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Global Optimization Toolbox

Hybrid Scheme in the Genetic Algorithm

Introduction

This example shows how to use a hybrid scheme to optimize a function using the Genetic Algorithm and another optimization method. GA can reach the region near an optimum point relatively quickly, but it can take many function evaluations to achieve convergence. A commonly used technique is to run GA for a small number of generations to get near an optimum point. Then the solution from GA is used as an initial point for another optimization solver that is faster and more efficient for local search.

Rosenbrock's Function

In this example we will optimize Rosenbrock's function (also known as Dejong's second function):

` f(x)= 100*(x(2)-x(1)^2)^2+(1-x(1))^2`

This function is notorious in optimization because of the slow convergence most methods exhibit when trying to minimize this function. This function has a unique minimum at the point x* = (1,1) where it has a function value f(x*) = 0.

We can view the code for this fitness function.

```type dejong2fcn.m ```
```function scores = dejong2fcn(pop) %DEJONG2FCN Compute DeJongs second function. %This function is also known as Rosenbrock's function % Copyright 2003-2004 The MathWorks, Inc. scores = zeros(size(pop,1),1); for i = 1:size(pop,1) p = pop(i,:); scores(i) = 100 * (p(1)^2 - p(2)) ^2 + (1 - p(1))^2; end ```

We use the function PLOTOBJECTIVE in the toolbox to plot the function DEJONG2FCN over the range = [-2 2;-2 2].

```plotobjective(@dejong2fcn,[-2 2;-2 2]); ```

Genetic Algorithm Solution

To start, we will use the Genetic Algorithm, GA, alone to find the minimum of Rosenbrock's function. We need to supply GA with a function handle to the fitness function dejong2fcn.m. Also, GA needs to know the how many variables are in the problem, which is two for this function.

```FitnessFcn = @dejong2fcn; numberOfVariables = 2; ```

Some plot functions can be selected to monitor the performance of the solver.

```options = gaoptimset('PlotFcns',{@gaplotbestf,@gaplotstopping}); ```

We set the random number stream for reproducibility, and run GA with the above inputs.

```rng('default') [x,fval] = ga(FitnessFcn,numberOfVariables,[],[],[],[],[],[],[],options) ```
```Optimization terminated: average change in the fitness value less than options.TolFun. x = 0.3454 0.1444 fval = 0.4913 ```

The global optimum is at x* = (1,1). GA found a point near the optimum, but could not get a more accurate answer with the default stopping criteria. By changing the stopping criteria, we might find a more accurate solution, but it may take many more function evaluations to reach x* = (1,1). Instead, we can use a more efficient local search that starts where GA left off. The hybrid function field in GA provides this feature automatically.

```fminuncOptions = optimoptions(@fminunc,'Display','iter','Algorithm','quasi-newton'); options = gaoptimset(options,'HybridFcn',{@fminunc, fminuncOptions}); ```
```[x,fval] = ga(@dejong2fcn,numberOfVariables,[],[],[],[],[],[],[],options) ```
```Optimization terminated: average change in the fitness value less than options.TolFun. First-order Iteration Func-count f(x) Step-size optimality 0 3 0.664192 28.1 1 12 0.489131 0.000402247 0.373 2 21 0.48383 91 1.22 3 24 0.422036 1 6.7 4 33 0.225633 0.295475 7.36 5 39 0.221682 0.269766 10.1 6 45 0.126376 10 7.96 7 48 0.0839643 1 0.457 8 54 0.0519836 0.5 4.56 9 57 0.0387946 1 5.37 10 60 0.0149721 1 0.85 11 63 0.00959914 1 3.45 12 66 0.0039939 1 0.662 13 69 0.00129755 1 0.348 14 72 0.000288982 1 0.634 15 75 2.29621e-05 1 0.0269 16 78 3.34554e-07 1 0.0139 17 81 5.38696e-10 1 0.000801 18 84 1.72147e-11 1 7.21e-06 Local minimum found. Optimization completed because the size of the gradient is less than the selected value of the function tolerance. x = 1.0000 1.0000 fval = 1.7215e-11 ```