Find minima of multiple functions using genetic algorithm
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(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,
...)
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
. For details on writing FITNESSFCN
,
see Compute Objective Functions. NVARS
is
the dimension of the optimization problem (number of decision variables). 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 $$A\ast x\le b$$, see Linear Inequality Constraints.
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 $$Aeq\ast x=beq$$ as well as the linear inequalities $$A\ast x\le b$$, see Linear Equality Constraints.
(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 $$LB\le x\le UB$$, see Bound Constraints.
Use empty matrices for LB
and UB
if
no bounds exist. 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,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. gamultiobj
minimizes
the fitnessfcn
such that C(x)
≤ 0
and Ceq(x) = 0
. (Set LB=[]
and UB=[]
if
no bounds exist.)
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 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:
fitnessfcn  Fitness functions 
nvars  Number of design variables 
Aineq 

bineq 

Aeq 

beq 

lb  Lower bound on 
ub  Upper bound on 
nonlcon  Nonlinear constraint function (optional) 
solver 

rngstate  Optional field to reset the state of the random number generator 
options  Options structure created using 
Create the structure problem
by
exporting a problem from Optimization app, as described in Importing and Exporting Your Work in
the Optimization Toolbox™ documentation.
[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.
EXITFLAG Value  Exit Condition 

1  Average change in value of the spread over 
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 
[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS,
...)
returns a structure OUTPUT
with
the following fields:
OUTPUT Field  Meaning 

problemtype  Type of problem:

rngstate  State of the MATLAB^{®} random number generator, just
before the algorithm started. You can use the values in 
generations  Total number of generations, excluding HybridFcn iterations 
funccount  Total number of function evaluations 
message  gamultiobj termination message 
averagedistance  Average "distance," which by default is the standard deviation of the norm of the difference between Pareto front members and their mean 
spread  Combination of the "distance," and a measure of the movement of the points on the Pareto front between the final two iterations 
maxconstraint  Maximum constraint violation at the final Pareto set 
[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
.
This example optimizes two objectives defined by Schaffer's second function, which has two objectives and a scalar input argument. The Pareto front is disconnected. Define this function in a file:
function y = schaffer2(x) % y has two columns % Initialize y for two objectives and for all x y = zeros(length(x),2); % ready for vectorization % 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 Paretooptimal 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.
The gamultiobjfitness
example
solves a simple problem with one decision variable and two objectives.
The gamultiobjoptionsdemo
example
shows how to set options for multiobjective optimization.
[1] Deb, Kalyanmoy. MultiObjective Optimization Using Evolutionary Algorithms. John Wiley & Sons, 2001.
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