Constrained Minimization Problem

We want to minimize a simple fitness function of two variables x1 and x2

   min f(x) = 100 * (x1^2 - x2) ^2 + (1 - x1)^2;
    x

such that the following two nonlinear constraints and bounds are satisfied

   x1*x2 + x1 - x2 + 1.5 <=0, (nonlinear constraint)
   10 - x1*x2 <=0,            (nonlinear constraint)
   0 <= x1 <= 1, and          (bound)
   0 <= x2 <= 13              (bound)

The above fitness function is known as 'cam' as described in L.C.W. Dixon and G.P. Szego (eds.), Towards Global Optimisation 2, North-Holland, Amsterdam, 1978.

Coding the Fitness Function

We create a MATLAB file named simple_fitness.m with the following code in it:

   function y = simple_fitness(x)
    y = 100 * (x(1)^2 - x(2)) ^2 + (1 - x(1))^2;

The Genetic Algorithm function ga assumes the fitness function will take one input x where x has as many elements as number of variables in the problem. The fitness function computes the value of the function and returns that scalar value in its one return argument y.

Coding the Constraint Function

We create a MATLAB file named simple_constraint.m with the following code in it:

   function [c, ceq] = simple_constraint(x)
   c = [1.5 + x(1)*x(2) + x(1) - x(2);
   -x(1)*x(2) + 10];
   ceq = [];

The ga function assumes the constraint function will take one input x where x has as many elements as number of variables in the problem. The constraint function computes the values of all the inequality and equality constraints and returns two vectors c and ceq respectively.

Minimizing Using ga

To minimize our fitness function using the ga function, we need to pass in a function handle to the fitness function as well as specifying the number of variables as the second argument. Lower and upper bounds are provided as LB and UB respectively. In addition, we also need to pass in a function handle to the nonlinear constraint function.

ObjectiveFunction = @simple_fitness;
nvars = 2;    % Number of variables
LB = [0 0];   % Lower bound
UB = [1 13];  % Upper bound
ConstraintFunction = @simple_constraint;
[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],LB,UB, ...
    ConstraintFunction)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance
 and constraint violation is less than options.ConstraintTolerance.

x = 1×2

    0.8122   12.3104

fval = 1.3574e+04

Note that for our constrained minimization problem, the ga function changed the mutation function to mutationadaptfeasible. The default mutation function, mutationgaussian, is only appropriate for unconstrained minimization problems.

ga Operators for Constrained Minimization

The ga solver handles linear constraints and bounds differently from nonlinear constraints. All the linear constraints and bounds are satisfied throughout the optimization. However, ga may not satisfy all the nonlinear constraints at every generation. If ga converges to a solution, the nonlinear constraints will be satisfied at that solution.

ga uses the mutation and crossover functions to produce new individuals at every generation. The way the ga satisfies the linear and bound constraints is to use mutation and crossover functions that only generate feasible points. For example, in the previous call to ga, the default mutation function mutationgaussian will not satisfy the linear constraints and so the mutationadaptfeasible is used instead. If you provide a custom mutation function, this custom function must only generate points that are feasible with respect to the linear and bound constraints. All the crossover functions in the toolbox generate points that satisfy the linear constraints and bounds.

We specify mutationadaptfeasible as the MutationFcn for our minimization problem by creating options with the optimoptions function.

options = optimoptions(@ga,'MutationFcn',@mutationadaptfeasible);
% Next we run the GA solver.
[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],LB,UB, ...
    ConstraintFunction,options)

Optimization terminated: average change in the fitness value less than options.FunctionTolerance
 and constraint violation is less than options.ConstraintTolerance.

x = 1×2

    0.8122   12.3103

fval = 1.3573e+04

Adding Visualization

Next we use optimoptions to select two plot functions. The first plot function is gaplotbestf, which plots the best and mean score of the population at every generation. The second plot function is gaplotmaxconstr, which plots the maximum constraint violation of nonlinear constraints at every generation. We can also visualize the progress of the algorithm by displaying information to the command window using the Display option.

options = optimoptions(options,'PlotFcn',{@gaplotbestf,@gaplotmaxconstr}, ...
    'Display','iter');
% Next we run the GA solver.
[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],LB,UB, ...
    ConstraintFunction,options)

                              Best       Max        Stall
Generation  Func-count        f(x)     Constraint  Generations

    1           2524       13579.8    2.099e-07      0

    2           4986       13578.2    1.686e-05      0

    3           7918         14031            0      0

    4          17371       13573.5    0.0009928      0
Optimization terminated: average change in the fitness value less than options.FunctionTolerance
 and constraint violation is less than options.ConstraintTolerance.

x = 1×2

    0.8122   12.3103

fval = 1.3574e+04

Providing a Start Point

A start point for the minimization can be provided to ga function by specifying the InitialPopulationMatrix option. The ga function will use the first individual from InitialPopulationMatrix as a start point for a constrained minimization. Refer to the documentation for a description of specifying an initial population to ga.

X0 = [0.5 0.5]; % Start point (row vector)
options.InitialPopulationMatrix = X0;
% Next we run the GA solver.
[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],LB,UB, ...
    ConstraintFunction,options)

                              Best       Max        Stall
Generation  Func-count        f(x)     Constraint  Generations

    1           2520       13578.1     0.000524      0

    2           4982       13578.2    1.024e-05      0

    3           7914       14030.5            0      0

    4          17379       13708.4    0.0001674      0
Optimization terminated: average change in the fitness value less than options.FunctionTolerance
 and constraint violation is less than options.ConstraintTolerance.

x = 1×2

    0.8089   12.3626

fval = 1.3708e+04