## Genetic Algorithm Options

### Options for Genetic Algorithm

Set options for ga by using optimoptions.

options = optimoptions('ga','Option1','value1','Option2','value2');
• Some options are listed in italics. These options do not appear in the listing that optimoptions returns. To see why 'optimoptions hides these option values, see Options that optimoptions Hides.

• Ensure that you pass options to the solver. Otherwise, patternsearch uses the default option values.

[x,fval] = ga(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

### Plot Options

PlotFcn specifies the plot function or functions called at each iteration by ga or gamultiobj. Set the PlotFcn option to be a built-in plot function name or a handle to the plot function. You can stop the algorithm at any time by clicking the button on the plot window. For example, to display the best function value, 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('patternsearch',...
'PlotFcn', {@plotfun1, @plotfun2, ...});

where @plotfun1, @plotfun2, and so on are function handles to the plot functions. If you specify more than one plot function, all plots appear as subplots in the same window. Right-click any subplot to obtain a larger version in a separate figure window.

Available plot functions for ga or for gamultiobj:

• 'gaplotscorediversity' plots a histogram of the scores at each generation.

• 'gaplotstopping' plots stopping criteria levels.

• '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.

• 'gaplotscores' plots the scores of the individuals at each generation.

• 'gaplotdistance' plots the average distance between individuals at each generation.

• 'gaplotselection' plots a histogram of the parents.

• 'gaplotmaxconstr' plots the maximum nonlinear constraint violation at each generation. For ga, available only when the NonlinearConstraintAlgorithm option is 'auglag' (default for non-integer problems). Therefore, not available for integer-constrained problems, as they use the 'penalty' nonlinear constraint algorithm.

• You can also create and use your own plot function. Structure of the Plot Functions describes the structure of a custom plot function. Pass any custom function as a function handle.

The following plot functions are available for ga only:

• 'gaplotbestf' plots the best score value and mean score versus generation.

• 'gaplotbestindiv' plots the vector entries of the individual with the best fitness function value in each generation.

• 'gaplotexpectation' plots the expected number of children versus the raw scores at each generation.

• 'gaplotrange' plots the minimum, maximum, and mean score values in each generation.

The following plot functions are available for gamultiobj only:

• 'gaplotpareto' plots the Pareto front for the first two or three objective functions.

• 'gaplotparetodistance' plots a bar chart of the distance of each individual from its neighbors.

• '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.

• 'gaplotspread' plots the average spread as a function of iteration number.

#### 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 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.

PopulationType specifies the type of input to the fitness function. Types and their restrictions are:

• 'doubleVector' — Use this option if the individuals in the population have type double. Use this option for mixed integer programming. This is the default.

• '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 CreationFcn and MutationFcn, use 'gacreationuniform' and 'mutationuniform' or handles to custom functions. For CrossoverFcn, use 'crossoverscattered', 'crossoversinglepoint', 'crossovertwopoint', or a handle to a custom function. You cannot use a HybridFcn, and ga ignores all constraints, including bounds, linear constraints, and nonlinear constraints.

• 'custom' — Indicates a custom population type. In this case, you must also use a custom CrossoverFcn and MutationFcn. You must provide either a custom creation function or an InitialPopulationMatrix. You cannot use a HybridFcn, and ga ignores all constraints, including bounds, linear constraints, and nonlinear constraints.

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. The default is '50 when numberOfVariables <= 5, else 200'.

If you set PopulationSize 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. Note that this option is not useful. See Migration Options.

CreationFcn specifies the function that creates the initial population for ga. Choose from:

• [] uses the default creation function for your problem type.

• '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.

• 'gacreationlinearfeasible' is the default when there are linear constraints and no integer constraints. This choice creates a random initial population that satisfies all bounds and linear constraints. If there are linear constraints, 'gacreationlinearfeasible' creates many individuals on the boundaries of the constraint region, and creates a well-dispersed population. 'gacreationlinearfeasible' ignores 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.

• 'gacreationnonlinearfeasible' is the default creation function for the 'penalty' nonlinear constraint algorithm. For details, see Constraint Parameters.

• 'gacreationuniformint' is the default creation function for ga when the problem has integer constraints. This function applies an artificial bound to unbounded components, generates individuals uniformly at random within the bounds, and then enforces integer constraints.

Note

When your problem has integer constraints, ga and gamultiobj enforce that integer constraints, bounds, and all linear constraints are feasible at each iteration. For nondefault mutation, crossover, creation, and selection functions, ga and gamultiobj apply extra feasibility routines after the functions operate.

• 'gacreationsobol' is the default creation function for gamultiobj when the problem has integer constraints. The creation function uses a quasirandom Sobol sequence to generate a well-dispersed initial population. The population is feasible with respect to bounds, linear constraints, and integer constraints.

• A function handle lets you write your own creation function, which must generate data of the type that you specify in PopulationType. For example,

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 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.

InitialPopulationMatrix specifies an initial population for the genetic algorithm. The default value is [], in which case ga uses the default CreationFcn to create an initial population. If you enter a nonempty array in the InitialPopulationMatrix, the array must have no more than PopulationSize rows, and exactly nvars columns, where nvars is the number of variables, the second input to ga or gamultiobj. If you have a partial initial population, meaning fewer than PopulationSize rows, then the genetic algorithm calls CreationFcn to generate the remaining individuals.

InitialScoreMatrix specifies initial scores for the initial population. The initial scores can also be partial. If your problem has nonlinear constraints then the algorithm does not use InitialScoreMatrix.

InitialPopulationRange specifies the range of the vectors in the initial population that is generated by the gacreationuniform creation function. You can set InitialPopulationRange to be a matrix with two rows and nvars 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 InitialPopulationRange to be a 2-by-1 vector, each entry is expanded to a constant row of length nvars. If you do not specify an InitialPopulationRange, the default is [-10;10] ([-1e4+1;1e4+1] for integer-constrained problems), modified to match any existing bounds. 'gacreationlinearfeasible' ignores InitialPopulationRange. See Set 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.

FitnessScalingFcn specifies the function that performs the scaling. The options are

• 'fitscalingrank' — The default fitness scaling function, 'fitscalingrank', 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.

• 'fitscalingprop' — Proportional scaling makes the scaled value of an individual proportional to its raw fitness score.

• 'fitscalingtop' — Top scaling scales the top individuals equally. You can modify the top scaling using an additional parameter:

options = optimoptions('ga',...
'FitnessScalingFcn',{@fitscalingtop,quantity})

quantity 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 ...].

• 'fitscalingshiftlinear' — Shift linear scaling scales the raw scores so that the expectation of the fittest individual is equal to a constant called rate multiplied by the average score. You can modify the rate parameter:

options = optimoptions('ga','FitnessScalingFcn',...
{@fitscalingshiftlinear, rate})

The default value of rate is 2.

• A function handle lets you write your own scaling function.

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 explains how to provide additional parameters to the function.

### Selection Options

Selection options specify how the genetic algorithm chooses parents for the next generation.

The SelectionFcn option specifies the selection function.

gamultiobj uses only the 'selectiontournament' selection function.

For ga the options are:

• 'selectionstochunif' — The ga default selection function, 'selectionstochunif', 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.

• '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.

• '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.

• '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.

• 'selectiontournament' — Tournament selection chooses each parent by choosing size players at random and then choosing the best individual out of that set to be a parent. size must be at least 2. The default value of size is 4. Set size to a different value as follows:

options = optimoptions('ga','SelectionFcn',...
{@selectiontournament,size})

When NonlinearConstraintAlgorithm is Penalty, ga uses 'selectiontournament' with size 2.

• Note

When your problem has integer constraints, ga and gamultiobj enforce that integer constraints, bounds, and all linear constraints are feasible at each iteration. For nondefault mutation, crossover, creation, and selection functions, ga and gamultiobj apply extra feasibility routines after the functions operate.

• A function handle enables you to write your own selection function.

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 explains how to provide additional parameters to the function.

### Reproduction Options

Reproduction options specify how the genetic algorithm creates children for the next generation.

EliteCount specifies the number of individuals that are guaranteed to survive to the next generation. Set EliteCount 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.

CrossoverFraction specifies the fraction of the next generation, other than elite children, that are produced by crossover. Set CrossoverFraction to be a fraction between 0 and 1. The default value is 0.8.

See "Setting the Crossover Fraction" in Vary Mutation and Crossover 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. Specify the mutation function in the MutationFcn option.

MutationFcn options:

• 'mutationgaussian' — The default mutation function for ga for unconstrained problems, 'mutationgaussian', 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, and by the InitialPopulationRange option. Set scale and shrink as follows:

options = optimoptions('ga','MutationFcn', ...
{@mutationgaussian, scale, shrink})
• The scale parameter determines the standard deviation at the first generation. If you set InitialPopulationRange 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 InitialPopulationRange to be a vector v with two rows and nvars 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 InitialPopulationRange 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 InitialPopulationRange to be a vector with two rows and nvars 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.

Caution

Do not use mutationgaussian when you have bounds or linear constraints. Otherwise, your population will not necessarily satisfy the constraints. Instead, use 'mutationadaptfeasible' or a custom mutation function that satisfies linear constraints.

• '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,

options = optimoptions('ga','MutationFcn', {@mutationuniform, rate})

Caution

Do not use mutationuniform when you have bounds or linear constraints. Otherwise, your population will not necessarily satisfy the constraints. Instead, use 'mutationadaptfeasible' or a custom mutation function that satisfies linear constraints.

• 'mutationadaptfeasible', the default mutation function for gamultiobj and for ga when there are noninteger 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.

• 'mutationpower' is the default mutation function for ga and gamultiobj when the problem has integer constraints. Power mutation mutates a parent, x, via the following. For each component of the parent, the ith component of the child is given by:

mutationChild(i) = x(i) - s(x(i) - lb(i)) if t < r

= x(i) + s(ub(i) - x(i)) if t >= r.

Here, t is the scaled distance of x(i) from the ith component of the lower bound, lb(i). s is a random variable drawn from a power distribution and r is a random number drawn from a uniform distribution.

This function can handle lb(i) = ub(i). New children are generated with the ith component set to lb(i), which equals ub(i). For more information on this crossover function see section 2.1 of the following reference:

Kusum Deep, Krishna Pratap Singsh, M. L. Kansal, C. Mohan. A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation, 212 (2009), 505–518.

Note

When your problem has integer constraints, ga and gamultiobj enforce that integer constraints, bounds, and all linear constraints are feasible at each iteration. For nondefault mutation, crossover, creation, and selection functions, ga and gamultiobj apply extra feasibility routines after the functions operate.

• 'mutationpositivebasis' — This mutation function is similar to orthogonal MADS steps, modified for linear constraints and bounds.

• A function handle enables you to write your own mutation function.

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 nvars.

Passing Extra Parameters 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.

CrossoverFcn specifies the function that performs the crossover. You can choose from the following functions:

• '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

When your problem has linear constraints, 'crossoverscattered' can give a poorly distributed population. In this case, use a different crossover function, such as 'crossoverintermediate'.

• 'crossoversinglepoint' chooses a random integer n between 1 and nvars 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

When your problem has linear constraints, 'crossoversinglepoint' can give a poorly distributed population. In this case, use a different crossover function, such as 'crossoverintermediate'.

• 'crossovertwopoint' selects two random integers m and n between 1 and nvars. 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

When your problem has linear constraints, 'crossovertwopoint' can give a poorly distributed population. In this case, use a different crossover function, such as 'crossoverintermediate'.

• '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 nvars. The default value of ratio is a vector of all 1's. Set the ratio parameter as follows.

options = optimoptions('ga','CrossoverFcn', ...
{@crossoverintermediate, ratio});

'crossoverintermediate' 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.

• 'crossoverlaplace' is the default crossover function when the problem has integer constraints. The Laplace crossover generates children using either of the following formulae (chosen at random):

xOverKid = p1 + bl*abs(p1 – p2)

xOverKid = p2 + bl*abs(p1 – p2)

Here, p1, p2 are the parents of xOverKid and bl is a random number generated from a Laplace distribution. For more information on this crossover function see section 2.1 of the following reference:

Kusum Deep, Krishna Pratap Singsh, M. L. Kansal, C. Mohan. A real coded genetic algorithm for solving integer and mixed integer optimization problems. Applied Mathematics and Computation, 212 (2009), 505–518.

• '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. The default value of ratiois 1.2. Set the ratio parameter as follows.

options = optimoptions('ga','CrossoverFcn',...
{@crossoverheuristic,ratio});

If parent1 and parent2 are the parents, and parent1 has the better fitness value, the function returns the child

child = parent2 + ratio * (parent1 - parent2);

Caution

When your problem has linear constraints, 'crossoverheuristic' can give a poorly distributed population. In this case, use a different crossover function, such as 'crossoverintermediate'.

• 'crossoverarithmetic' creates children that are the weighted arithmetic mean of two parents. Children are always feasible with respect to linear constraints and bounds.

• Note

When your problem has integer constraints, ga and gamultiobj enforce that integer constraints, bounds, and all linear constraints are feasible at each iteration. For nondefault mutation, crossover, creation, and selection functions, ga and gamultiobj apply extra feasibility routines after the functions operate.

• A function handle enables you to write your own crossover function.

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 PopulationSize and the number of columns is nvars.

The function returns xoverKids—the crossover offspring—as a matrix where rows correspond to the children. The number of columns of the matrix is nvars.

Passing Extra Parameters 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.

### Migration Options

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 PopulationSize 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 options:

• MigrationDirection — Migration can take place in one or both directions.

• If you set MigrationDirection to 'forward', migration takes place toward the last subpopulation. That is, the nth subpopulation migrates into the (n+1)th subpopulation.

• If you set MigrationDirection to '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.

• MigrationInterval — Specifies how many generation pass between migrations. For example, if you set MigrationInterval to 20, migration takes place every 20 generations.

• MigrationFraction — Specifies how many individuals move between subpopulations. MigrationFraction 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 MigrationFraction 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

• InitialPenalty — Specifies an initial value of the penalty parameter that is used by the nonlinear constraint algorithm. InitialPenalty must be greater than or equal to 1, and has a default of 10.

• PenaltyFactor — Increases the penalty parameter when the problem is not solved to required accuracy and constraints are not satisfied. PenaltyFactor 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
SolverOptsfmincon options, created using optimoptions or optimset.
UseParallelWhen true, run fmincon in parallel on initial points; default is false.
NumStartPtsNumber 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 gamultiobj 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 the HybridFcn option. Do not use with integer problems. The choices are

• [] — No hybrid function.

• 'fminsearch' — Uses the MATLAB® function fminsearch to perform unconstrained minimization.

• 'patternsearch' — Uses a pattern search to perform constrained or unconstrained minimization.

• 'fminunc' — Uses the Optimization Toolbox™ function fminunc to perform unconstrained minimization.

• '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 the HybridFcn option.

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:

• MaxGenerations — Specifies the maximum number of iterations for the genetic algorithm to perform. The default is 100*numberOfVariables.

• 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.

• FitnessLimit — The algorithm stops if the best fitness value is less than or equal to the value of FitnessLimit. Does not apply to gamultiobj.

• MaxStallGenerations — The algorithm stops if the average relative change in the best fitness function value over MaxStallGenerations is less than or equal to FunctionTolerance. (If the StallTest option is 'geometricWeighted', then the test is for a geometric weighted average relative change.) For a problem with nonlinear constraints, MaxStallGenerations 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 MaxStallGenerations is less than FunctionTolerance, and the final spread is smaller than the average spread over the last MaxStallGenerations, 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.

• MaxStallTime — The algorithm stops if there is no improvement in the best fitness value for an interval of time in seconds specified by MaxStallTime, as measured by tic and toc.

• FunctionTolerance — The algorithm stops if the average relative change in the best fitness function value over MaxStallGenerations is less than or equal to FunctionTolerance. (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 MaxStallGenerations is less than FunctionTolerance, and the final spread is smaller than the average spread over the last MaxStallGenerations, 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.

• ConstraintTolerance — The ConstraintTolerance 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 and Stall Generations for an example.

### Output Function Options

Output functions are functions that the genetic algorithm calls at each generation. Unlike other solvers, a ga output function can not only read the values of the state of the algorithm, but also modify those values. An output function can also halt the solver according to conditions you 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 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.

#### Changing the State Structure

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

'Display' specifies how much information is displayed at the command line while the genetic algorithm is running. The available options are

• 'final' (default) — The reason for stopping is displayed.

• 'off' or the equivalent 'none' — No output is displayed.

• 'iter' — Information is displayed at each iteration.

• '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.

Both 'iter' 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, 'iter' and 'diagnose' do not display the Mean f(x), but additionally display:

• Max Constraint — Maximum nonlinear constraint violation

In addition, 'iter' and 'diagnose' display problem information before the iterative display, such as problem type and which creation, mutation, crossover, and selection functions ga or gamultiobj is using.

### 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. Set the 'UseVectorized' and 'UseParallel' options with optimoptions.

• When 'UseVectorized' is false (default), ga calls the fitness function on one individual at a time as it loops through the population. (This assumes 'UseParallel' is at its default value of false.)

• When 'UseVectorized' is true, ga calls the fitness function on the entire population at once, 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 UseParallel is true, ga calls the fitness function in parallel, using the parallel environment you established (see How to Use Parallel Processing in Global Optimization Toolbox). Set UseParallel to false (default) 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 = falseUseVectorized = true
UseParallel = falseSerialVectorized
UseParallel = trueParallelVectorized