Can I use this fitness function? - optimization

Question

I am working on a project using a genetic algorithm, and I am trying to formulate a fitness function, my questions are:
What is the effect of fitness formula choice on a GA?
It is possible to make the fitness function equals directly the number of violation (in case of minimisation)?

Answer 1

What is the effect of fitness formula choice on a GA
The fitness function plays a very important role in guiding GA.
Good fitness functions will help GA to explore the search space effectively and efficiently. Bad fitness functions, on the other hand, can easily make GA get trapped in a local optimum solution and lose the discovery power.
Unfortunately every problem has its own fitness function.
For classification tasks error measures (euclidean, manhattan...) are widely adopted. You can also use entropy based approaches.
For optimization problems, you can use a crude model of the function you are investigating.
Extensive literature is available on the characteristics of fitness function (e.g. {2}, {3}, {5}).
From an implementation point of view, some additional mechanisms have to be taken into consideration: linear scaling, sigma truncation, power scaling... (see {1}, {2}).
Also the fitness function can be dynamic: changing during the evolution to help search space exploration.
It is possible to make the fitness function equals directly the number of violation (in case of minimisation)?
Yes, it's possible but you have to consider that it could be a too coarse grained fitness function.
If the fitness function is too coarse (*), it doesn't have enough expressiveness to guide the search and the genetic algorithm will get stuck in local minima a lot more often and may never converge on a solution.
Ideally a good fitness function should have the capacity to tell you what the best direction to go from a given point is: if the fitness of a point is good, a subset of its neighborhood should be better.
So no large plateau (a broad flat region that doesn't give a search direction and induces a random walk).
(*) On the other hand a perfectly smooth fitness function could be a sign you are using the wrong type of algorithm.
A naive example: you look for parameters a, b, c such that
g(x) = a * x / (b + c * sqrt(x))
is a good approximation of n given data points (x_i, y_i)
You could minimize this fitness function:
| 0 if g(x_i) == y_i
E1_i = |
| 1 otherwise
f1(a, b, c) = sum (E1_i)
i
and it could work, but the search isn't aimed. A better choice is:
E2_i = (y_i - g(x_i)) ^ 2
f1(a, b, c) = sum (E2_i)
i
now you have a "search direction" and greater probability of success.
Further details:
Genetic Algorithms: what fitness scaling is optimal? by Vladik Kreinovich, Chris Quintana
Genetic Algorithms in Search, Optimization and Machine Learning by Goldberg, D. (1989, Addison-Wesley)
The Royal Road for Genetic Algorithms: Fitness Landscapes and GA Performance by Melanie Mitchell, Stephanie Forrest, John H Holland.
Avoiding the pitfalls of noisy fitness functions with genetic algorithms by Fiacc Larkin, Conor Ryan (ISBN: 978-1-60558-325-9)
Essentials of Metaheuristics by Sean Luke