gamultiobj

Find Pareto front of multiple fitness functions using genetic algorithm

Description

example

x = gamultiobj(fun,nvars) finds x on the Pareto Front of the objective functions defined in fun. 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 explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary.

example

x = gamultiobj(fun,nvars,A,b) finds a local Pareto set x subject to the linear inequalities $A\ast x\le b$. See Linear Inequality Constraints. gamultiobj supports linear constraints only for the default PopulationType option ('doubleVector').

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

example

x = gamultiobj(fun,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 lb  x  ub, see Bound Constraints. 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(fun,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 fun 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(fun,nvars,A,b,Aeq,beq,lb,ub,options) or x = gamultiobj(fun,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 a structure.

example

x = gamultiobj(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon) or x = gamultiobj(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, gamultiobj does not accept nonlinear equality constraints, only nonlinear inequality constraints.

x = gamultiobj(problem) finds the Pareto set for problem, where problem is a structure described in problem.

[x,fval] = gamultiobj(___), for any input variables, returns a matrix fval, the value of all the fitness functions defined in fun 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,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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Find 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.

Plot the solution points.

plot(x(:,1),x(:,2),'ko')
xlabel('x(1)')
ylabel('x(2)')
title('Pareto Points in Parameter Space') 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 $x\left(1\right)+x\left(2\right)\le 1/2$.

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.

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--')
xlabel('x(1)')
ylabel('x(2)')
title('Pareto Points in Parameter Space')
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 $0\le x\le 2\pi$.

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: average change in the spread of Pareto solutions less than options.FunctionTolerance.
x = 18×1

4.7124
4.7124
3.1415
3.6733
3.9845
3.4582
3.9098
4.4409
4.0846
3.8686
⋮

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') Find and plot the Pareto front for the two-objective Schaffer's second function. This function has a disconnected 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: maximum number of generations exceeded. Create a two-objective function in two problem variables.

rng default % For reproducibility
M = diag([-1 -1]) + randn(2)/4; % Two problem variables
fun = @(x)[(x').^2 / 30 + M*x']; % Two objectives

Specify that the second variable must be an integer.

intcon = 2;

Specify problem bounds, the gaplotpareto plot function, and a population size of 100.

lb = [0 0];
ub = [100 50];
options = optimoptions("gamultiobj","PlotFcn","gaplotpareto",...
"PopulationSize",100);

Find the Pareto set for the problem.

nvars = 2;
A = [];
b = [];
Aeq = [];
beq = [];
nonlcon = [];
[x,fval] = gamultiobj(fun,nvars,A,b,Aeq,beq,lb,ub,nonlcon,intcon,options);
Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance. List ten of the solutions, and notice that the second variable is integer-valued.

x(1:10,:)
ans = 10×2

8.3393   28.0000
12.9927   49.0000
7.1611   27.0000
7.0210   18.0000
0.0004   12.0000
9.0989   44.0000
9.3974   29.0000
0.5537   17.0000
6.4010   17.0000
7.0531   31.0000

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 kur_multiobjective function has the following code.

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

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, corresponding 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.

fun 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 fun, see Compute Objective Functions.

If you set the UseVectorized option to true, then fun accepts a matrix of size n-by-nvars, where the matrix represents n individuals. fun 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. The solver passes row vectors of length nvars to fun.

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.

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.

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. The gamultiobj function attempts to satisfy c(x) <= 0 for all entries of c.

• ceq(x) is the row vector nonlinear equality constraints at x. The gamultiobj function 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 the following:

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

Data Types: char | function_handle | string

Optimization options, specified as the output of optimoptions or a structure. See option details in Genetic Algorithm Options.

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 the solver handles options for integer constraints differently.

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

Options for 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 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.

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('gamultiobj','PlotFcn',@gaplotpareto)

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.

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

Data Types: double

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 'gamultiobj' options Options created using optimoptions or an options structure

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

Data Types: struct

Output Arguments

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Pareto points, returned as an m-by-nvars 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 the 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 ValueStopping Condition
1

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

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, 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 ). 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 DistanceMeasureFcn, control the elitism. ParetoFraction limits the number of individuals on the Pareto front (elite members). The distance function, selected by DistanceMeasureFcn, 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. For details, see gamultiobj Algorithm.

Alternative Functionality

App

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

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

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