Genetic Algorithms as a Tool for General Optimization

Angel Kuri2001

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

SGA

The SGA starts from a set of binary strings

Evaluation

Every candidate solution is evaluated

Evaluation

Strings are sorted best to worst

Selection

Every string is asigned a probability proportional to its performance

Selection

Probability of Selection

Crossover

A couple of individuals is randomly chosen; a locus is also randomly chosen

Crossover

Genetic contents are exchanged

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

Deterministic Coupling

Annular Crossover

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

Numerical Representation

Numerical Example

Sample Run

Sample Run

Sample Run

Sample Run

Sample Run

Sample Run

Non-Numerical ExampleTraveling Salesman Problem

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 1:

Identify cities off limits

Repair Algorithm

Step 2:

Identify repeated cities:

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

Repair Algorithm

One possible repaired individual:

Non-Numerical ExampleTraveling Salesman Problem

Non-Numerical Example

Non-Numerical Example

Non-Numerical Example

Non-Numerical Example

Traveling Salesman Problem(Solution)

Non-Numerical Example

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.

Now See the Program!