ga

Find minimum of function using genetic algorithm

Syntax

x = ga(fun,nvars)

x = ga(fun,nvars,A,b)

x = ga(fun,nvars,A,b,Aeq,beq)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options)

x = ga(problem)

[x,fval] = ga(___)

[x,fval,exitflag,output] = ga(___)

[x,fval,exitflag,output,population,scores] = ga(___)

Description

x = ga(fun,nvars) finds a local unconstrained minimum, x, to the objective function, fun. nvars is the dimension (number of design variables) of fun.

Note

Passing Extra Parameters explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary.

x = ga(fun,nvars,A,b) finds a local minimum x to fun, subject to the linear inequalities A*x ≤ b. ga evaluates the matrix product A*x as if x is transposed (A*x').

x = ga(fun,nvars,A,b,Aeq,beq) finds a local minimum x to fun, subject to the linear equalities Aeq*x = beq and A*x ≤ b. (Set A=[] and b=[] if no linear inequalities exist.) ga evaluates the matrix product Aeq*x as if x is transposed (Aeq*x').

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that a solution is found in the range lb ≤ x ≤ ub. (Set Aeq=[] and beq=[] if no linear equalities exist.)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the constraints defined in nonlcon. The function nonlcon accepts x and returns vectors C and Ceq, representing the nonlinear inequalities and equalities respectively. ga minimizes the fun such that C(x) ≤ 0 and Ceq(x) = 0. (Set lb=[] and ub=[] if no bounds exist.)

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the default optimization parameters replaced by values in options. (Set nonlcon=[] if no nonlinear constraints exist.) Create options using optimoptions.

x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon) or x = ga(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options) requires that the variables listed in intcon take integer values.

Note

When there are integer constraints, ga does not accept nonlinear equality constraints, only nonlinear inequality constraints.

x = ga(problem) finds the minimum for problem, a structure described in problem.

[x,fval] = ga(___), for any previous input arguments, also returns fval, the value of the fitness function at x.

[x,fval,exitflag,output] = ga(___) also returns exitflag, an integer identifying the reason the algorithm terminated, and output, a structure that contains output from each generation and other information about the performance of the algorithm.

[x,fval,exitflag,output,population,scores] = ga(___) also returns a matrix population, whose rows are the final population, and a vector scores, the scores of the final population.

Examples

Optimize a Nonsmooth Function Using ga

The ps_example.m file ships with your software. Plot the function.

xi = linspace(-6,2,300);
yi = linspace(-4,4,300);
[X,Y] = meshgrid(xi,yi);
Z = ps_example([X(:),Y(:)]);
Z = reshape(Z,size(X));
surf(X,Y,Z,'MeshStyle','none')
colormap 'jet'
view(-26,43)
xlabel('x(1)')
ylabel('x(2)')
title('ps\_example(x)')

Find the minimum of this function using ga.

rng default % For reproducibility
x = ga(@ps_example,2)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

x = 1×2

   -4.6793   -0.0860

Minimize a Nonsmooth Function with Linear Constraints

Use the genetic algorithm to minimize the ps_example function on the region x(1) + x(2) >= 1 and x(2) <= 5 + x(1).

First, convert the two inequality constraints to the matrix form A*x <= b. In other words, get the x variables on the left-hand side of the inequality, and make both inequalities less than or equal:

-x(1) -x(2) <= -1

-x(1) + x(2) <= 5

A = [-1,-1;
    -1,1];
b = [-1;5];

Solve the constrained problem using ga.

rng default % For reproducibility
fun = @ps_example;
x = ga(fun,2,A,b)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

x = 1×2

    0.9990   -0.0000

The constraints are satisfied to within the default value of the constraint tolerance, 1e-3. To see this, compute A*x' - b, which should have negative components.

disp(A*x' - b)

    0.0010
   -5.9990

Minimize a Nonsmooth Function with Linear Equality and Inequality Constraints

Use the genetic algorithm to minimize the ps_example function on the region x(1) + x(2) >= 1 and x(2) == 5 + x(1).

First, convert the two constraints to the matrix form A*x <= b and Aeq*x = beq. In other words, get the x variables on the left-hand side of the expressions, and make the inequality into less than or equal form:

-x(1) -x(2) <= -1

-x(1) + x(2) == 5

A = [-1 -1];
b = -1;
Aeq = [-1 1];
beq = 5;

Solve the constrained problem using ga.

rng default % For reproducibility
fun = @ps_example;
x = ga(fun,2,A,b,Aeq,beq)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

x = 1×2

   -2.0005    2.9995

Check that the constraints are satisfied to within the default value of ConstraintTolerance, 1e-3.

disp(A*x' - b)

   1.0000e-03

disp(Aeq*x' - beq)

  -1.5494e-05

Optimize with Linear Constraints and Bounds

Use the genetic algorithm to minimize the ps_example function on the region x(1) + x(2) >= 1 and x(2) == 5 + x(1). In addition, set bounds 1 <= x(1) <= 6 and -3 <= x(2) <= 8.

First, convert the two linear constraints to the matrix form A*x <= b and Aeq*x = beq. In other words, get the x variables on the left-hand side of the expressions, and make the inequality into less than or equal form:

-x(1) -x(2) <= -1

-x(1) + x(2) == 5

A = [-1 -1];
b = -1;
Aeq = [-1 1];
beq = 5;

Set bounds lb and ub.

lb = [1 -3];
ub = [6 8];

Solve the constrained problem using ga.

rng default % For reproducibility
fun = @ps_example;
x = ga(fun,2,A,b,Aeq,beq,lb,ub)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

x = 1×2

    1.0000    5.9999

Check that the linear constraints are satisfied to within the default value of ConstraintTolerance, 1e-3.

disp(A*x' - b)

   -5.9999

disp(Aeq*x' - beq)

  -6.1065e-05

Optimize with Nonlinear Constraints Using ga

Use the genetic algorithm to minimize the ps_example function on the region $2{\mathit{x}}_1^2+{\mathit{x}}_2^2\le 3$ and $$(x_1+1{)}^2=(x_2/2{)}^4$$.

To do so, first write a function ellipsecons.m that returns the inequality constraint in the first output, c, and the equality constraint in the second output, ceq. Save the file ellipsecons.m to a folder on your MATLAB® path.

type ellipsecons

function [c,ceq] = ellipsecons(x)

c = 2*x(1)^2 + x(2)^2 - 3;
ceq = (x(1)+1)^2 - (x(2)/2)^4;

Include a function handle to ellipsecons as the nonlcon argument.

nonlcon = @ellipsecons;
fun = @ps_example;
rng default % For reproducibility
x = ga(fun,2,[],[],[],[],[],[],nonlcon)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance
 and constraint violation is less than options.ConstraintTolerance.

x = 1×2

   -0.9766    0.0362

Check that the nonlinear constraints are satisfied at x. The constraints are satisfied when c ≤ 0 and ceq = 0 to within the default value of ConstraintTolerance, 1e-3.

[c,ceq] = nonlcon(x)

c = -1.0911

ceq = 5.4645e-04

Minimize with Nondefault Options

Use the genetic algorithm to minimize the ps_example function on the region x(1) + x(2) >= 1 and x(2) == 5 + x(1) using a constraint tolerance that is smaller than the default.

First, convert the two constraints to the matrix form A*x <= b and Aeq*x = beq. In other words, get the x variables on the left-hand side of the expressions, and make the inequality into less than or equal form:

-x(1) -x(2) <= -1

-x(1) + x(2) == 5

A = [-1 -1];
b = -1;
Aeq = [-1 1];
beq = 5;

To obtain a more accurate solution, set a constraint tolerance of 1e-6. And to monitor the solver progress, set a plot function.

options = optimoptions('ga','ConstraintTolerance',1e-6,'PlotFcn', @gaplotbestf);

Solve the minimization problem.

rng default % For reproducibility
fun = @ps_example;
x = ga(fun,2,A,b,Aeq,beq,[],[],[],options)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

x = 1×2

   -2.0000    3.0000

Check that the linear constraints are satisfied to within 1e-6.

disp(A*x' - b)

   9.9967e-07

disp(Aeq*x' - beq)

  -2.9701e-08

Minimize a Nonlinear Function with Integer Constraints

Use the genetic algorithm to minimize the ps_example function subject to the constraint that x(1) is an integer.

intcon = 1;
rng default % For reproducibility
fun = @ps_example;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = [];
x = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,intcon)

Optimization terminated: average change in the penalty fitness value less than options.FunctionTolerance
and constraint violation is less than options.ConstraintTolerance.

x = 1×2

   -5.0000   -0.0834

Obtain the Solution and Function Value

Use the genetic algorithm to minimize an integer-constrained nonlinear problem. Obtain both the location of the minimum and the minimum function value.

intcon = 1;
rng default % For reproducibility
fun = @ps_example;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = [];
[x,fval] = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,intcon)

Optimization terminated: average change in the penalty fitness value less than options.FunctionTolerance
and constraint violation is less than options.ConstraintTolerance.

x = 1×2

   -5.0000   -0.0834

fval = -1.8344

Compare this result to the solution of the problem with no constraints.

[x,fval] = ga(fun,2)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

x = 1×2

   -4.6906   -0.0078

fval = -1.9918

Obtain Diagnostic Information

Use the genetic algorithm to minimize the ps_example function constrained to have x(1) integer-valued. To understand the reason the solver stopped and how ga searched for a minimum, obtain the exitflag and output results. Also, plot the minimum observed objective function value as the solver progresses.

intcon = 1;
rng default % For reproducibility
fun = @ps_example;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = [];
options = optimoptions('ga','PlotFcn', @gaplotbestf);
[x,fval,exitflag,output] = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options)

Optimization terminated: average change in the penalty fitness value less than options.FunctionTolerance
and constraint violation is less than options.ConstraintTolerance.

x = 1×2

   -5.0000   -0.0834

fval = -1.8344

exitflag = 1

output = struct with fields:
      problemtype: 'integerconstraints'
         rngstate: [1x1 struct]
      generations: 86
        funccount: 3311
          message: 'Optimization terminated: average change in the penalty fitness value less than options.FunctionTolerance...'
    maxconstraint: 0
       hybridflag: []

Obtain Final Population and Scores

Use the genetic algorithm to minimize the ps_example function constrained to have x(1) integer-valued. Obtain all outputs, including the final population and vector of scores.

intcon = 1;
rng default % For reproducibility
fun = @ps_example;
A = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
nonlcon = [];
[x,fval,exitflag,output,population,scores] = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,intcon);

Optimization terminated: average change in the penalty fitness value less than options.FunctionTolerance
and constraint violation is less than options.ConstraintTolerance.

Examine the first 10 members of the final population and their corresponding scores. Notice that x(1) is integer-valued for all these population members. The integer ga algorithm generates only integer-feasible populations.

disp(population(1:10,:))

   1.0e+03 *

   -0.0050   -0.0001
   -0.0050   -0.0001
   -1.6420    0.0027
   -1.5070    0.0010
   -0.4540    0.0104
   -0.2530   -0.0011
   -0.1210   -0.0003
   -0.1040    0.1314
   -0.0140   -0.0010
    0.0160   -0.0002

disp(scores(1:10))

   1.0e+06 *

   -0.0000
   -0.0000
    2.6798
    2.2560
    0.2016
    0.0615
    0.0135
    0.0099
    0.0001
    0.0000

Input Arguments

fun — Objective function

function handle | function name

Objective function, specified as a function handle or function name. Write the objective function to accept a row vector of length nvars and return a scalar value.

When the 'UseVectorized' option is true, write fun to accept a pop-by-nvars matrix, where pop is the current population size. In this case, fun returns a vector the same length as pop containing the fitness function values. Ensure that fun does not assume any particular size for pop, since ga can pass a single member of a population even in a vectorized calculation.

Example: fun = @(x)(x-[4,2]).^2

Data Types: char | function_handle | string

nvars — Number of variables

positive integer

Number of variables, specified as a positive integer. The solver passes row vectors of length nvars to fun.

Example: 4

Data Types: double

A — Linear inequality constraints

real matrix

Linear inequality constraints, specified as a real matrix. A is an M-by-nvars matrix, where M is the number of inequalities.

A encodes the M linear inequalities

A*x <= b,

where x is the column vector of nvars variables x(:), and b is a column vector with M elements.

For example, to specify

x₁ + 2x₂ ≤ 10
3x₁ + 4x₂ ≤ 20
5x₁ + 6x₂ ≤ 30,

give these constraints:

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the control variables sum to 1 or less, give the constraints A = ones(1,N) and b = 1.

Data Types: double

b — Linear inequality constraints

real vector

Linear inequality constraints, specified as a real vector. b is an M-element vector related to the A matrix. If you pass b as a row vector, solvers internally convert b to the column vector b(:).

b encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and A is a matrix of size M-by-N.

For example, to specify

x₁ + 2x₂ ≤ 10
3x₁ + 4x₂ ≤ 20
5x₁ + 6x₂ ≤ 30,

give these constraints:

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the control variables sum to 1 or less, give the constraints A = ones(1,N) and b = 1.

Data Types: double

Aeq — Linear equality constraints

real matrix

Linear equality constraints, specified as a real matrix. Aeq is an Me-by-nvars matrix, where Me is the number of equalities.

Aeq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and beq is a column vector with Me elements.

For example, to specify

x₁ + 2x₂ + 3x₃ = 10
2x₁ + 4x₂ + x₃ = 20,

give these constraints:

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the control variables sum to 1, give the constraints Aeq = ones(1,N) and beq = 1.

Data Types: double

beq — Linear equality constraints

real vector

Linear equality constraints, specified as a real vector. beq is an Me-element vector related to the Aeq matrix. If you pass beq as a row vector, solvers internally convert beq to the column vector beq(:).

beq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and Aeq is a matrix of size Meq-by-N.

For example, to specify

x₁ + 2x₂ + 3x₃ = 10
2x₁ + 4x₂ + x₃ = 20,

give these constraints:

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the control variables sum to 1, give the constraints Aeq = ones(1,N) and beq = 1.

Data Types: double

lb — Lower bounds

[] (default) | real vector or array

Lower bounds, specified as a real vector or array of doubles. lb represents the lower bounds element-wise in lb ≤ x ≤ ub.

Internally, ga converts an array lb to the vector lb(:).

Example: lb = [0;-Inf;4] means x(1) ≥ 0, x(3) ≥ 4.

Data Types: double

ub — Upper bounds

[] (default) | real vector or array

Upper bounds, specified as a real vector or array of doubles. ub represents the upper bounds element-wise in lb ≤ x ≤ ub.

Internally, ga converts an array ub to the vector ub(:).

Example: ub = [Inf;4;10] means x(2) ≤ 4, x(3) ≤ 10.

Data Types: double

nonlcon — Nonlinear constraints

function handle | function name

Nonlinear constraints, specified as a function handle or function name. nonlcon is a function that accepts a vector or array x and returns two arrays, c(x) and ceq(x).

• c(x) is the array of nonlinear inequality constraints at x. ga attempts to satisfy

  c(x) <= 0

  for all entries of c.

• ceq(x) is the array of nonlinear equality constraints at x. ga attempts to satisfy

  ceq(x) = 0

  for all entries of ceq.

For example,

x = ga(@myfun,4,A,b,Aeq,beq,lb,ub,@mycon)

where mycon is a MATLAB^® function such as

function [c,ceq] = mycon(x)
c = ...     % Compute nonlinear inequalities at x.
ceq = ...   % Compute nonlinear equalities at x.

For more information, see Nonlinear Constraints.

To learn how to use vectorized constraints, see Vectorized Constraints.

Note

ga does not enforce nonlinear constraints to be satisfied when the PopulationType option is set to 'bitString' or 'custom'.

If intcon is not empty, the second output of nonlcon (ceq) must be an empty entry ([]).

For information on how ga uses nonlcon, see Nonlinear Constraint Solver Algorithms.

Data Types: char | function_handle | string

options — Optimization options

output of optimoptions | structure

Optimization options, specified as the output of optimoptions or a structure.

optimoptions hides the options listed in italics. See Options that optimoptions Hides.

• Values in {} denote the default value.

• {}* represents the default when there are linear constraints, and for MutationFcn also when there are bounds.

• I* indicates default for integer constraints, or indicates special considerations for integer constraints.

• NM indicates that the option does not apply to gamultiobj.

Options for ga and gamultiobj

Option	Description	Values
ConstraintTolerance	Determines the feasibility with respect to nonlinear constraints. Also, max(sqrt(eps),ConstraintTolerance) determines feasibility with respect to linear constraints. For an options structure, use TolCon.	Positive scalar | {1e-3}
CreationFcn	Function that creates the initial population. Specify as a name of a built-in creation function or a function handle. See Population Options.	{'gacreationuniform'} | {'gacreationlinearfeasible'}* | 'gacreationnonlinearfeasible' | {'gacreationuniformint'}I* for ga | {'gacreationsobol'}I* for gamultiobj | Custom creation function
CrossoverFcn	Function that the algorithm uses to create crossover children. Specify as a name of a built-in crossover function or a function handle. See Crossover Options.	{'crossoverscattered'} for ga, {'crossoverintermediate'}* for gamultiobj | {'crossoverlaplace'}I* | 'crossoverheuristic' | 'crossoversinglepoint' | 'crossovertwopoint' | 'crossoverarithmetic' | Custom crossover function
CrossoverFraction	The fraction of the population at the next generation, not including elite children, that the crossover function creates.	Positive scalar | {0.8}
Display	Level of display.	'off' | 'iter' | 'diagnose' | {'final'}
DistanceMeasureFcn	Function that computes the distance measure of individuals. Specify as a name of a built-in distance measure function or a function handle. The value applies to the decision variable or design space (genotype) or to function space (phenotype). The default 'distancecrowding' is in function space (phenotype). For gamultiobj only. See Multiobjective Options. For an options structure, use a function handle, not a name.	{'distancecrowding'} means the same as {@distancecrowding,'phenotype'} | {@distancecrowding,'genotype'} | Custom distance function
EliteCount	NM Positive integer specifying how many individuals in the current generation are guaranteed to survive to the next generation. Not used in gamultiobj.	Positive integer | {ceil(0.05*PopulationSize)} | {0.05*(default PopulationSize)} for mixed-integer problems
FitnessLimit	NM If the fitness function attains the value of FitnessLimit, the algorithm halts.	Scalar | {-Inf}
FitnessScalingFcn	Function that scales the values of the fitness function. Specify as a name of a built-in scaling function or a function handle. Option unavailable for gamultiobj.	{'fitscalingrank'} | 'fitscalingshiftlinear' | 'fitscalingprop' | 'fitscalingtop' | Custom fitness scaling function
FunctionTolerance	The algorithm stops if the average relative change in the best fitness function value over MaxStallGenerations generations is less than or equal to FunctionTolerance. If StallTest is 'geometricWeighted', then the algorithm stops if the weighted average relative change is less than or equal to FunctionTolerance. For gamultiobj, the algorithm stops when the geometric average of the relative change in value of the spread over options.MaxStallGenerations generations is less than options.FunctionTolerance, and the final spread is less than the mean spread over the past options.MaxStallGenerations generations. See gamultiobj Algorithm. For an options structure, use TolFun.	Positive scalar | {1e-6} for ga, {1e-4} for gamultiobj
HybridFcn	I* Function that continues the optimization after ga terminates. Specify as a name or a function handle. Alternatively, a cell array specifying the hybrid function and its options. See ga Hybrid Function. For gamultiobj, the only hybrid function is @fgoalattain. See gamultiobj Hybrid Function. When the problem has integer constraints, you cannot use a hybrid function. See When to Use a Hybrid Function.	Function name or handle | 'fminsearch' | 'patternsearch' | 'fminunc' | 'fmincon' | {[]} or 1-by-2 cell array | {@solver, hybridoptions}, where solver = fminsearch, patternsearch, fminunc, or fmincon {[]}
InitialPenalty	NM I* Initial value of the penalty parameter	Positive scalar | {10}
InitialPopulationMatrix	Initial population used to seed the genetic algorithm. Has up to PopulationSize rows and N columns, where N is the number of variables. You can pass a partial population, meaning one with fewer than PopulationSize rows. In that case, the genetic algorithm uses CreationFcn to generate the remaining population members. See Population Options. For an options structure, use InitialPopulation.	Matrix | {[]}
InitialPopulationRange	Matrix or vector specifying the range of the individuals in the initial population. Applies to gacreationuniform creation function. ga shifts and scales the default initial range to match any finite bounds. For an options structure, use PopInitRange.	Matrix or vector | {[-10;10]} for unbounded components, {[-1e4+1;1e4+1]} for unbounded components of integer-constrained problems, {[lb;ub]} for bounded components, with the default range modified to match one-sided bounds
InitialScoresMatrix	Initial scores used to determine fitness. Has up to PopulationSize rows and Nf columns, where Nf is the number of fitness functions (1 for ga, greater than 1 for gamultiobj). You can pass a partial scores matrix, meaning one with fewer than PopulationSize rows. In that case, the solver fills in the scores when it evaluates the fitness functions. For an options structure, use InitialScores.	Column vector for single objective | matrix for multiobjective | {[]}
MaxGenerations	Maximum number of iterations before the algorithm halts. For an options structure, use Generations.	Positive integer |{100*numberOfVariables} for ga, {200*numberOfVariables} for gamultiobj
MaxStallGenerations	The algorithm stops if the average relative change in the best fitness function value over MaxStallGenerations generations is less than or equal to FunctionTolerance. If StallTest is 'geometricWeighted', then the algorithm stops if the weighted average relative change is less than or equal to FunctionTolerance. For gamultiobj, the algorithm stops when the geometric average of the relative change in value of the spread over options.MaxStallGenerations generations is less than options.FunctionTolerance, and the final spread is less than the mean spread over the past options.MaxStallGenerations generations. See gamultiobj Algorithm. For an options structure, use StallGenLimit.	Positive integer | {50} for ga, {100} for gamultiobj
MaxStallTime	NM The algorithm stops if there is no improvement in the objective function for MaxStallTime seconds, as measured by tic and toc. For an options structure, use StallTimeLimit.	Positive scalar | {Inf}
MaxTime	The algorithm stops after running for MaxTime seconds, 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. For an options structure, use TimeLimit.	Positive scalar | {Inf}
MigrationDirection	Direction of migration. See Migration Options.	'both' | {'forward'}
MigrationFraction	Scalar from 0 through 1 specifying the fraction of individuals in each subpopulation that migrates to a different subpopulation. See Migration Options.	Scalar | {0.2}
MigrationInterval	Positive integer specifying the number of generations that take place between migrations of individuals between subpopulations. See Migration Options.	Positive integer | {20}
MutationFcn	Function that produces mutation children. Specify as a name of a built-in mutation function or a function handle. See Mutation Options.	{'mutationgaussian'} for ga without constraints | {'mutationadaptfeasible'}* for gamultiobj and for ga with constraints | {'mutationpower'}I* | 'mutationpositivebasis' | 'mutationuniform' | Custom mutation function
NonlinearConstraintAlgorithm	Nonlinear constraint algorithm. See Nonlinear Constraint Solver Algorithms. Option unchangeable for gamultiobj. For an options structure, use NonlinConAlgorithm.	{'auglag'} for ga, {'penalty'} for gamultiobj
OutputFcn	Functions that ga calls at each iteration. Specify as a function handle or a cell array of function handles. See Output Function Options. For an options structure, use OutputFcns.	Function handle or cell array of function handles | {[]}
ParetoFraction	Scalar from 0 through 1 specifying the fraction of individuals to keep on the first Pareto front while the solver selects individuals from higher fronts, for gamultiobj only. See Multiobjective Options.	Scalar | {0.35}
PenaltyFactor	NM I* Penalty update parameter.	Positive scalar | {100}
PlotFcn	Function that plots data computed by the algorithm. Specify as a name of a built-in plot function, a function handle, or a cell array of built-in names or function handles. See Plot Options. For an options structure, use PlotFcns.	ga or gamultiobj: {[]} | 'gaplotdistance' | 'gaplotgenealogy' | 'gaplotselection' | 'gaplotscorediversity' |'gaplotscores' | 'gaplotstopping' | 'gaplotmaxconstr' | Custom plot function ga only: 'gaplotbestf' | 'gaplotbestindiv' | 'gaplotexpectation' | 'gaplotrange' gamultiobj only: 'gaplotpareto' | 'gaplotparetodistance' | 'gaplotrankhist' | 'gaplotspread'
PlotInterval	Positive integer specifying the number of generations between consecutive calls to the plot functions.	Positive integer | {1}
PopulationSize	Size of the population.	Positive integer | {50} when numberOfVariables <= 5, {200} otherwise | {min(max(10*nvars,40),100)} for mixed-integer problems
PopulationType	Data type of the population. Must be 'doubleVector' for mixed-integer problems.	'bitstring' | 'custom' | {'doubleVector'} ga ignores all constraints when PopulationType is set to 'bitString' or 'custom'. See Population Options.
SelectionFcn	Function that selects parents of crossover and mutation children. Specify as a name of a built-in selection function or a function handle. gamultiobj uses only 'selectiontournament'.	{'selectionstochunif'} for ga, {'selectiontournament'} for gamultiobj | 'selectionremainder' | 'selectionuniform' | 'selectionroulette' | Custom selection function
StallTest	NM Stopping test type.	'geometricWeighted' | {'averageChange'}
UseParallel	Compute fitness and nonlinear constraint functions in parallel. See Vectorize and Parallel Options (User Function Evaluation) and How to Use Parallel Processing in Global Optimization Toolbox.	true | {false}
UseVectorized	Specifies whether functions are vectorized. See Vectorize and Parallel Options (User Function Evaluation) and Vectorize the Fitness Function. For an options structure, use Vectorized with the values 'on' or 'off'.	true | {false}

Example: optimoptions('ga','PlotFcn',@gaplotbestf)

intcon — Integer variables

vector of positive integers

Integer variables, specified as a vector of positive integers taking values from 1 to nvars. Each value in intcon represents an x component that is integer-valued.

Note

When intcon is nonempty, nonlcon must return empty for ceq. For more information on integer programming, see Mixed Integer ga Optimization.

Example: To specify that the even entries in x are integer-valued, set intcon to 2:2:nvars

Data Types: double

problem — Problem description

structure

Problem description, specified as a structure containing these fields.

fitnessfcn	Fitness functions
nvars	Number of design variables
Aineq	A matrix for linear inequality constraints
Bineq	b vector for linear inequality constraints
Aeq	Aeq matrix for linear equality constraints
Beq	beq vector for linear equality constraints
lb	Lower bound on x
ub	Upper bound on x
nonlcon	Nonlinear constraint functions
intcon	Indices of integer variables
rngstate	Field to reset the state of the random number generator
solver	'ga'
options	Options created using optimoptions or an options structure

You must specify the fields fitnessfcn, nvars, and options. The remainder are optional for ga.

Data Types: struct

Output Arguments

x — Solution

real vector

Solution, returned as a real vector. x is the best point that ga located during its iterations.

fval — Objective function value at the solution

real number

Objective function value at the solution, returned as a real number. Generally, fval = fun(x).

exitflag — Reason ga stopped

integer

Reason that ga stopped, returned as an integer.

Exit Flag	Meaning
1	Without nonlinear constraints — Average cumulative change in value of the fitness function over MaxStallGenerations generations is less than FunctionTolerance, and the constraint violation is less than ConstraintTolerance.
	With nonlinear constraints — Magnitude of the complementarity measure (see Complementarity Measure) is less than sqrt(ConstraintTolerance), the subproblem is solved using a tolerance less than FunctionTolerance, and the constraint violation is less than ConstraintTolerance.
3	Value of the fitness function did not change in MaxStallGenerations generations and the constraint violation is less than ConstraintTolerance.
4	Magnitude of step smaller than machine precision and the constraint violation is less than ConstraintTolerance.
5	Minimum fitness limit FitnessLimit reached and the constraint violation is less than ConstraintTolerance.
0	Maximum number of generations MaxGenerations exceeded.
-1	Optimization terminated by an output function or plot function.
-2	No feasible point found.
-4	Stall time limit MaxStallTime exceeded.
-5	Time limit MaxTime exceeded.

When there are integer constraints, ga uses the penalty fitness value instead of the fitness value for stopping criteria.

output — Information about the optimization process

structure

Information about the optimization process, returned as a structure with these fields:

• problemtype — Problem type, one of:

  • 'unconstrained'

  • 'boundconstraints'

  • 'linearconstraints'

  • 'nonlinearconstr'

  • 'integerconstraints'

• rngstate — State of the MATLAB random number generator, just before the algorithm started. You can use the values in rngstate to reproduce the output of ga. See Reproduce Results.

• generations — Number of generations computed.

• funccount — Number of evaluations of the fitness function.

• message — Reason the algorithm terminated.

• maxconstraint — Maximum constraint violation, if any.

• hybridflag — Exit flag from the hybrid function. Relates to the HybridFcn options. Not applicable to gamultiobj.

population — Final population

matrix

Final population, returned as a PopulationSize-by-nvars matrix. The rows of population are the individuals.

scores — Final scores

column vector

Final scores, returned as a column vector.

• For non-integer problems, the final scores are the fitness function values of the rows of population.

• For integer problems, the final scores are the penalty fitness values of the population members. See Integer ga Algorithm.

More About

Complementarity Measure

In the Augmented Lagrangian nonlinear constraint solver, the complementarity measure is the norm of the vector whose elements are c_iλ_i, where c_i is the nonlinear inequality constraint violation, and λ_i is the corresponding Lagrange multiplier. See Augmented Lagrangian Genetic Algorithm.

Tips

• To write a function with additional parameters to the independent variables that can be called by ga, see Passing Extra Parameters.

• For problems that use the population type Double Vector (the default), ga does not accept functions whose inputs are of type complex. To solve problems involving complex data, write your functions so that they accept real vectors, by separating the real and imaginary parts.

Algorithms

For a description of the genetic algorithm, see How the Genetic Algorithm Works.

For a description of the mixed integer programming algorithm, see Integer ga Algorithm.

For a description of the nonlinear constraint algorithms, see Nonlinear Constraint Solver Algorithms.

Alternative Functionality

App

The Optimize Live Editor task provides a visual interface for ga.

References

[1] Goldberg, David E., Genetic Algorithms in Search, Optimization & Machine Learning, Addison-Wesley, 1989.

[2] A. R. Conn, N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent Augmented Lagrangian Algorithm for Optimization with General Constraints and Simple Bounds”, SIAM Journal on Numerical Analysis, Volume 28, Number 2, pages 545–572, 1991.

[3] A. R. Conn, N. I. M. Gould, and Ph. L. Toint. “A Globally Convergent Augmented Lagrangian Barrier Algorithm for Optimization with General Inequality Constraints and Simple Bounds”, Mathematics of Computation, Volume 66, Number 217, pages 261–288, 1997.

Extended Capabilities

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, set the 'UseParallel' option to true.

options = optimoptions('solvername','UseParallel',true)

For more information, see How to Use Parallel Processing in Global Optimization Toolbox.

Version History

Introduced before R2006a

R2019b: ga Performs Fewer Fitness Function Evaluations

Behavior changed in R2019b

When the fitness function is deterministic, ga does not reevaluate the fitness function on elite (current best) individuals. You can control this behavior by accessing the new state.EvalElites field and modifying it in a custom output function or custom plot function. Similarly, when the initial population has duplicate members, ga evaluates each unique member only once. You can control this behavior in a custom output function or custom plot function by accessing and modifying the new state.HaveDuplicates field. For details, see Custom Output Function for Genetic Algorithm or Custom Plot Function.

For details about the two new fields, see The State Structure.