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gamultiobj

Find Pareto front of multiple fitness functions using genetic algorithm

Syntax

x = gamultiobj(fitnessfcn,nvars)
x = gamultiobj(fitnessfcn,nvars,A,b)
x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq)
x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub)
x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,nonlcon)
x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,options)
x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = gamultiobj(problem)
[x,fval] = gamultiobj(___)
[x,fval,exitflag] = gamultiobj(___)
[x,fval,exitflag,output] = gamultiobj(___)
[x,fval,exitflag,output,population,scores] = gamultiobj(___)

Description

example

x = gamultiobj(fitnessfcn,nvars) finds x on the Pareto Front of the objective functions defined in fitnessfcn. nvars is the dimension of the optimization problem (number of decision variables). The solution x is local, which means it might not be on the global Pareto front.

Note

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

example

x = gamultiobj(fitnessfcn,nvars,A,b) finds a local Pareto set x subject to the linear inequalities Axb. See Linear Inequality Constraints (Optimization Toolbox). gamultiobj supports linear constraints only for the default PopulationType option ('doubleVector').

x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq) finds a local Pareto set x subject to the linear equalities Aeqx=beq and the linear inequalities Axb, see Linear Equality Constraints (Optimization Toolbox). (Set A = [] and b = [] if no inequalities exist.) gamultiobj supports linear constraints only for the default PopulationType option ('doubleVector').

example

x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables x so that a local Pareto set is found in the range LBxUB, see Bound Constraints (Optimization Toolbox). Use empty matrices for Aeq and beq if no linear equality constraints exist. gamultiobj supports bound constraints only for the default PopulationType option ('doubleVector').

x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,nonlcon) finds a Pareto set subject to the constraints defined in nonlcon. The function nonlcon accepts x and returns vectors c and ceq, representing the nonlinear inequalities and equalities respectively. gamultiobj minimizes fitnessfcn such that c(x)  0 and ceq(x) = 0. (Set lb = [] and ub = [] if no bounds exist.) gamultiobj supports nonlinear constraints only for the default PopulationType option ('doubleVector').

example

x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,options) or x = gamultiobj(fitnessfcn,nvars,A,b,Aeq,beq,lb,ub,nonlcon,options) finds a Pareto set x with the default optimization parameters replaced by values in options. Create options using optimoptions (recommended) or gaoptimset.

x = gamultiobj(problem) finds the Pareto set for problem, where problem is a structure. Create problem by exporting a problem from the Optimization app, as described in Importing and Exporting Your Work (Optimization Toolbox).

[x,fval] = gamultiobj(___), for any input variables, returns a matrix fval, the value of all the fitness functions defined in fitnessfcn for all the solutions in x. fval has nf columns, where nf is the number of objectives, and has the same number of rows as x.

[x,fval,exitflag] = gamultiobj(___) returns exitflag, an integer identifying the reason the algorithm stopped.

[x,fval,exitflag,output] = gamultiobj(___) returns exitflag, an integer identifying the reason the algorithm stopped, and output, a structure that contains information about the optimization process.

example

[x,fval,exitflag,output,population,scores] = gamultiobj(___) returns population, whose rows are the final population, and scores, the scores of the final population.

Examples

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Compute the Pareto front for a simple multiobjective problem. There are two objectives and two decision variables x.

fitnessfcn = @(x)[norm(x)^2,0.5*norm(x(:)-[2;-1])^2+2];

Find the Pareto front for this objective function.

rng default % For reproducibility
x = gamultiobj(fitnessfcn,2)
Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.
x = 

   -0.0072    0.0003
    0.0947   -0.0811
    1.0217   -0.6946
    1.1254   -0.0857
   -0.0072    0.0003
    0.4489   -0.2101
    1.8039   -0.8394
    0.5115   -0.6314
    1.5164   -0.7277
    1.7082   -0.7006

Plot the solution points.

plot(x(:,1),x(:,2),'ko')

To see the effect of a linear constraint on this problem, see Multiobjective Problem with Linear Constraint.

This example shows how to find the Pareto front for a multiobjective problem in the presence of a linear constraint.

There are two objective functions and two decision variables x.

fitnessfcn = @(x)[norm(x)^2,0.5*norm(x(:)-[2;-1])^2+2];

The linear constraint is .

A = [1,1];
b = 1/2;

Find the Pareto front.

rng default % For reproducibility
x = gamultiobj(fitnessfcn,2,A,b)
Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.
x = 

   -0.0012   -0.0029
    1.6403   -1.2085
    1.5748   -1.1211
    0.5255   -0.2921
   -0.0012   -0.0029
    1.3242   -0.9740
    0.9543   -0.4661
    1.2511   -0.7512
    1.2700   -0.8607
    1.3872   -1.0913

Plot the constrained solution and the linear constraint.

plot(x(:,1),x(:,2),'ko')
t = linspace(-1/2,2);
y = 1/2 - t;
hold on
plot(t,y,'b--')
hold off

To see the effect of removing the linear constraint from this problem, see Simple Multiobjective Problem.

Find the Pareto front for the two fitness functions sin(x) and cos(x) on the interval .

fitnessfcn = @(x)[sin(x),cos(x)];
nvars = 1;
lb = 0;
ub = 2*pi;
rng default % for reproducibility
x = gamultiobj(fitnessfcn,nvars,[],[],[],[],lb,ub)
Optimization terminated: maximum number of generations exceeded.
x = 

    3.1416
    4.7124
    3.6842
    4.3171
    3.2260
    3.1416
    3.8049
    3.7220
    4.5960
    3.4626

Plot the solution. gamultiobj finds points along the entire Pareto front.

plot(sin(x),cos(x),'r*')
xlabel('sin(x)')
ylabel('cos(x)')
title('Pareto Front')
legend('Pareto front')

Compute and plot the Pareto front for the two-objective Schaffer's second function. This function has a disconected Pareto front.

Copy this code to a function file on your MATLAB® path.

function y = schaffer2(x) % y has two columns

% Initialize y for two objectives and for all x
y = zeros(length(x),2);

% Evaluate first objective. 
% This objective is piecewise continuous.
for i = 1:length(x)
    if x(i) <= 1
        y(i,1) = -x(i);
    elseif x(i) <=3 
        y(i,1) = x(i) -2; 
    elseif x(i) <=4 
        y(i,1) = 4 - x(i);
    else 
        y(i,1) = x(i) - 4;
    end
end

% Evaluate second objective
y(:,2) = (x -5).^2;

Plot the two objectives.

x = -1:0.1:8;
y = schaffer2(x);
plot(x,y(:,1),'r',x,y(:,2),'b');
xlabel x
ylabel 'schaffer2(x)'
legend('Objective 1','Objective 2')

The two objective functions compete for x in the ranges [1,3] and [4,5]. But, the Pareto-optimal front consists of only two disconnected regions, corresponding to the x in the ranges [1,2] and [4,5]. There are disconnected regions because the region [2,3] is inferior to [4,5]. In that range, objective 1 has the same values, but objective 2 is smaller.

Set bounds to keep population members in the range .

lb = -5;
ub = 10;

Set options to view the Pareto front as gamultiobj runs.

options = optimoptions('gamultiobj','PlotFcn',@gaplotpareto);

Call gamultiobj.

rng default % For reproducibility
[x,fval,exitflag,output] = gamultiobj(@schaffer2,1,[],[],[],[],lb,ub,options);
Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.

Run a simple multiobjective problem and obtain all available outputs.

Set the random number generator for reproducibility.

rng default

Set the fitness functions to kur_multiobjective, a function that has three control variables and returns two fitness function values.

fitnessfcn = @kur_multiobjective;
nvars = 3;

The code for the kur_multiobjective function is the following.

function y = kur_multiobjective(x)
%KUR_MULTIOBJECTIVE Objective function for a multiobjective problem. 
%   The Pareto-optimal set for this two-objective problem is nonconvex as
%   well as disconnected. The function KUR_MULTIOBJECTIVE computes two
%   objectives and returns a vector y of size 2-by-1.
%
%   Reference: Kalyanmoy Deb, "Multi-Objective Optimization using
%   Evolutionary Algorithms", John Wiley & Sons ISBN 047187339 

%   Copyright 2007 The MathWorks, Inc.


% Initialize for two objectives 
y = zeros(2,1);

% Compute first objective
for i = 1:2
  y(1) = y(1)  - 10*exp(-0.2*sqrt(x(i)^2 + x(i+1)^2));
end

% Compute second objective
for i = 1:3
   y(2) = y(2) +  abs(x(i))^0.8 + 5*sin(x(i)^3);
end

This function also appears in the example Multiobjective Genetic Algorithm Options.

Set lower and upper bounds on all variables.

ub = [5 5 5];
lb = -ub;

Find the Pareto front and all other outputs for this problem.

[x,fval,exitflag,output,population,scores] = gamultiobj(fitnessfcn,nvars,...
    [],[],[],[],lb,ub);
Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.

Examine the sizes of some of the returned variables.

sizex = size(x)
sizepopulation = size(population)
sizescores = size(scores)
sizex =

    18     3


sizepopulation =

    50     3


sizescores =

    50     2

The returned Pareto front contains 18 points. There are 50 members of the final population. Each population row has three dimensions, coresponding to the three decision variables. Each scores row has two dimensions, corresponding to the two fitness functions.

Input Arguments

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Fitness functions to optimize, specified as a function handle or function name.

fitnessfcn is a function that accepts a real row vector of doubles x of length nvars and returns a real vector F(x) of objective function values. For details on writing fitnessfcn, see Compute Objective Functions.

If you set the UseVectorized option to true, then fitnessfcn accepts a matrix of size n-by-nvars, where the matrix represents n individuals. fitnessfcn returns a matrix of size n-by-m, where m is the number of objective functions. See Vectorize the Fitness Function.

Example: @(x)[sin(x),cos(x)]

Data Types: char | function_handle | string

Number of variables, specified as a positive integer. gamultiobj passes row vectors of length nvars to fitnessfcn.

Example: 4

Data Types: double

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

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, give constraints A = [1,2;3,4;5,6] and b = [10;20;30] to specify these sums:

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

Example: To set the sum of the x-components to 1 or less, take A = ones(1,N) and b = 1.

Data Types: double

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 nvars variables x(:), and A is a matrix of size M-by-nvars.

For example, give constraints A = [1,2;3,4;5,6] and b = [10;20;30] to specify these sums:

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

Example: To set the sum of the x-components to 1 or less, take A = ones(1,N) and b = 1.

Data Types: double

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

Aeq encodes the Me linear equalities

Aeq*x = beq,

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

For example, give constraints Aeq = [1,2,3;2,4,1] and beq = [10;20] to specify these sums:

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

Example: To set the sum of the x-components to 1, take Aeq = ones(1,N) and beq = 1.

Data Types: double

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 nvars variables x(:), and Aeq is a matrix of size Meq-by-N.

For example, give constraints Aeq = [1,2,3;2,4,1] and beq = [10;20] to specify these sums:

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

Example: To set the sum of the x-components to 1, take Aeq = ones(1,N) and beq = 1.

Data Types: double

Lower bounds, specified as a real vector or real array. If numel(lb) = nvars, then lb specifies that x(i) >= lb(i) for all i.

If numel(lb) < nvars, then lb specifies that x(i) >= lb(i) for 1 <= i <= numel(lb).

In this case, solvers issue a warning.

Example: To specify all x-components as positive, set lb = zeros(nvars,1).

Data Types: double

Upper bounds, specified as a real vector or real array. If numel(ub) = nvars, then ub specifies that x(i) <= ub(i) for all i.

If numel(ub) < nvars, then ub specifies that x(i) <= ub(i) for 1 <= i <= numel(ub).

In this case, solvers issue a warning.

Example: To specify all x-components as less than one, set ub = ones(nvars,1).

Data Types: double

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

  • c(x) is the row vector of nonlinear inequality constraints at x. gamultiobj attempts to satisfy c(x) <= 0 for all entries of c.

  • ceq(x) is the row vector nonlinear equality constraints at x. gamultiobj attempts to satisfy ceq(x) = 0 for all entries of ceq.

If you set the UseVectorized option to true, then nonlcon accepts a matrix of size n-by-nvars, where the matrix represents n individuals. nonlcon returns a matrix of size n-by-mc in the first argument, where mc is the number of nonlinear inequality constraints. nonlcon returns a matrix of size n-by-mceq in the second argument, where mceq is the number of nonlinear equality constraints. See Vectorize the Fitness Function.

For example, x = gamultiobj(@myfun,nvars,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 (Optimization Toolbox).

Internally, gamultiobj sets the scores of infeasible population members to Inf. This setting ensures that infeasible population members are ranked worst for reproduction. See Selection, which describes how the scaled fitness value enters into creating the next generation, and Fitness Scaling Options, which describes the scaling of fitness values.

Data Types: char | function_handle | string

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

Create options by using optimoptions (recommended), or gaoptimset, or by exporting options from the Optimization app. For details, see Importing and Exporting Your Work (Optimization Toolbox).

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 that ga handles options for integer constraints differently; this notation does not apply to gamultiobj.

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

Options for ga, Integer ga, and gamultiobj

OptionDescriptionValues
ConstraintTolerance

Determines the feasibility with respect to nonlinear constraints. Also, max(sqrt(eps),ConstraintTolerance) determines feasibility with respect to linear constraints.

For gaoptimset, use TolCon.

Positive scalar | {1e-3}

CreationFcn

I* Handle to the function that creates the initial population. See Population Options.

{@gacreationuniform} | {@gacreationlinearfeasible}* | Custom creation function

CrossoverFcn

I* Handle to the function that the algorithm uses to create crossover children. See Crossover Options.

{@crossoverscattered} for ga, {@crossoverintermediate}* for gamultiobj | @crossoverheuristic | @crossoversinglepoint | @crossovertwopoint | @crossoverarithmetic | Custom crossover function

CrossoverFraction

The fraction of the population at the next generation, not including elite children, that is created by the crossover function.

Positive scalar | {0.8}

Display

Level of display.

'off' | 'iter' | 'diagnose' | {'final'}

DistanceMeasureFcn

Handle to the function that computes distance measure of individuals. The value applies to decision variable or design space (genotype) or to function space (phenotype). For gamultiobj only. See Multiobjective Options.

{@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

Handle to the function that scales the values of the fitness function. 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 gaoptimset, use TolFun.

Positive scalar | {1e-6} for ga, {1e-4} for gamultiobj

HybridFcn

I* Handle to a function that continues the optimization after ga terminates.

Alternatively, a cell array specifying the hybrid function and its options structure. See ga Hybrid Function.

For gamultiobj, the only hybrid function is @fgoalattain. See gamultiobj Hybrid Function.

Function 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 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 gaoptimset, 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 gaoptimset, 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

I* Initial scores used to determine fitness. Has up to PopulationSize rows and has 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 gaoptimset, use InitialScores.

Column vector for single objective | matrix for multiobjective | {[]}

MaxGenerations

Maximum number of iterations before the algorithm halts.

For gaoptimset, 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 gaoptimset, 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 gaoptimset, use StallTimeLimit.

Positive scalar | {Inf}

MaxTime

The algorithm stops after running after 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 gaoptimset, use TimeLimit.

Positive scalar | {Inf}

MigrationDirection

Direction of migration. See Migration Options

'both' | {'forward'}

MigrationFraction

Scalar between 0 and 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

I* Handle to the function that produces mutation children. See Mutation Options.

{@mutationgaussian} for ga, {@mutationadaptfeasible}* for gamultiobj | @mutationuniform | Custom mutation function

NonlinearConstraintAlgorithm

Nonlinear constraint algorithm. See Nonlinear Constraint Solver Algorithms. Option unchangeable for gamultiobj.

For gaoptimset, use NonlinConAlgorithm.

{'auglag'} for ga, {'penalty'} for gamultiobj

OutputFcn

Functions that ga calls at each iteration. See Output Function Options.

For gaoptimset, use OutputFcns.

Function handle or cell array of function handles | {[]}

ParetoFraction

I* Scalar between 0 and 1 specifying the fraction of individuals to keep on the first Pareto front while the solver selects individuals from higher fronts, for gamultiobj only.

Scalar | {0.35}

PenaltyFactor

NM I* Penalty update parameter.

Positive scalar | {100}

PlotFcn

Function handle or cell array of function handles that plot data computed by the algorithm. See Plot Options.

For gaoptimset, 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

I* Handle to the function that selects parents of crossover and mutation children.

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.

true | {false}

UseVectorized

Specifies whether functions are vectorized. See Vectorize and Parallel Options (User Function Evaluation) and Vectorize the Fitness Function.

For gaoptimset, use Vectorized with the values 'on' or 'off'.

true | {false}

Example: optimoptions('gamultiobj','PlotFcn',@gaplotpareto)

Multiobjective problem, 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 function

rngstate

Optional field to reset the state of the random number generator

solver

'gamultiobj'

options

Options created using optimoptions, gaoptimset, or exported from the Optimization app

Create problem by exporting a problem from the Optimization app, as described in Importing and Exporting Your Work (Optimization Toolbox).

Data Types: struct

Output Arguments

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Pareto points, returned as an m-by-nvar array, where m is the number of points on the Pareto front. Each row of x represents one point on the Pareto front.

Function values on Pareto front, returned as an m-by-nf array. m is the number of points on the Pareto front, and nf is the number of fitness functions. Each row of fval represents the function values at one Pareto point in x.

Reason gamultiobj stopped, returned as an integer.

exitflag ValueExit Condition
1

Average change in value of the spread over options.MaxStallGenerations generations less than options.FunctionTolerance, and the final spread is less than the average spread over the past options.MaxStallGenerations generations.

0

Maximum number of generations exceeded.

-1

Optimization terminated by an output function or plot function.

-2

No feasible point found.

-5

Time limit exceeded.

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

output FieldMeaning
problemtype

Type of problem:

  • 'unconstrained' — No constraints

  • 'boundconstraints' — Only bound constraints

  • 'linearconstraints' — Linear constraints, with or without bound constraints

  • 'nonlinearconstr' — Nonlinear constraints, with or without other types of constraints

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 gamultiobj. See Reproduce Results.

generationsTotal number of generations, excluding HybridFcn iterations.
funccountTotal number of function evaluations.
messagegamultiobj exit message.
averagedistanceAverage “distance,” which by default is the standard deviation of the norm of the difference between Pareto front members and their mean.
spreadCombination of the “distance,” and a measure of the movement of the points on the Pareto front between the final two iterations.
maxconstraintMaximum constraint violation at the final Pareto set.

Final population, returned as an n-by-nvars array, where n is the number of members of the population.

Scores of the final population, returned as an n-by-nf array. n is the number of members of the population, and nf is the number of fitness functions.

When there are nonlinear constraints, gamultiobj sets the scores of infeasible population members to Inf.

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Pareto Front

A Pareto front is a set of points in parameter space (the space of decision variables) that have noninferior fitness function values.

In other words, for each point on the Pareto front, you can improve one fitness function only by degrading another. For details, see What Is Multiobjective Optimization?

As in Local vs. Global Optima (Optimization Toolbox), it is possible for a Pareto front to be local, but not global. “Local” means that the Pareto points can be noninferior compared to nearby points, but points farther away in parameter space could have lower function values in every component.

Algorithms

gamultiobj uses a controlled, elitist genetic algorithm (a variant of NSGA-II [1]). An elitist GA always favors individuals with better fitness value (rank). A controlled elitist GA also favors individuals that can help increase the diversity of the population even if they have a lower fitness value. It is important to maintain the diversity of population for convergence to an optimal Pareto front. Diversity is maintained by controlling the elite members of the population as the algorithm progresses. Two options, ParetoFraction and DistanceFcn, control the elitism. ParetoFraction limits the number of individuals on the Pareto front (elite members). The distance function, selected by DistanceFcn, helps to maintain diversity on a front by favoring individuals that are relatively far away on the front. The algorithm stops if the spread, a measure of the movement of the Pareto front, is small.

References

[1] Deb, Kalyanmoy. Multi-Objective Optimization Using Evolutionary Algorithms. Chichester, England: John Wiley & Sons, 2001.

Introduced in R2007b

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