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%% Multiobjective Genetic Algorithm Options
% This example shows how to create and manage options for the
% multiobjective genetic algorithm function |gamultiobj| using
% |optimoptins| in Global Optimization Toolbox.
% Copyright 2007-2015 The MathWorks, Inc.
%% Setting Up a Problem for |gamultiobj|
% |gamultiobj| finds a local Pareto front for multiple objective functions
% using the genetic algorithm. For this example, we will use |gamultiobj| to
% obtain a Pareto front for two objective functions described in the MATLAB
% file |kur_multiobjective.m|. It is a real-valued function that consists of
% two objectives, each of three decision variables. We also impose bound
% constraints on the decision variables -5 <= x(i) <= 5, i = 1,2,3.
type kur_multiobjective.m
%%
% We need to provide a fitness function, the number of variables, and bound
% constraints in the problem to |gamultiobj| function. Refer to the help for
% the |gamultiobj| function for the syntax. Here we also want to plot the
% Pareto front in every generation using the plot function |@gaplotpareto|.
% We use |optimoptions| function to specify this plot function.
FitnessFunction = @kur_multiobjective; % Function handle to the fitness function
numberOfVariables = 3; % Number of decision variables
lb = [-5 -5 -5]; % Lower bound
ub = [5 5 5]; % Upper bound
A = []; % No linear inequality constraints
b = []; % No linear inequality constraints
Aeq = []; % No linear equality constraints
beq = []; % No linear equality constraints
options = optimoptions(@gamultiobj,'PlotFcn',@gaplotpareto);
%%
% Run the |gamultiobj| solver and display the number of solutions found on
% the Pareto front and the number of generations.
[x,Fval,exitFlag,Output] = gamultiobj(FitnessFunction,numberOfVariables,A, ...
b,Aeq,beq,lb,ub,options);
fprintf('The number of points on the Pareto front was: %d\n', size(x,1));
fprintf('The number of generations was : %d\n', Output.generations);
%%
% The Pareto plot displays two competing objectives. For this problem,
% the Pareto front is known to be disconnected. The solution from
% |gamultiobj| can capture the Pareto front even if it is disconnected. Note
% that when you run this example, your result may be different from the results
% shown because |gamultiobj| uses random number generators.
%% Elitist Multiobjective Genetic Algorithm
% The multiobjective genetic algorithm (|gamultiobj|) works on a population
% using a set of operators that are applied to the population. A population
% is a set of points in the design space. The initial population is
% generated randomly by default. The next generation of the population is
% computed using the non-dominated rank and a distance measure of the
% individuals in the current generation.
%%
% A non-dominated rank is assigned to each individual using the relative
% fitness. Individual 'p' dominates 'q' ('p' has a lower rank than 'q') if
% 'p' is strictly better than 'q' in at least one objective and 'p' is no
% worse than 'q' in all objectives. This is same as saying 'q' is dominated
% by 'p' or 'p' is non-inferior to 'q'. Two individuals 'p' and 'q' are
% considered to have equal ranks if neither dominates the other. The
% distance measure of an individual is used to compare individuals with
% equal rank. It is a measure of how far an individual is from the
% other individuals with the same rank.
%%
% The multiobjective GA function |gamultiobj| uses a controlled elitist
% genetic algorithm (a variant of NSGA-II [1]). An elitist GA always favors
% individuals with better fitness value (rank) whereas, 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 very
% important to maintain the diversity of population for convergence to an
% optimal Pareto front. This is done by controlling the elite members of
% the population as the algorithm progresses. Two options 'ParetoFraction'
% and 'DistanceFcn' are used to control the elitism. The Pareto fraction
% option limits the number of individuals on the Pareto front (elite
% members) and the distance function helps to maintain diversity on a
% front by favoring individuals that are relatively far away on the front.
%% Specifying Multiobjective GA Options
% We can chose the default distance measure function, |distancecrowding|,
% that is provided in the toolbox or write our own function to calculate
% the distance measure of an individual. The crowding distance measure
% function in the toolbox takes an optional argument to calculate distance
% either in function space (phenotype) or design space (genotype). If
% |'genotype'| is chosen, then the diversity on a Pareto front is based on
% the design space. The default choice is |'phenotype'| and, in that case, the
% diversity is based on the function space. Here we choose |'genotype'| for
% our distance function. We will directly modify the value of the parameter
% |DistanceMeasureFcn|.
options.DistanceMeasureFcn = {@distancecrowding,'genotype'};
%%
% The Pareto fraction has a default value of 0.35 i.e., the solver will try
% to limit the number of individuals in the current population that are on
% the Pareto front to 35 percent of the population size. Here we set the
% Pareto fraction to 0.5.
options = optimoptions(options,'ParetoFraction',0.5);
%%
% Run the |gamultiobj| solver and display the number of solutions found on
% the Pareto front and the average distance measure of solutions.
[x,Fval,exitFlag,Output] = gamultiobj(FitnessFunction,numberOfVariables,A, ...
b,Aeq,beq,lb,ub,options);
fprintf('The number of points on the Pareto front was: %d\n', size(x,1));
fprintf('The average distance measure of the solutions on the Pareto front was: %g\n', Output.averagedistance);
fprintf('The spread measure of the Pareto front was: %g\n', Output.spread);
%%
% A smaller average distance measure indicates that the solutions on the
% Pareto front are evenly distributed. However, if the Pareto front is
% disconnected, then the distance measure will not indicate the true spread
% of solutions.
%% Modifying the Stopping Criteria
% |gamultiobj| uses three different criteria to determine when to stop the
% solver. The solver stops when any one of the stopping criteria is met. It
% stops when the maximum number of generations is reached; by default this
% number is |'200*numberOfVariables'|. |gamultiobj| also stops if the
% average change in the spread of the Pareto front over the
% |MaxStallGenerations| generations (default is 100) is less than tolerance
% specified in |options.FunctionTolerance|. The third criterion is the
% maximum time limit in seconds (default is |Inf|). Here we modify the
% stopping criteria to change the |FunctionTolerance| from 1e-4 to 1e-3 and
% increase |MaxStallGenerations| to 150.
options = optimoptions(options,'FunctionTolerance',1e-3,'MaxStallGenerations',150);
%%
% Run the |gamultiobj| solver and display the number of solutions found on
% the Pareto front and the number of generations.
[x,Fval,exitFlag,Output] = gamultiobj(FitnessFunction,numberOfVariables,A, ...
b,Aeq,beq,lb,ub,options);
fprintf('The number of points on the Pareto front was: %d\n', size(x,1));
fprintf('The number of generations was : %d\n', Output.generations);
%% Multiobjective GA Hybrid Function
% We will use a hybrid scheme to find an optimal Pareto front for our
% multiobjective problem. |gamultiobj| can reach the region near an optimal
% Pareto front relatively quickly, but it can take many function
% evaluations to achieve convergence. A commonly used technique is to run
% |gamultiobj| for a small number of generations to get near an optimum
% front. Then the solution from |gamultiobj| is used as an initial point for
% another optimization solver that is faster and more efficient for a local
% search. We use |fgoalattain| as the hybrid solver with |gamultiobj|.
% |fgoalattain| solves the goal attainment problem, which is one formulation
% for minimizing a multiobjective optimization problem.
%%
% The hybrid functionality in multiobjective function |gamultiobj| is
% slightly different from that of the single objective function GA. In
% single objective GA the hybrid function starts at the best point returned
% by GA. However, in |gamultiobj| the hybrid solver will start at all the
% points on the Pareto front returned by |gamultiobj|. The new individuals
% returned by the hybrid solver are combined with the existing population
% and a new Pareto front is obtained. It may be useful to see the syntax of
% |fgoalattain| function to better understand how the output from |gamultiobj|
% is internally converted to the input of |fgoalattain| function. |gamultiobj|
% estimates the pseudo weights (required input for |fgoalattain|) for each
% point on the Pareto front and runs the hybrid solver starting from each
% point on the Pareto front. Another required input, goal, is a vector
% specifying the goal for each objective. |gamultiobj| provides this input
% as the extreme points from the Pareto front found so far.
%%
% Here we run |gamultiobj| without the hybrid function.
[x,Fval,exitFlag,Output] = gamultiobj(FitnessFunction,numberOfVariables,A, ...
b,Aeq,beq,lb,ub,options);
fprintf('The number of points on the Pareto front was: %d\n', size(x,1));
fprintf('The average distance measure of the solutions on the Pareto front was: %g\n', Output.averagedistance);
fprintf('The spread measure of the Pareto front was: %g\n', Output.spread);
%%
% Here we use |fgoalattain| as the hybrid function. We also reset the random
% number generators so that we can compare the results with the previous
% run (without the hybrid function).
options = optimoptions(options,'HybridFcn',@fgoalattain);
% Reset the random state (only to compare with previous run)
RandStream.getGlobalStream.State = Output.rngstate.state;
% Run the GAMULTIOBJ solver with hybrid function.
[x,Fval,exitFlag,Output,Population,Score] = gamultiobj(FitnessFunction,numberOfVariables,A, ...
b,Aeq,beq,lb,ub,options);
fprintf('The number of points on the Pareto front was: %d\n', size(x,1));
fprintf('The average distance measure of the solutions on the Pareto front was: %g\n', Output.averagedistance);
fprintf('The spread measure of the Pareto front was: %g\n', Output.spread);
%%
% If the Pareto fronts obtained by |gamultiobj| alone and by using the
% hybrid function are close, we can compare them using the spread and
% the average distance measures. The average distance of the solutions on
% the Pareto front can be improved by using a hybrid function. The spread
% is a measure of the change in two fronts and that can be higher when
% hybrid function is used. This indicates that the front has changed
% considerably from that obtained by |gamultiobj| with no hybrid function.
%%
% It is certain that using the hybrid function will result in a optimal
% Pareto front but we may lose the diversity of the solution (because
% |fgoalattain| does not try to preserve the diversity). This can be
% indicated by a higher value of the average distance measure and the
% spread of the front. We can further improve the average distance measure
% of the solutions and the spread of the Pareto front by running |gamultiobj|
% again with the final population returned in the last run. Here, we should
% remove the hybrid function.
options = optimoptions(options,'HybridFcn',[]); % No hybrid function
% Provide initial population and scores
options = optimoptions(options,'InitialPopulationMatrix',Population,'InitialScoresMatrix',Score);
% Run the GAMULTIOBJ solver with hybrid function.
[x,Fval,exitFlag,Output,Population,Score] = gamultiobj(FitnessFunction,numberOfVariables,A, ...
b,Aeq,beq,lb,ub,options);
fprintf('The number of points on the Pareto front was: %d\n', size(x,1));
fprintf('The average distance measure of the solutions on the Pareto front was: %g\n', Output.averagedistance);
fprintf('The spread measure of the Pareto front was: %g\n', Output.spread);
%% References
% [1] Kalyanmoy Deb, "Multi-Objective Optimization using Evolutionary
% Algorithms", John Wiley & Sons ISBN 047187339.