Genetic Algorithms

Genetic Algorithms

Exercise In this Exercise

GAs

1. 2. 3. 4. 5. 6. 7.

Theory (in brief) Things you have to consider GAs and Matlab Part #1 (fitness function, variables, representation, plots) Part #2 (population diversity – size – range, fitness scaling) Part #3 (selection, elitism, mutation) Part #4 (global vs. local minima)

Duration: 120 min

1. Theory (in brief)

(5 min)

A Genetic Algorithm is an optimization technique that is based on the evolution theory. Instead of searching for a solution to a problem in the "state space" (like the traditional search algorithms do), a GA works in the "solution space" and builds (or better, "breeds") new, hopefully better solutions based on existing ones. The general idea behind GAs is that we can build a better solution if we somehow combine the "good" parts of other solutions (schemata theory), just like nature does by combining the DNA of living beings. The overall idea of a GA is depicted in Figure 1 (you should refer to the theory for the very details). Pi

selection

Pi+1

elitism

 Γ

parents

mutation Α

mating

descendants

Figure 1: The outline of a Genetic Algorithm

2. Things you have to consider (and be aware of)

(5 min)

The first thing you must do in order to use a GA is to decide if it is possible to automatically build solutions to your problem. For example, in the Traveling Salesman Problem, every route that passes through the cities in question is potentially a solution, although probably not the optimal one. You must be able to do that because a GA requires an initial population P of solutions.

Kokkoras F. | Paraskevopoulos K.

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International Hellenic University

Genetic Algorithms

Then you must decide what "gene" representation you will use. You have a few alternatives like binary, integer, double, permutation, etc. with the binary and double being the most commonly used since they are the most flexible. After having selected the representation you must decide in order:     

the method to select parents from the population P (Cost Roulette Wheel, Stochastic Universal Sampling, Rank Roulette Wheel, Tournament Selection, etc.) the way these parents will "mate" to create descendants (to many methods to mention here – just note that your available options are a result of the representation decided earlier) the mutation method (optional but useful – again, options are representation depended) the method you will use to populate the next generation (Pi+1) (age based, quality based, etc. – you probably use elitism as well) the algorithm's termination condition (number of generations, time limit, acceptable quality threshold, improvement stall, etc. – combination of these is commonly used)

3. GAs and Matlab

(10 min)

Figure 2: GAs in Matlab's Optimization Toolbox

Matlab provides an optimization toolbox that includes a GA-based solver. You start the toolbox by typing optimtool in the Matlab's command line and pressing enter. As soon as the optimization window appears, you select the solver ga – Genetic Algorithm and you are ready to go. Matlab does


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not provide every method available in the literature in every step but it does have a lot of options for fine tuning and also provides hooks for customization. The user should program (by writing m files) any extended functionality required. Take your time and explore the window. If you mesh up with the settings, before you proceed close the window and run the toolbox again. Matlab R2008a (v.7.6) was used for the tutorial. Earlier versions are OK as soon as the proper toolbox is presented and installed.

4. Part #1 (fitness function, variables, representation, plots)

(15 min)

The first thing you have to do is to provide the fitness function, that is, the function that calculates the quality of each member of the population (or in plain mathematics, the function you have to optimize). Let's use one provided by Matlab: type @rastriginsfcn in the proper field and set the Number of variables to 2. The representation used is defined in the Options-Population section. The default selection Double Vector is fine. To have an idea of what we are looking for, check the equation of this function and its plot on the right. Ras(x,y)=20+x2+y2-10(cos2πx+cos2πy) We want to find the absolute minimum which is 0 at (0,0). Note that by default, only minimization is supported. If for example you want to maximize the f1(x,y) function then built and minimize the following custom function: f2(x,y) = - f1(x,y). Although you are ready to run, let's ask for some plots, so we will be able to better figure out what happens. Go in the Options section, scroll down to the Plot functions and check Best Fitness and Distance checkboxes. Now you are ready (the default settings in everything else is adequate). Press the Start button. The algorithm starts, the plots are pop-up and soon you have the results at the bottom left of the window. The best fitness function value (the smallest one since we minimize) and the termination condition met are printed, together with the solution (Final Point – it is very close to (0,0)). Since the method is stochastic, don't expect to be able to reproduce any result found in a different run. Now check the two plots on the left. It is obvious that the population converges, since the average distance between individuals (solutions) in term of the fitness value is reduced, as the generations pass. This is a measure of the diversity of a population. It is hard to avoid convergence but keeping it low or postponing its appearance is better. Having diversity in the population allows the GA to search better in the solution space.


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Check also the fitness value as it gradually gets smaller. This is required, it is an indication that optimization takes place. Not only the fitness value of the best individual was reduced but the mean (average) fitness of the population was also reduced (that is, in terms of the fitness value, the whole population was improved –we have better solutions in the population, at the end). All the above together are a good indication that the GA did its job well but we are really happy only because we know where the solution is (at (0,0) with fitness value 0). Note however that the nature of the GA prevents it from finding the best solution (0,0). It can go very close to this value but getting exactly to (0,0) is hard and could be done only by luck. This is OK since in this kind of problems (optimization) we are happy even with a good (and not the perfect one) solution. If not, the hybrid function option should be used (not discussed here).

Generally speaking, to get the best results from the GA requires experimentation with the different options. Let's see how some of these affect the performance of the GA.

5. Part #2 (population diversity – size – range, fitness scaling)

(35 min)

The performance of a GA is affected by the diversity of the initial population. If the average distance between individuals is large, the diversity is high; if the average distance is small, the diversity is low. You should experiment to get the right amount of diversity. If the diversity is too high or too low, the genetic algorithm might not perform well. We will demonstrate this in the following. By default, the Optimization Tool creates a random initial population using a creation function. You can limit this by setting the Initial range field in Population options. Set it to (1; 1.1). By this we actually make it harder for the GA to search equally well in all the solutions space. We do not prevent it though. The genetic algorithm can find the solution even if it does not lie in the initial range, provided that the populations have enough diversity. Note: The initial range only restricts the range of the points in the initial population by specifying the lower and upper bounds. Subsequent generations can contain points whose entries do not lie in the initial range. If you want to bound all the individuals in all generations in a range, then you can use the lower and upper bound fields in the constraints panel, on the left.

Leave the rest settings as in Part #1 except Options-Stopping Criteria-Stall Generations which should be set to 100. This will let the algorithm run for 100 generation providing us with better results (and plots). Now click the Start button. The GA returns the best fitness function value of approximately 2 and displays the plots in the figure on the right. The upper plot, which displays the best fitness at each generation, shows little progress in lowering the fitness value (black dots). The lower plot shows the average distance between individuals at each generation, which is a good measure of the diversity of a population. For this setting of initial range, there is too little diversity for the algorithm to make progress. The algorithm was trapped in a local minimum due to the initial range restriction!


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Next, set Initial range to [1; 100] and run the algorithm again. The GA returns the best fitness value of approximately 3.3 and displays the following plots: This time, the genetic algorithm makes progress, but because the average distance between individuals is so large, the best individuals are far from the optimal solution. Note though that if we let the GA to run for more generations (by setting Generations and Stall Generations in Stopping Criteria to 200) it will eventually find a better solution. Note: If you try this, please leave the settings in their initial values before you proceed (default and 100, respectively).

Finally, set Initial range to [1; 2] and run the GA. This returns the best fitness value of approximately 0.012 and displays the plots that follow. The diversity in this case is better suited to the problem, so the genetic algorithm returns a much better result than in the previous two cases. In all the examples above, we had the Population Size (Options-Population) set to 20 (the default). This value determines the size of the population at each generation. Increasing the population size enables the genetic algorithm to search more points and thereby obtain a better result. However, the larger the population size, the longer the genetic algorithm takes to compute each generation. Note though that you should set Population Size to be at least the value of Number of variables, so that the individuals in each population span the space being searched. You can experiment with different settings for Population Size that return good results without taking a prohibitive amount of time to run. Finally, another parameter that affects the diversity of the population (remember, it's vital to have good diversity in the population) is the Fitness Scaling (in Options). If the fitness values vary too widely Figure 3, the individuals with the lowest values (recall that we minimize) reproduce too rapidly, taking over the population pool too quickly and preventing the GA from searching other areas of the solution space. On the other hand, if the values vary only a little, all individuals have approximately the same chance of reproduction and the search will progress very slowly.


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Figure 3: Raw fitness values (lower is better) vary too widely on the left. Scaled values (right) do not alter the selection advantage of the good individuals (except that now bigger is better). They just reduce the diversity we have on the left. This prevents the GA from converging too early.

The Fitness Scaling adjusts the fitness values (scaled values) before the selection step of the GA. This is done without changing the ranking order, that is, the best individual based on the raw fitness value remains the best in the scaled rank, as well. Only the values are changed, and thus the probability of an individual to get selected for mating by the selection procedure. This prevents the GA from converging too fast which allows the algorithm to better search the solution space.

6. Part #3 (selection, elitism, mutation)

(35 min)

We continue this GA tutorial using the Rastrigin's function. Use the following settings leaving everything else in its default value (Fitness function: @rastriginsfcn, Number of Variables: 2, Initial Range: [1; 20], Plots: Best Fitness, Distance). The Selection panel in Options controls the Selection Function, that is, how individuals are selected to become parents. Note that this mechanism works on the scaled values, as described in the previous section. Most well-known methods are presented (uniform, roulette and tournament). An individual can be selected more than once as a parent, in which case it contributes its genes to more than one child.

Figure 4: Stochastic uniform selection method. For 6 parents we step the selection line with steps equal to 15/6.

The default selection option, Stochastic Uniform, lays out a line (Figure 4) in which each parent corresponds to a section of the line of length proportional to its scaled value. The algorithm moves


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along the line in steps of equal size. At each step, the algorithm allocates a parent from the section it lands on. For example, assume a population of 4 individuals with scaled values 7, 4, 3 and 1. The individual with the scaled value of 7 is the best and should contribute its genes more than the rest. We create a line of length 1+3+4+7=15. Now, let's say that we need to select 6 individuals for parents. We step over this line in steps of 15/6 and select the individual we land in (Figure 4). The Reproduction panel in Options control how the GA creates the next generation. Here you specify the amount of elitism and the fraction of the population of the next generation that is generated through mating (the rest is generated by mutation). The options are: 

Elite Count: the number of individuals with the best fitness values in the current generation that are guaranteed to survive to the next generation. These individuals are called elite children. The default value of Elite count is 2. Try to solve the Rastrigin's problem by changing only this parameter. Try values of 10, 3 and 1. You will get results like those depicted in Figure 5. It is obvious that you should keep this value low. 1 (or 2 - depending on the population size) is OK. (Why?)

Figure 5: Elite count 10 (left), 3 (middle) and 1 (right). Too much elitism results in early convergence which can make the search less effective.



Crossover Fraction: the fraction of individuals in the next generation, other than elite children, that are created by crossover (that is, mating). The rest are generated by mutation. A crossover fraction of 1 means that all children other than elite individuals are crossover children. A crossover fraction of 0 means that all children are mutation children. The following example shows that neither of these extremes is an effective strategy for optimizing a function. You will now change the problem (you better restart the optimization toolbox to have everything set to default values). You will optimize this function: f(x1, x2, ..., x10) = |X1| + |X2| + ... +|X10| Use the following settings: o Fitness Function: @(x) sum(abs(x)) o Number of variables: 10 o Initial range: [-1; 1]. o Plots: Best fitness and Distance Run the example with the default value of 0.8 for Crossover fraction, in the Options > Reproduction panel. This returns the best fitness value of approximately 0.25 and displays plots


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like those in Figure 6 (left). Note though that for another fitness function, a different setting for Crossover fraction might yield the best result.

Figure 6: Plots for Crossover fraction set to 0.8 (left) and 1 (right).

To see how the genetic algorithm performs when there is no mutation, set Crossover fraction to 1.0 and click Start. This returns the best fitness value of approximately 1.1 and displays plots similar to the one in Figure 6 (right). In this case, the algorithm selects genes from the individuals in the initial population and recombines them. The algorithm cannot create any new genes because there is no mutation. The algorithm generates the best individual that it can using these genes at generation number ~15, where the best fitness plot becomes level. After this, it creates new copies of the best individual, which are then are selected for the next generation. By generation number ~19, all individuals in the population are the same, namely, the best individual. When this occurs, the average distance between individuals is 0. Since the algorithm cannot improve the best fitness value after generation ~15, it terminates because the average change to the fitness function is less what is set to the termination conditions. To see how the genetic algorithm performs when there is no crossover, set Crossover fraction to 0 and click Start. This returns the best fitness value of approximately ~2.7 and displays plots like that on the right. In this case, all children are generated though mutation. The random changes that the algorithm applies never improve the fitness value of the best individual at the first generation. While it improves the individual genes of other individuals, as you can see in the upper plot by the decrease in the mean value of the fitness function, these improved genes are never combined with the genes of the best individual because there is no


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crossover. As a result, the best fitness plot is level and the algorithm stalls at generation number 50.

7. Part #4 (global vs. local minima)

(15 min)

Optimization algorithms sometimes return a local minimum instead of the global one, that is, a point where the function value is smaller than the nearby points, but possibly greater than one at a distant point in the solution space. The genetic algorithm can sometimes overcome this deficiency with the right settings. As an example, consider the following function which has the plot depicted on the right: ( )

{

(

(

)

)(

)

The function has two local minima, one at x = 0, where the function value is –1, and the other at x = 21, where the function value is about -1.37. Since the latter value is smaller, the global minimum occurs at x = 21. Let us now see how we can define custom fitness functions. Go to Matlab and select File>New>M-File. Define the function in the editor window as shown is the picture. Then Save the file to your desktop using the suggested filename two_min (do not change it!). Now in the Matlab's main toolbar, set the current directory to the Desktop. This way, your M file will be visible to the Matlab. In the Optimization Toolbox set Fitness function to @two_min, Number of variables to 1, Stopping criteria>Stall Generations to 100 and click Start. The genetic algorithm returns a point very close to the local minimum at x = 0. The problem here is the default initial range of [0; 1] (in the Options > Population panel). This range is not large enough to explore points near the global minimum at x = 21. One way to make the GA explore a wider range of points (that is, to increase the diversity of the populations) is to increase the Initial range. It does not have to include the point x=21, but it must be large enough so that the algorithm will be able to generates individuals near x = 21. So, set Initial range to [0; 15] and click Start once again. Now the GA returns a point very close to 21.


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