## Genetic Algorithm Options

### Optimization App vs. Command Line

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 `options`

For example:

• Population type is the label of the option in the Optimization app.

• `PopulationType` is the corresponding field of `options`.

### Plot Options

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 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 built-in plot function name or a 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 a cell array of built-in plot function names or a cell array of function handles:

`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.

#### Structure of the Plot Functions

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. For details, see Output Function Options.

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'`.

#### The State Structure

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, there are no integer variables, `flag` is not `'interrupt'`, and `NonlinearConstraintAlgorithm` is `'auglag'`.

• `NonlinEq` — Nonlinear equality constraints at current point, present only when a nonlinear constraint function is specified, there are no integer variables, `flag` is not `'interrupt'`, and `NonlinearConstraintAlgorithm` is `'auglag'`.

• `EvalElites` — Logical value indicating whether `ga` evaluates the fitness function of elite individuals. Initially, this value is `true`. In the first generation, if the elite individuals evaluate to their previous values (which indicates that the fitness function is deterministic), then this value becomes `false` by default for subsequent iterations. When `EvalElites` is `false`, `ga` does not reevaluate the fitness function of elite individuals. You can override this behavior in a custom plot function or custom output function by changing the output `state.EvalElites`.

• `HaveDuplicates` — Logical value indicating whether `ga` adds duplicate individuals for the initial population. `ga` uses a small relative tolerance to determine whether an individual is duplicated or unique. If `HaveDuplicates` is `true`, then `ga` locates the unique individuals and evaluates the fitness function only once for each unique individual. `ga` copies the fitness and constraint function values to duplicate individuals. `ga` repeats the test in each generation until all individuals are unique. The test takes order `n*m*log(m)` operations, where `m` is the population size and `n` is `nvars`. To override this test in a custom plot function or custom output function, set the output `state.HaveDuplicates` to `false`.

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` — Standard deviation (not 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

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 `Bit string` population are vectors of type `double`, not strings or characters.

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 `'gacreationuniform'` when you have linear constraints. Otherwise, your population might not satisfy the linear constraints.

• `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

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 Options

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.

### Selection Options

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 `expectation(:,1)`. For example, `edit selectionstochunif` or any of the other built-in selection functions.

• For `gamultiobj`, `expectation` is a matrix whose first column is the negative of 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`.

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.

### Reproduction Options

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

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, σk, 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, σi,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 `mutationgaussian` when you have bounds or linear constraints. Otherwise, your population will not necessarily satisfy the constraints.

• `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 `mutationuniform` when you have bounds or linear constraints. Otherwise, your population will not necessarily satisfy the constraints.

• `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

• `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

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 `'crossoverscattered'` when you have linear constraints. Otherwise, your population will not necessarily satisfy the constraints.

• `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 `'crossoversinglepoint'` when you have linear constraints. Otherwise, your population will not necessarily satisfy the constraints.

• `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 `'crossovertwopoint'` when you have linear constraints. Otherwise, your population will not necessarily satisfy the constraints.

• `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

• `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. `ga` currently does not support this form. In subpopulations, each worker hosts a number of individuals. These individuals are a subpopulation. The worker evolves the subpopulation independently of other workers, except when migration causes some individuals to travel between workers.

Because `ga` does not currently support this form of parallel processing, there is no benefit to setting `PopulationSize` to a vector, or to setting the `MigrationDirection`, `MigrationInterval`, or `MigrationFraction` options.

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 nth 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

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).

#### Augmented Lagrangian Genetic 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`.

#### Penalty Algorithm

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.

NameValue
`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

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.

### Note

The fraction of individuals on the first Pareto front can exceed `ParetoFraction`. This occurs when there are too few individuals of other ranks in step 6 of Iterations.

• `DistanceMeasureFcn` — Defines a handle to the function that computes distance measure of individuals, computed in decision variable space (genotype, also termed design variable space) 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 decision 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`.

### Hybrid Function Options

#### `ga` Hybrid Function

A 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, `ga` throws an error.

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');`
Include the hybrid options in the Genetic Algorithm `options` as follows:
`options = optimoptions('ga',options,'HybridFcn',{@fminunc,hybridopts}); `
`hybridopts` must exist before you set `options`.

See Hybrid Scheme in the Genetic Algorithm for an example. See When to Use a Hybrid Function.

#### `gamultiobj` Hybrid Function

A 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:

1. Compute the maximum and minimum of each objective function at the solutions. For objective j at solution k, let

`$\begin{array}{c}{F}_{\mathrm{max}}\left(j\right)=\underset{k}{\mathrm{max}}{F}_{k}\left(j\right)\\ {F}_{\mathrm{min}}\left(j\right)=\underset{k}{\mathrm{min}}{F}_{k}\left(j\right).\end{array}$`
2. Compute the total weight at each solution k,

`$w\left(k\right)=\sum _{j}\frac{{F}_{\mathrm{max}}\left(j\right)-{F}_{k}\left(j\right)}{1+{F}_{\mathrm{max}}\left(j\right)-{F}_{\mathrm{min}}\left(j\right)}.$`
3. Compute the weight for each objective function j at each solution k,

`$p\left(j,k\right)=w\left(k\right)\frac{{F}_{\mathrm{max}}\left(j\right)-{F}_{k}\left(j\right)}{1+{F}_{\mathrm{max}}\left(j\right)-{F}_{\mathrm{min}}\left(j\right)}.$`
4. For each solution k, perform the goal attainment problem with goal vector Fk(j) and weight vector p(j,k).

### Stopping Criteria Options

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. Does not apply to `gamultiobj`.

• 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 geometric average of the relative change in the spread of the Pareto solutions over Stall generations is less than Function tolerance, and the final spread is smaller than the average spread over the last Stall generations, then the algorithm stops. The geometric average coefficient is ½. The spread is a measure of the movement of the Pareto front. See gamultiobj Algorithm.

• 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 geometric average of the relative change in the spread of the Pareto solutions over Stall generations is less than Function tolerance, and the final spread is smaller than the average spread over the last Stall generations, then the algorithm stops. The geometric average coefficient is ½. The spread is a measure of the movement of the Pareto front. See gamultiobj Algorithm.

• 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 Function Options

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 of function handles:

`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.

#### Structure of the Output Function

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 `state.StopFlag` to a nonempty character vector, such as `'y'`.

The output function has the following input arguments:

• `options` — Options

• `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 for the `'auglag'` nonlinear constraint algorithm. When `flag` is `'interrupt'`:

• The values of `state` fields apply to the subproblem iterations.

• `ga` does not accept changes in `options`, and ignores `optchanged`.

• The `state.NonlinIneq` and `state.NonlinEq` fields are not available.

• `'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 as 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 `Population` field, then be sure to update the `Score` field, and possibly the `Best`, `NonlinIneq`, or `NonlinEq` fields, so that they contain consistent information.

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.

### Display to Command Window 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`

### Vectorize and Parallel Options (User Function Evaluation)

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 in Global Optimization Toolbox). At the command line, set `UseParallel` to `false` to compute serially.

### Note

You cannot simultaneously use vectorized and parallel computations. If you set `'UseParallel'` to `true` and `'UseVectorized'` to `true`, `ga` evaluates your fitness and constraint functions in a vectorized manner, not in parallel.

How Fitness and Constraint Functions Are Evaluated

`UseVectorized` = `false``UseVectorized` = `true`
`UseParallel` = `false`SerialVectorized
`UseParallel` = `true`ParallelVectorized

Watch now