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`ga`

Suppose you want to minimize the simple fitness function of two variables
*x*_{1} and
*x*_{2},

$$\underset{x}{\mathrm{min}}f(x)=100{\left({x}_{1}^{2}-{x}_{2}\right)}^{2}+{(1-{x}_{1})}^{2}$$

subject to the following nonlinear inequality constraints and bounds

$$\begin{array}{ll}{x}_{1}\cdot {x}_{2}+{x}_{1}-{x}_{2}+1.5\le 0\hfill & \text{(nonlinearconstraint)}\hfill \\ 10-{x}_{1}\cdot {x}_{2}\le 0\hfill & \text{(nonlinearconstraint)}\hfill \\ 0\le {x}_{1}\le 1\hfill & \text{(bound)}\hfill \\ 0\le {x}_{2}\le 13\hfill & \text{(bound)}\hfill \end{array}$$

Begin by creating the fitness and constraint functions. First, create a file named
`simple_fitness.m`

as
follows:

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

`simple_fitness.m`

ships with Global Optimization
Toolbox software.)The genetic algorithm function, `ga`

, assumes the fitness function
will take one input `x`

, where `x`

has as many
elements as the 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`

.

Then create a file, `simple_constraint.m`

, containing the
constraints

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 the
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.

To minimize the fitness function, you need to pass a function handle to the fitness
function as the first argument to the `ga`

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,
you also need to pass 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; rng(1,'twister') % for reproducibility [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 = 0.8123 12.3137 fval = 1.3581e+04

For problems without integer constraints, the genetic algorithm 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.

If there are integer constraints, `ga`

does not enforce the
feasibility of linear constraints, and instead adds any linear constraint violations to
the penalty function. See Integer ga Algorithm.

`ga`

uses the mutation and crossover functions to produce new
individuals at every generation. `ga`

satisfies linear and bound
constraints by using mutation and crossover functions that only generate feasible
points. For example, in the previous call to `ga`

, the mutation
function `mutationguassian`

does not necessarily obey the bound
constraints. So when there are bound or linear constraints, the default
`ga`

mutation function is
`mutationadaptfeasible`

. 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 included crossover functions generate points that
satisfy the linear constraints and bounds except the
`crossoverheuristic`

function.

To see the progress of the optimization, use the `optimoptions`

function to create options that 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. You can also visualize the progress of the
algorithm by displaying information to the command window using the
`'Display'`

option.

options = optimoptions('ga','PlotFcn',{@gaplotbestf,@gaplotmaxconstr},'Display','iter');

Rerun the `ga`

solver.

[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],... LB,UB,ConstraintFunction,options)

Best Max Stall Generation Func-count f(x) Constraint Generations 1 2670 13603.6 0 0 2 5282 13578.2 5.718e-06 0 3 7994 14033.9 0 0 4 11794 13573.7 0.0009577 0 Optimization terminated: average change in the fitness value less than options.FunctionTolerance and constraint violation is less than options.ConstraintTolerance. x = 0.8122 12.3104 fval = 1.3574e+04

You can provide a start point for the minimization to the `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.

X0 = [0.5 0.5]; % Start point (row vector) options = optimoptions('ga',options,'InitialPopulationMatrix',X0);

Now, rerun the `ga`

solver.

[x,fval] = ga(ObjectiveFunction,nvars,[],[],[],[],... LB,UB,ConstraintFunction,options)

Best Max Stall Generation Func-count f(x) Constraint Generations 1 2670 13578.1 0.0005269 0 2 5282 13578.2 1.815e-05 0 3 8494 14031.3 0 0 4 14356 14054.9 0 0 5 18706 13573.5 0.0009986 0 Optimization terminated: average change in the fitness value less than options.FunctionTolerance and constraint violation is less than options.ConstraintTolerance. x = 0.8122 12.3103 fval = 1.3573e+04