# Minimize Rastrigins' Function Using `ga`

, Problem-Based

This example shows how to minimize a function with multiple minima using the genetic algorithm in the problem-based approach. For two variables `x`

and `y`

, Rastrigin's function is defined as follows.

ras = @(x, y) 20 + x.^2 + y.^2 - 10*(cos(2*pi*x) + cos(2*pi*y));

Plot the function scaled by 10 in each direction.

rf3 = @(x, y) ras(x/10, y/10); fsurf(rf3,[-30 30],"ShowContours","on") title("rastriginsfcn([x/10,y/10])") xlabel("x") ylabel("y")

The function has many local minima and a global minimum value of 0 that is attained at `x`

= 0, `y`

= 0. See What Is Global Optimization?

Create optimization variables `x`

and `y`

. Specify that the variables are bounded by $$\pm 100$$.

x = optimvar("x","LowerBound",-100,"UpperBound",100); y = optimvar("y","LowerBound",-100,"UpperBound",100);

Create an optimization problem with the objective function `rastriginsfcn(x)`

.

`prob = optimproblem("Objective",ras(x,y));`

**Note:** If you have a nonlinear function that is not composed of polynomials, rational expressions, and elementary functions such as `exp`

, then convert the function to an optimization expression by using `fcn2optimexpr`

. See Convert Nonlinear Function to Optimization Expression and Supported Operations for Optimization Variables and Expressions.

Create `ga`

options to use the `gaplotbestf`

plot function.

options = optimoptions("ga","PlotFcn","gaplotbestf");

Solve the problem using `ga`

as the solver.

rng default % For reproducibility [sol,fval] = solve(prob,"Solver","ga","Options",options)

Solving problem using ga. Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

`sol = `*struct with fields:*
x: 0.9950
y: 0.9950

fval = 1.9899

Is the resulting function value the lowest minimum? Perform the search again. Because `ga`

is a stochastic algorithm, the results can differ.

[sol2,fval2] = solve(prob,"Solver","ga","Options",options)

Solving problem using ga. Optimization terminated: average change in the fitness value less than options.FunctionTolerance.

`sol2 = `*struct with fields:*
x: 0.9950
y: -4.9289e-06

fval2 = 0.9950

The second solution is better because it has a lower function value. A solution returned by `ga`

is not guaranteed to be a global solution.

## See Also

`ga`

| `fcn2optimexpr`

| `solve`