genetic algorithms as a tool for general optimization angel kuri 2001
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Optimization
Genetic Algorithms are meta-heuristics based on an analogy with evolutionary processes, where three ideas outstand:
a) That any optimizable problem is amenable to an (usually digital) encoding process.
b) That a problem thusly encoded is more easily tackled than its non-encoded couterpart.
c) That genetic operators called natural selection, crossover and mutation are enough to fully and efficiently explore the solution landscape.
Optimization(Any problem is encodable)
For this to be true we have to restrict the solution landscapes to a finite subset of the original ones.
This is not a source of serious conceptual or practical trouble since, in all cases, the computational solution of any given problem is digitally encoded.
Optimization (An encoded problem is more easily tackled)
This dictum is not obvious. However, the ample evidence supporting it leaves little room for doubt.
More importantly, theoretical arguments seem to leave no doubt:
a) Genetic Algorithms always converge to the best solution
b) Convergence is usually quite efficient
Optimization(Genetic operators yield efficient performance)
Our aim when designing Gas is to make the abovementioned convergence as efficient as possible.
Two sources of inefficiency have been clearly determined:
a) Deceptive functions
b) Spurious correlation
Optimization
Before discussing how to overcome the aforementioned limitations we shall analyze a canonical GA (or SGA).
Then we will describe a non-standard GA (VGA) which was designed to avoid the pifalls of the SGA.
The Simple Genetic Algorithm
Although there are many possible varia-tions, the best understood and more widely treated sort of GA is called “Simple”.
Simple Genetic Algorithm
It considers:» Genetic haploid representation» Proportional selection» 1-point crossover» Mutation with low probability» No elitism» Binary strings» Constant sized population» Bounded number of generations
Crossover
The process is repeated m/2 times, yielding m descendants.
The idea is that best genes survive giving rise to better fit individuals.
Mutation
Once the new population is set, some genes are mutated.
Mutation is effected on a very low probability basis (typically, from 0.1% to 0.5%).
Population’s Behavior
m H t m H tf H
fp
H
lp o Hc m( , ) ( , )
( )[
( )( )]
_
1 1
1
The number of schemata “m” of a given schema (H) at time “t” is given by:
SGA
yprobabilitmutationp
yprobabilitcrossoverp
orderschemaHo
lengthdefiningH
m
c
)(
)(
where:
HoffitnessHf )(
populationtheoffitnessaveragef _
SGA
We now illustrate with a simple example. Keep in mind that the purpose of the example
is to transmit the feeling of how a SGA works. The point to be stressed is that this Gas solve
equally well trivial problems as well as very complex ones.
We illustrate both kinds.
Illustrating the SGA
Problem:What is the largest possible value for the
expression ?
To make the question meaningful we further impose:
32 2 x
120 10 x
Encoding
In this case, the GA’s population consists of a set of binary strings of length 10 which we assume to be encoded in positional weighted binary.
SGA’s Limitations
The SGA, as described, does not converge. In fact, if run indefinitely, it will find and loose the solution an infinite number of times.
Furthermore, in some cases it will get stuck in a local optimum.
Vasconcelos Algorithm
To overcome the limitations of a SGA we introduced the so called Vasconcelos GA.
It displays:
a) Deterministic (i -> n-i+1) coupling
b) Annular crossover
c) Uniform mutation
d) Full elitism
Application Example
Optimization of a function subject to constraints» Linear functions» Linear constraints» Non-linear functions» Non-linear constraints
Fitness Function
where s is the number of constraints which are satisfied and
otherwises
fulfilledareconditionsallifxxxf
)1025(10
96)( 79
21
40 s
Non-Numerical Example
In this type of problem it is needed to repair the solutions, since invalid proposals arise on two accounts:
a) Because no two cities may be repeated
b) Because not all binary combinations are necessarily utilized. Hence, some individuals are unfeasible.
Repair Algorithm
To following individual is unfeasible on two accounts:
a) One city is off limits (7)
b) One city repeats itself (2)
Repair Algorithm
Step 3:
Establish a list of candidates for replacement.
In this case, the list is:
a) 1
b) 2
c) 5
Repair Algorithm
for i=1 to number of cities to repair
a) randomly select a vacant position
b) randomly select a candidate from the list
c) eliminate the selected candidate from the list
d) Fill in the selected position
endfor
Conclusions
In the problems we illustrated there is essentially no change in the GA itself. That is, the same algorithm works on essentially different probems.
The methodology, therefore, is applicable to a wide range of possible probelms.
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