations.
Fraction (MigrationFraction
)
— Specifies how many individuals move between subpopulations. Fraction specifies
the fraction of the smaller of the two subpopulations that moves.
For example, if individuals migrate from a subpopulation of 50 individuals
into a subpopulation of 100 individuals and you set Fraction to 0.1
,
the number of individuals that migrate is 0.1*50=5.
Constraint parameters refer to the nonlinear constraint solver. For details on the algorithm, see Nonlinear Constraint Solver Algorithms.
Choose between the nonlinear constraint algorithms by setting
the NonlinearConstraintAlgorithm
option to 'auglag'
(Augmented
Lagrangian) or 'penalty'
(Penalty algorithm).
Initial penalty (InitialPenalty
)
— Specifies an initial value of the penalty parameter that
is used by the nonlinear constraint algorithm. Initial penalty must
be greater than or equal to 1
, and has a default
of 10
.
Penalty factor (PenaltyFactor
)
— Increases the penalty parameter when the problem is not solved
to required accuracy and constraints are not satisfied. Penalty
factor must be greater than 1
, and has
a default of 100
.
The penalty algorithm uses the gacreationnonlinearfeasible
creation
function by default. This creation function uses fmincon
to
find feasible individuals. gacreationnonlinearfeasible
starts fmincon
from
a variety of initial points within the bounds from the InitialPopulationRange
option.
Optionally, gacreationnonlinearfeasible
can run fmincon
in
parallel on the initial points.
You can specify tuning parameters for gacreationnonlinearfeasible
using
the following name-value pairs.
Name | Value |
---|---|
SolverOpts | fmincon options, created using optimoptions or optimset . |
UseParallel | When true , run fmincon in
parallel on initial points; default is false . |
NumStartPts | Number of start points, a positive integer up to sum(PopulationSize) in
value. |
Include the name-value pairs in a cell array along with @gacreationnonlinearfeasible
.
options = optimoptions('ga','CreationFcn',{@gacreationnonlinearfeasible
,...
'UseParallel',true,'NumStartPts',20});
Multiobjective options define parameters characteristic of the multiobjective genetic algorithm. You can specify the following parameters:
ParetoFraction
— Sets the
fraction of individuals to keep on the first Pareto front while the
solver selects individuals from higher fronts. This option is a scalar
between 0 and 1.
DistanceMeasureFcn
— Defines
a handle to the function that computes distance measure of individuals,
computed in decision variable or design space (genotype) or in function
space (phenotype). For example, the default distance measure function
is @distancecrowding
in function space, which is
the same as @distancecrowding,'phenotype'
.
"Distance" measures a crowding of each individual in a population. Choose between the following:
@distancecrowding
, or the equivalent @distancecrowding,'phenotype'
—
Measure the distance in fitness function space.
@distancecrowding,'genotype'
—
Measure the distance in control variable space.
@distancefunction
— Write
a custom distance function using the following template.
function distance = distancefunction(pop,score,options) % Uncomment one of the following two lines, or use a combination of both % y = score; % phenotype % y = pop; % genotype popSize = size(y,1); % number of individuals numData = size(y,2); % number of dimensions or fitness functions distance = zeros(popSize,1); % allocate the output % Compute distance here
gamultiobj
passes the population in pop
,
the computed scores for the population in scores
,
and the options in options
. Your distance function
returns the distance from each member of the population to a reference,
such as the nearest neighbor in some sense. For an example, edit the
built-in file distancecrowding.m
.
ga
Hybrid FunctionA hybrid function is another minimization function that runs
after the genetic algorithm terminates. You can specify a hybrid function
in Hybrid function (HybridFcn
)
options. Do not use with integer problems. The choices are
[]
— No hybrid function.
fminsearch
(@fminsearch
)
— Uses the MATLAB^{®} function fminsearch
to
perform unconstrained minimization.
patternsearch
(@patternsearch
)
— Uses a pattern search to perform constrained or unconstrained
minimization.
fminunc
(@fminunc
)
— Uses the Optimization Toolbox™ function fminunc
to perform unconstrained minimization.
fmincon
(@fmincon
)
— Uses the Optimization Toolbox function fmincon
to perform constrained minimization.
Note:
Ensure that your hybrid function accepts your problem constraints.
Otherwise, |
You can set separate options for the hybrid function. Use optimset
for fminsearch
,
or optimoptions
for fmincon
, patternsearch
,
or fminunc
. For example:
hybridopts = optimoptions('fminunc','Display','iter','Algorithm','quasi-newton');
options
structure
as follows:options = optimoptions('ga',options,'HybridFcn',{@fminunc,hybridopts});
hybridopts
must
exist before you set options
.See Include a Hybrid Function for an example.
gamultiobj
Hybrid FunctionA hybrid function is another minimization function that runs
after the multiobjective genetic algorithm terminates. You can specify
the hybrid function fgoalattain
in Hybrid
function (HybridFcn
) options.
In use as a multiobjective hybrid function, the solver does the following:
Compute the maximum and minimum of each objective function at the solutions. For objective j at solution k, let
$$\begin{array}{c}{F}_{\mathrm{max}}(j)=\underset{k}{\mathrm{max}}{F}_{k}(j)\\ {F}_{\mathrm{min}}(j)=\underset{k}{\mathrm{min}}{F}_{k}(j).\end{array}$$
Compute the total weight at each solution k,
$$w(k)={\displaystyle \sum _{j}\frac{{F}_{\mathrm{max}}(j)-{F}_{k}(j)}{1+{F}_{\mathrm{max}}(j)-{F}_{\mathrm{min}}(j)}.}$$
Compute the weight for each objective function j at each solution k,
$$p(j,k)=w(k)\frac{{F}_{\mathrm{max}}(j)-{F}_{k}(j)}{1+{F}_{\mathrm{max}}(j)-{F}_{\mathrm{min}}(j)}.$$
For each solution k, perform the goal attainment problem with goal vector F_{min} and weight vector p(j,k).
For more information, see section 9.6 of Deb [3].
Stopping criteria determine what causes the algorithm to terminate. You can specify the following options:
Generations (MaxGenerations
)
— Specifies the maximum number of iterations for the genetic
algorithm to perform. The default is 100*numberOfVariables
.
Time limit (MaxTime
)
— Specifies the maximum time in seconds the genetic algorithm
runs before stopping, as measured by tic
and toc
. This limit is enforced after each
iteration, so ga
can exceed the limit when an
iteration takes substantial time.
Fitness limit (FitnessLimit
)
— The algorithm stops if the best fitness value is less than
or equal to the value of Fitness limit.
Stall generations (MaxStallGenerations
)
— The algorithm stops if the average relative change in the
best fitness function value over Stall generations is
less than or equal to Function tolerance. (If
the Stall Test (StallTest
)
option is 'geometricWeighted'
, then the test is
for a geometric weighted average relative change.)
For a problem with nonlinear constraints, Stall generations applies
to the subproblem (see Nonlinear Constraint Solver Algorithms).
For gamultiobj
, if the weighted average
relative change in the spread of the Pareto
solutions over Stall generations is less than Function
tolerance, and the spread is smaller than the average spread
over the last Stall generations, then the algorithm
stops. The spread is a measure of the movement
of the Pareto front.
Stall time limit (MaxStallTime
)
— The algorithm stops if there is no improvement in the best
fitness value for an interval of time in seconds specified by Stall
time limit, as measured by tic
and toc
.
Function tolerance (FunctionTolerance
)
— The algorithm stops if the average relative change in the
best fitness function value over Stall generations is
less than or equal to Function tolerance. (If
the StallTest
option is 'geometricWeighted'
,
then the test is for a geometric weighted average
relative change.)
For gamultiobj
, if the weighted average
relative change in the spread of the Pareto
solutions over Stall generations is less than Function
tolerance, and the spread is smaller than the average spread
over the last Stall generations, then the algorithm
stops. The spread is a measure of the movement
of the Pareto front.
Constraint tolerance (ConstraintTolerance
)
— The Constraint tolerance is not used
as stopping criterion. It is used to determine the feasibility with
respect to nonlinear constraints. Also, max(sqrt(eps),ConstraintTolerance)
determines
feasibility with respect to linear constraints.
See Set Maximum Number of Generations for an example.
Output functions are functions that the genetic algorithm calls
at each generation. Unlike all other solvers, a ga
output
function can not only read the values of the state of the algorithm,
but can modify those values.
To specify the output function using the Optimization app,
Select Custom function.
Enter @myfun
in the text box, where myfun
is
the name of your function. Write myfun
with appropriate
syntax.
To pass extra parameters in the output function, use Anonymous Functions (Optimization Toolbox).
For multiple output functions, enter a cell array
of output function handles: {@myfun1,@myfun2,...}
.
At the command line, set
options = optimoptions('ga','OutputFcn',@myfun);
For multiple output functions, enter a cell array:
options = optimoptions('ga','OutputFcn',{@myfun1,@myfun2,...});
To see a template that you can use to write your own output functions, enter
edit gaoutputfcntemplate
at the MATLAB command line.
For an example, see Custom Output Function for Genetic Algorithm.
Your output function must have the following calling syntax:
[state,options,optchanged] = myfun(options,state,flag)
MATLAB passes the options
, state
,
and flag
data to your output function, and the
output function returns state
, options
,
and optchanged
data.
Note:
To stop the iterations, set |
The output function has the following input arguments:
options
— Options structure
state
— Structure containing
information about the current generation. The State Structure describes the fields of state
.
flag
— Current status of
the algorithm:
'init'
— Initialization
state
'iter'
— Iteration state
'interrupt'
— Iteration
of a subproblem of a nonlinearly constrained problem
When flag
is 'interrupt'
,
the values of optimvalues
fields apply to the subproblem
iterations.
When flag
is 'interrupt'
, ga
does
not accept changes in options
, and ignores optchanged
.
'done'
— Final state
Passing Extra Parameters (Optimization Toolbox) explains how to provide additional parameters to the function.
The output function returns the following arguments to ga
:
state
— Structure containing
information about the current generation. The State Structure describes the fields of state
.
To stop the iterations, set state.StopFlag
to a
nonempty character vector, such as 'y'
.
options
— Options structure
modified by the output function. This argument is optional.
optchanged
— Boolean flag
indicating changes to options
. To change options
for
subsequent iterations, set optchanged
to true
.
Caution: Changing the state structure carelessly can lead to inconsistent or erroneous results. Usually, you can achieve the same or better state modifications by using mutation or crossover functions, instead of changing the state structure in a plot function or output function. |
ga
output functions can change the state
structure
(see The State Structure). Be careful
when changing values in this structure, as you can pass inconsistent
data back to ga
.
Tip:
If your output structure changes the |
To update the Score
field after changing
the Population
field, first calculate the fitness
function values of the population, then calculate the fitness scaling
for the population. See Fitness Scaling Options.
Level of display ('Display'
)
specifies how much information is displayed at the command line while
the genetic algorithm is running. The available options are
Off
('off'
)
— No output is displayed.
Iterative
('iter'
)
— Information is displayed at each iteration.
Diagnose
('diagnose'
)
— Information is displayed at each iteration. In addition,
the diagnostic lists some problem information and the options that
have been changed from the defaults.
Final
('final'
)
— The reason for stopping is displayed.
Both Iterative
and Diagnose
display
the following information:
Generation
— Generation
number
f-count
— Cumulative number
of fitness function evaluations
Best f(x)
— Best fitness
function value
Mean f(x)
— Mean fitness
function value
Stall generations
— Number
of generations since the last improvement of the fitness function
When a nonlinear constraint function has been specified, Iterative
and Diagnose
do
not display the Mean f(x)
, but will additionally
display:
Max Constraint
— Maximum
nonlinear constraint violation
The default value of Level of display is
Off
in the Optimization app
'final'
in options created using optimoptions
You can choose to have your fitness and constraint functions
evaluated in serial, parallel, or in a vectorized fashion. These options
are available in the User function evaluation section
of the Options pane of the Optimization app,
or by setting the 'UseVectorized'
and 'UseParallel'
options
with optimoptions
.
When Evaluate fitness and constraint functions ('UseVectorized'
)
is in serial (false
), ga
calls
the fitness function on one individual at a time as it loops through
the population. (At the command line, this assumes 'UseParallel'
is
at its default value of false
.)
When Evaluate fitness and constraint functions ('UseVectorized'
)
is vectorized (true
), ga
calls
the fitness function on the entire population at once, i.e., in a
single call to the fitness function.
If there are nonlinear constraints, the fitness function and the nonlinear constraints all need to be vectorized in order for the algorithm to compute in a vectorized manner.
See Vectorize the Fitness Function for an example.
When Evaluate fitness and constraint functions (UseParallel
)
is in parallel (true
), ga
calls
the fitness function in parallel, using the parallel environment you
established (see How to Use Parallel Processing). At the command line, set UseParallel
to false
to
compute serially.
Note:
You cannot simultaneously use vectorized and parallel computations.
If you set |
How Fitness and Constraint Functions Are Evaluated
UseVectorized = false | UseVectorized = true | |
---|---|---|
UseParallel = false | Serial | Vectorized |
UseParallel = true | Parallel | Vectorized |