The question you need to ask yourself are the following:

How are you going to encode your 4 variables?

How are you going to measure the fitness of your solutions?

By your question, you appear to have your hear set on genetic algorithms. They are many others that may be suitable. I would recommend you to perhaps look into them. You seems to have only 4 variables and this suggests some multi-objectives, so perhaps an ant colony optimisation may be helpful. Without, any more information about the complexity of the problem you are trying to solve, I would suggest to look at some deterministic methods too. 256 possible solutions is not a large search space. Integer programming may help too. Perhaps you may consider statically comparing several methods.

If you were to use to compare a mathematical expression against a solution from your evolutionary algorithm then I would suggest you use a form of genetic programming. A graph, a register or a grammar form of this active field is likely not to bloat and you are likely to be able to read your generated equation and understand them better. From your question this aspect is not clear, whether you are looking forward a solution or a mathematical expression.

If you were to focus on the solution, then use a fitness function that compare against a known minima. Then the distance away from this minima can be minimised.

Dear Aymen,

1) I agree with @Patricia.  It seems that you have already set your mind on using GA. Of course, GA might work very well in your case. However, in general, the optimizarion tool is not selected beforehand but is rather taylored according to the characteristics of the problem at hand. In other words, many other tools exist that could be better suited to your problem.

2) My understanding (please correct me if I am wrong) is that you are not able to compute your objective function Y for any X. It seems that you only know Y for a limited set of values of X = [X1, X2, X2, X4], which you call your panel of data. If this is the case, then your problem must be divided in several distinct steps:

Step 1: Carefully select a suitable parametric model M to represent the data at hand. Of course, the type of model and the number of ajustable parameters will depend highly on your problem.

Step 2: Find the best-fit values for the free parameters. This is a separate optimization problem with a separate objective function.

step 3: Use the ajusted model as a surrogate objective function to represent (approximatively) the function Y. Then, find the values of X1, X2, X3, X4 that maximize it. Keep in mind that you are not maximizing the true Y but an appromimation of it.

Clearly, the optimization tools used in step 2 and Step 3 need not be the same. Moreover, the quality of the final result will depend, of course, on the optimisation tool selected in step 3 but also on the adequateness of:

a) the model selected in step 1

b) the objective function used in step 2 and the corresponding optimization tool.

Hope this helps.

optimal values of x that maximizes Y. but what are the constaints? using GA only your objective function will get maximized. if no constaints the x will take its upper bound value. it doesnt make sense. from  regression you will get an equation of Y in linear / non linear combination of X (depends on type of regression you are gonna use). then if you want to optimize it using any optimization methods, you want to have constraints. 

other wise all the optimization techniques will give you a result of Y with the upper bounds of X

salam

can you send your fitness function y in math form with their x variables like:

max .y=x1+x2+x3+x4 like that and if you have constraints on x1,x2,x3,x4 like x1 must be less than any number and greater than any number