GA Constrained optimization using genetic algorithm.
GA attempts to solve problems of the following forms:
min F(X) subject to: A*X <= B, Aeq*X = Beq (linear constraints)
X C(X) <= 0, Ceq(X) = 0 (nonlinear constraints)
LB <= X <= UB
X(i) integer, where i is in the index
vector INTCON (integer constraints)
Note: If INTCON is not empty, then no equality constraints are allowed.
That is:-
* Aeq and Beq must be empty
* Ceq returned from NONLCON must be empty
X = GA(FITNESSFCN,NVARS) finds a local unconstrained minimum X to the
FITNESSFCN using GA. NVARS is the dimension (number of design
variables) of the FITNESSFCN. FITNESSFCN accepts a vector X of size
1-by-NVARS, and returns a scalar evaluated at X.
X = GA(FITNESSFCN,NVARS,A,b) finds a local minimum X to the function
FITNESSFCN, subject to the linear inequalities A*X <= B. Linear
constraints are not satisfied when the PopulationType option is set to
'bitString' or 'custom'. See the documentation for details.
X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq) finds a local minimum X to the
function FITNESSFCN, subject to the linear equalities Aeq*X = beq as
well as A*X <= B. (Set A=[] and B=[] if no inequalities exist.) Linear
constraints are not satisfied when the PopulationType option is set to
'bitString' or 'custom'. See the documentation for details.
X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub) defines a set of lower and
upper bounds on the design variables, X, so that a solution is found in
the range lb <= X <= ub. 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. Linear constraints are not satisfied when the
PopulationType option is set to 'bitString' or 'custom'. See the
documentation for details.
X = GA(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 the vectors C and Ceq, representing the
nonlinear inequalities and equalities respectively. GA minimizes
FITNESSFCN such that C(X)<=0 and Ceq(X)=0. (Set lb=[] and/or ub=[] if
no bounds exist.) Nonlinear constraints are not satisfied when the
PopulationType option is set to 'bitString' or 'custom'. See the
documentation for details.
X = GA(FITNESSFCN,NVARS,A,b,Aeq,beq,lb,ub,NONLCON,options) minimizes
with the default optimization parameters replaced by values in OPTIONS.
OPTIONS can be created with the OPTIMOPTIONS function. See OPTIMOPTIONS
for details. For a list of options accepted by GA refer to the
documentation.
X = GA(FITNESSFCN,NVARS,A,b,[],[],lb,ub,NONLCON,INTCON) requires that
the variables listed in INTCON take integer values. Note that GA does
not solve problems with integer and equality constraints. Pass empty
matrices for the Aeq and beq inputs if INTCON is not empty.
X = GA(FITNESSFCN,NVARS,A,b,[],[],lb,ub,NONLCON,INTCON,options)
minimizes with integer constraints and the default optimization
parameters replaced by values in OPTIONS. OPTIONS can be created with
the OPTIMOPTIONS function. See OPTIMOPTIONS for details.
X = GA(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a structure
that has the following fields:
fitnessfcn: <Fitness function>
nvars: <Number of design variables>
Aineq: <A matrix for inequality constraints>
bineq: <b vector for inequality constraints>
Aeq: <Aeq matrix for equality constraints>
beq: <beq vector for equality constraints>
lb: <Lower bound on X>
ub: <Upper bound on X>
nonlcon: <Nonlinear constraint function>
intcon: <Index vector for integer variables>
options: <Options created with optimoptions('ga',...)>
rngstate: <State of the random number generator>
[X,FVAL] = GA(FITNESSFCN, ...) returns FVAL, the value of the fitness
function FITNESSFCN at the solution X.
[X,FVAL,EXITFLAG] = GA(FITNESSFCN, ...) returns EXITFLAG which
describes the exit condition of GA. Possible values of EXITFLAG and the
corresponding exit conditions are
1 Average change in value of the fitness function over
options.MaxStallGenerations generations less than
options.FunctionTolerance and constraint violation less than
options.ConstraintTolerance.
3 The value of the fitness function did not change in
options.MaxStallGenerations generations and constraint violation
less than options.ConstraintTolerance.
4 Magnitude of step smaller than machine precision and constraint
violation less than options.ConstraintTolerance. This exit
condition applies only to nonlinear constraints.
5 Fitness limit reached and constraint violation less than
options.ConstraintTolerance.
0 Maximum number of generations exceeded.
-1 Optimization terminated by the output or plot function.
-2 No feasible point found.
-4 Stall time limit exceeded.
-5 Time limit exceeded.
[X,FVAL,EXITFLAG,OUTPUT] = GA(FITNESSFCN, ...) returns a
structure OUTPUT with the following information:
rngstate: <State of the random number generator before GA started>
generations: <Total generations, excluding HybridFcn iterations>
funccount: <Total function evaluations>
maxconstraint: <Maximum constraint violation>, if any
message: <GA termination message>
[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = GA(FITNESSFCN, ...) returns the
final POPULATION at termination.
[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORES] = GA(FITNESSFCN, ...) returns
the SCORES of the final POPULATION.
Example:
Unconstrained minimization of Rastrigins function:
function scores = myRastriginsFcn(pop)
scores = 10.0 * size(pop,2) + sum(pop.^2 - 10.0*cos(2*pi .* pop),2);
numberOfVariables = 2
x = ga(@myRastriginsFcn,numberOfVariables)
Display plotting functions while GA minimizes
options = optimoptions('ga','PlotFcn',...
{@gaplotbestf,@gaplotbestindiv,@gaplotexpectation,@gaplotstopping});
[x,fval,exitflag,output] = ga(fitfcn,2,[],[],[],[],[],[],[],options)
An example with inequality constraints and lower bounds
A = [1 1; -1 2; 2 1]; b = [2; 2; 3]; lb = zeros(2,1);
fitfcn = @(x)0.5*x(1)^2 + x(2)^2 -x(1)*x(2) -2*x(1) - 6.0*x(2);
% Use mutation function which can handle constraints
options = optimoptions('ga','MutationFcn',@mutationadaptfeasible);
[x,fval,exitflag] = ga(fitfcn,2,A,b,[],[],lb,[],[],options);
If FITNESSFCN or NONLCON are parameterized, you can use anonymous
functions to capture the problem-dependent parameters. Suppose you want
to minimize the fitness given in the function myfit, subject to the
nonlinear constraint myconstr, where these two functions are
parameterized by their second argument a1 and a2, respectively. Here
myfit and myconstr are MATLAB file functions such as
function f = myfit(x,a1)
f = exp(x(1))*(4*x(1)^2 + 2*x(2)^2 + 4*x(1)*x(2) + 2*x(2) + a1);
and
function [c,ceq] = myconstr(x,a2)
c = [1.5 + x(1)*x(2) - x(1) - x(2);
-x(1)*x(2) - a2];
% No nonlinear equality constraints:
ceq = [];
To optimize for specific values of a1 and a2, first assign the values
to these two parameters. Then create two one-argument anonymous
functions that capture the values of a1 and a2, and call myfit and
myconstr with two arguments. Finally, pass these anonymous functions to
GA:
a1 = 1; a2 = 10; % define parameters first
% Mutation function for constrained minimization
options = optimoptions('ga','MutationFcn',@mutationadaptfeasible);
x = ga(@(x)myfit(x,a1),2,[],[],[],[],[],[],@(x)myconstr(x,a2),options)
Example: Solving a mixed-integer optimization problem
An example of optimizing a function where a subset of the variables are
required to be integers:
% Define the objective and call GA. Here variables x(2) and x(3) will
% be integer.
fun = @(x) (x(1) - 0.2)^2 + (x(2) - 1.7)^2 + (x(3) -5.1)^2;
x = ga(fun,3,[],[],[],[],[],[],[],[2 3])
See also OPTIMOPTIONS, FITNESSFUNCTION, GAOUTPUTFCNTEMPLATE, PATTERNSEARCH, @.
Documentation for ga
doc ga
