Genetic Algorithm and Direct Search Toolbox™

gamultiobj - Find minima of multiple 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,options)
X = gamultiobj(problem)
[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = gamultiobj(FITNESSFCN, ...)
[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] = gamultiobj(FITNESSFCN, ...)

Description

gamultiobj implements the genetic algorithm at the command line to minimize a multicomponent objective function.

X = gamultiobj(FITNESSFCN,NVARS) finds a local Pareto set X of the objective functions defined in FITNESSFCN. NVARS is the dimension of the optimization problem (number of decision variables). FITNESSFCN accepts a vector X of size 1-by-NVARS and returns a vector of size 1-by-numberOfObjectives evaluated at a decision variable. X is a matrix with NVARS columns. The number of rows in X is the same as the number of Pareto solutions. All solutions in a Pareto set are equally optimal; it is up to the designer to select a solution in the Pareto set depending on the application.

X = gamultiobj(FITNESSFCN,NVARS,A,b) finds a local Pareto set X of the objective functions defined in FITNESSFCN, subject to the linear inequalities . Linear constraints are supported only for the default PopulationType option ('doubleVector'). Other population types, e.g., 'bitString' and 'custom', are not supported.

X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq) finds a local Pareto set X of the objective functions defined in FITNESSFCN, subject to the linear equalities as well as the linear inequalities . (Set A=[] and b=[] if no inequalities exist.) Linear constraints are supported only for the default PopulationType option ('doubleVector'). Other population types, e.g., 'bitString' and 'custom', are not supported.

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 . Use empty matrices for LB and UB if no bounds exist. Set LB(i) = -Inf if X(i) is unbounded below; set UB(i) = Inf if X(i) is unbounded above. Bound constraints are supported only for the default PopulationType option ('doubleVector'). Other population types, e.g., 'bitString' and 'custom', are not supported.

X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,options) finds a Pareto set X with the default optimization parameters replaced by values in the structure options. options can be created with the gaoptimset function.

X = gamultiobj(problem) finds the Pareto set for problem, where problem is a structure containing the following fields:

Create the structure problem by exporting a problem from Optimization Tool, as described in Importing and Exporting Your Work in the Optimization Toolbox™ User's Guide.

[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...) returns a matrix FVAL, the value of all the objective functions defined in FITNESSFCN at all the solutions in X. FVAL has numberOfObjectives columns and same number of rows as does X.

[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...) returns EXITFLAG, which describes the exit condition of gamultiobj. Possible values of EXITFLAG and the corresponding exit conditions are listed in this table.

[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS, ...) returns a structure OUTPUT with the following fields:

[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = gamultiobj(FITNESSFCN, ...) returns the final POPULATION at termination.

[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] = gamultiobj(FITNESSFCN, ...) returns the SCORE of the final POPULATION.

Example

GA Multiobjective

This example optimizes two objectives defined by Schaffer's second function: a vector-valued function of two components and one input argument. The Pareto front is disconnected. Define this function in an M-file:

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;

First, plot the two objectives:

x = -1:0.1:8;
y = schaffer2(x);

plot(x,y(:,1),'.r'); hold on
plot(x,y(:,2),'.b');

The two component functions compete in the range [1, 3] and [4, 5]. But the Pareto-optimal front consists of only two disconnected regions: [1, 2] and [4, 5]. This is because the region [2, 3] is inferior to [1, 2].

Next, impose a bound constraint on x, setting

lb = -5;
ub = 10;

The best way to view the results of the genetic algorithm is to visualize the Pareto front directly using the @gaplotpareto option. To optimize Schaffer's function, a larger population size than the default (15) is needed, because of the disconnected front. This example uses 60. Set the optimization options as:

options = gaoptimset('PopulationSize',60,'PlotFcns',...
@gaplotpareto);

Now call gamultiobj, specifying one independent variable and only the bound constraints:

[x,f,exitflag] = gamultiobj(@schaffer2,1,[],[],[],[],...
lb,ub,options);

Optimization terminated: average change in the spread of
Pareto solutions less than options.TolFun.

exitflag
exitflag = 1

The vectors x, f(:,1), and f(:,2) respectively contain the Pareto set and both objectives evaluated on the Pareto set.

Demos

The gamultiobjfitness demo solves a simple problem with one decision variable and two objectives.

The gamultiobjoptionsdemo demo shows how to set options for multiobjective optimization.

References

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

See Also

ga, gaoptimget, gaoptimset, patternsearch, @ (Special Characters), rand, randn

ga

gaoptimget