# Threshold selecting: best possible probability distribution for crossover selection in genetic algorithms.

**ABSTRACT** The paper considers the problem of selecting individuals in the current population in Genetic Algorithms for crossover to find a solution of high fitness of a given combinatorial optimization problem. Many different schemes have been considered in literature as possible selection strategies, such as Windowing, Exponential reduction, Linear transformation or normalization and Binary Tournament selection. It is shown that if one wishes to maximize any linear function of the final state probabilities, e.g. the fitness of the best individual of the final population of the algorithm, then the best probability distribution for selecting individuals in each generation is a rectangular distribution over the individuals sorted by their fitness values. This means uniform probabilities have to be assigned to a group of the best individuals of the population but probabilities equal to zero to individuals with fitness ranks higher than a fixed cutoff, which is equal to a certain rank in the sorted fitness vector. The considered strategy is called Threshold Selecting. The proof applies basic arguments of Markov chains and linear optimization and makes only a few assumptions on the underlying principles and hence applies to a large class of Genetic Algorithms.

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**ABSTRACT:**Genetic algorithms are a widely studied area of research. This paper proposes an exhaustive analysis of recombi-nation and mutation schemes for genetic algorithms in a generic way. Besides an often one sided analysis of advantages and disad-vantages of algorithms, this paper tries to highlight all relevant influence for the election of a suitable algorithm. Intention is an independent inspection of influence characteristics. Entropy tests as well as convergence tests are accomplished and several other kinds of influence like population pool size are taken into account. To my best knowledge, there is no research papers that deal with the genetic algorithms at this abstract level. The main intention of this work is to provide theoretical background for further genetic design. A wide variety of analyzes are performed to build an adequate basis for comparison.01/2010; - SourceAvailable from: cs.allegheny.edu
##### Conference Paper: Empirically studying the role of selection operators duringsearch-based test suite prioritization.

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**ABSTRACT:**Regression test suite prioritization techniques reorder test cases so that, on average, more faults will be revealed earlier in the test suite's execution than would otherwise be possible. This paper presents a genetic algorithm-based test prioritization method that employs a wide variety of mutation, crossover, selection, and transformation operators to reorder a test suite. Leveraging statistical analysis techniques, such as tree model construction through binary recursive partitioning and kernel density estimation, the paper's empirical results highlight the unique role that the selection operators play in identifying an effective ordering of a test suite. The study also reveals that, while truncation selection consistently outperformed the tournament and roulette operators in terms of test suite effectiveness, increasing selection pressure consistently produces the best results within each class of operator. After further explicating the relationship between selection intensity, termination condition, fitness landscape, and the quality of the resulting test suite, this paper demonstrates that the genetic algorithm-based prioritizer is superior to random search and hill climbing and thus suitable for many regression testing environments.Genetic and Evolutionary Computation Conference, GECCO 2010, Proceedings, Portland, Oregon, USA, July 7-11, 2010; 01/2010

Page 1

Threshold Selecting: Best Possible Probability Distribution

for Crossover Selection in Genetic Algorithms

Jörg Lässig

Chemnitz University of Techn.

Reichenhainer Str. 70,R.A209

D-09126 Chemnitz, Germany

joerg.laessig@cs.tu-

chemnitz.de

Karl Heinz Hoffmann

Chemnitz University of Techn.

Reichenhainer Str. 70, R. 356

D-09126 Chemnitz, Germany

hoffmann@physik.tu-

chemnitz.de

Mihaela En˘ achescu

Stanford University

Terman Eng. Center, R.323

Stanford, CA 94305-4036,

United States

mihaela@cs.stanford.edu

ABSTRACT

The paper considers the problem of selecting individuals in

the current population in Genetic Algorithms for crossover

to find a solution of high fitness of a given combinatorial

optimization problem.

Many different schemes have been considered in literature

as possible crossover selection strategies, such as Windowing,

Exponential reduction, Linear transformation or normaliza-

tion and Binary tournament selection.

It is shown that if one wishes to maximize any linear func-

tion of the final state probabilities, e.g. the fitness of the best

individual of the final population of the algorithm, then the

best probability distribution for selecting individuals in each

generation is a rectangular distribution over the individuals

sorted by their fitness values.

This means uniform probabilities have to be assigned to

a group of the best individuals of the population but proba-

bilities equal to zero to individuals with fitness ranks higher

than a fixed cutoff, which is equal to a certain rank in

the sorted fitness vector. The considered strategy is called

Threshold Selecting.

The proof applies basic arguments of Markov chains and

linear optimization and makes only a few assumptions on

the underlying principles and hence applies to a large class

of Genetic Algorithms.

Track: Genetic Algorithms.

Categories and Subject Descriptors

I.2.8 [Artificial Intelligence]: Problem Solving, Control

Methods, and Search—heuristic methods

General Terms

Algorithms, Theory

Keywords

Genetic Algorithms, Crossover Selection, Markov Process,

Master Equation, Threshold Selecting

Copyright is held by the author/owner(s).

GECCO’08, July 12–16, 2008, Atlanta, Georgia, USA.

ACM 978-1-60558-131-6/08/07.

1.INTRODUCTION

Designing a Genetic Algorithm (GA) for a certain given

problem, there are many degrees of freedom to be fixed but

often the choice of certain parameters or operators relies on

experimental tests and the experience of the programmer.

Such choices are e.g.:

• representation of a solution in the state space as an

artificial genome,

• choice of a crossover operator to form a new population

in each iteration,

• choice of a mutation rate,

• choice of a selection scheme over the individuals of a

population for crossover.

Today GAs are in broad practical application to problems

in many different fields as science, engineering or economics

(see e.g. [2, 7, 12, 14, 16, 18]) and excellent experimental

results have been obtained. Despite interesting theoretical

progress in the last years [3, 5, 6, 4, 19, 22] exact proves for

optimal choices of design criteria are still missing.

This paper focuses on the last of the design criteria above,

also called parent selection. In all variants of GAs some form

of the selection operator must be present [3]. A wide variety

of selection strategies have been proposed in the literature.

In general, m individuals of the current population of size

n have to be selected for crossover into a mating pool. In-

dividuals with higher fitness are more likely to receive more

than one copy and less fit individuals are more likely to re-

ceive no copies. In different replacement schemes the size

of the pool differs. After selecting the mating pool some

crossover scheme takes individuals from that pool and pro-

duces new outcome, until the pool is exhausted. Regarding

the crossover scheme no further restrictions are necessary

for our considerations concerning the optimal choice of a

selection strategy as discussed in the sections below.

The behavior of the GA very much depends on how in-

dividuals are chosen to go into the mating pool [20]. The

simplest approach is that the reproduction probability of

an individual of the population is proportional directly to

its fitness (roulette-wheel selection). Other approaches are

windowing, where first the fitness of the worst individual is

subtracted from each individual fitness, exponential, where

the square root of one plus the fitness is taken, linear trans-

formation, where a linear function of the fitness is computed,

e.g. f′= ̺ · f + ϕ, linear ranking selection, where a linear

function over a fitness ranking of the individuals is applied,

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and binary tournament selection, where two individuals are

selected with uniform probability in a preselection and the

individual with better fitness is then submitted to the mat-

ing pool. See e.g. [3, 6, 20] for an overview on different

selection schemes.

Each of these choices is reported to have strengths and

weaknesses. The selection strategy has to be chosen such

that the population evolves towards ”better“ overall fitness.

For example, the fitness of the fittest individual in the final

population might be required to be as high as possible.

In the following it is proven that Threshold Selecting is op-

timal in a certain sense defined below. In Threshold Select-

ing the selection is based on fitness ranks, and the selection

probability on the ranks is rectangular, i.e. it includes one

or more individual(s) with the highest fitness value(s) with

the same non-vanishing probability but introduces a cutoff

rank γ so that all individuals with higher ranks are selected

with probability zero.

Table 1 [20] gives an example on the different methods

for a population of four individuals with exemplary fitness

values 50, 25, 15, 10.

Table 1: Comparison of different selection strategies

Rank of the individuals

Rough fitness

Roulette-wheel

Windowing

Exponential

Linear transformation (2f +1)

Linear ranking selection

Binary tournament selection

Threshold Selecting (γ = 3)

1234

50.0

0.5

0.6

0.365 0.261 0.205 0.169

0.495 0.25

0.40.3

0.438 0.312 0.188 0.062

0.30.3

25.0

0.25

0.25

15.0

0.15

0.083 0.0

10.0

0.1

0.152 0.103

0.20.1

0.30.0

2. TECHNIQUE

The proof technique is based on the fact that the selection

probability distributions assigning probabilities to n ordered

objects can be seen as vectors in a n dimensional space. Spe-

cial assumptions of the problem structure restrict the space

of possible solutions to a simplex. Then, due do the linearity

of applicable objective functions on the selection probabili-

ties, the problem reduces to the task to find the minimum of

a linear function on a simplex, which must be a vertex in the

general case by the fundamental theorem of linear program-

ming. As will be shown, the vertices are exactly equivalent

to the rectangular distributions mentioned above.

This technique has been already applied to show for the

acceptance rule in Monte Carlo methods as Simulated An-

nealing [17], Threshold Accepting [8] or Tsallis Statistics [9,

10, 23] that Threshold Accepting is provably a best possible

choice [11].

Further the stochastic optimization algorithm Extremal

Optimization [1] has been investigated [13, 15]. Extremal

Optimization also works by simulating random walkers as

the methods mentioned before, but needs a special structure

of the problem under consideration: every state is specified

by several degrees of freedom, each of which can be assigned

a fitness. Each iteration chooses one degree of freedom for

change based on fitness values. It has been shown that a

rectangular distribution is the best choice in each iteration

of Extremal Optimization.

3.DEFINITIONS

We consider combinatorial optimization problems with a

finite state space Ω of states α ∈ Ω, which are the possible

solutions for the problem. A fitness function f(α) describes

the quality of the solution α and has to be maximized, i.e.

the states with a higher fitness are better. Note that there

is only a finite number of possible values for f(α).

GAs consider populations (or pools) of states. If there are

n states in a population, then each generation of the GA

is equivalent to a generalized state α := (α1,α2,...,αn) ∈

Ωn= Ω with n finite. A generalized fitness function f(α)

has to be defined as well, which is usually done by f(α) :=

max{f(αi) | i = 1,2,...,n}.

To obtain good solutions GAs proceed by randomly select-

ing a start population, and then evolving it by a selection

and subsequent crossover operation. Mutations are also pos-

sible, but are of no importance here. We here confine our-

selves to selection steps, where the probability to enter the

mating pool is based on the fitness ranks of the population

members. The possible mating pools are again described by

generalized states ¯ γ := (γ1,γ2,...,γm), albeit not of size

n but of size m. The bar notation is used to differentiate

between the population and the mating pool.

For the choice of the m individuals for the crossover step in

the GA, m time dependent probability distributions di,t(k),

i = 1,2,...,m are defined over the ranks k.

structure at time t exactly m ranks kl1,kl2,...,klmare cho-

sen by the GA and hence m individuals from the current

population according to di,t,i = 1,2,...,m. Technically,

each of the individual members βi of the current population

β is assigned a rank ki based on its fitness: The individuals

of a population can be ordered according to their fitness in

a ranking ki ∈ N⋆

n= {1,2,...,n}:

Given this

ki ≤ kj ⇐⇒ f(αi) ≥ f(αj) ∀ pairs (i,j) .

The following assumptions are adopted for the selection

probabilities di,t(k):

(A1) Each step of the algorithm is independent of the former

steps.

(A2) In each step t, 1 ≥ di,t(1) ≥ di,t(2) ≥ ··· ≥ di,t(n) ≥ 0

holds for i = 1,2,...,m, i.e. it is more probable to

recombine individuals with lower rank (higher fitness)

than individuals with a higher rank (lower fitness).

(A3)Pn

k=1di,t(k) = 1 for i = 1,2,...m, i.e. the distribu-

tions are normalized.

Due to the random nature of the selection process there

is a transition probability

ΛS

¯ γβ= d1,t(kl1)d2,t(kl2)···dm,t(klm)(1)

to obtain the mating pool ¯ γ = (βl1,βl2,...,βlm) from the

population β = (β1,β2,...,βn).

In the crossover step an operator C¯ γ is applied to the cur-

rent population β. The operator C¯ γ is not deterministic

but determines the fixed probabilities ΛC

population α ∈ Ω from β ∈ Ω and ¯ γ as intermediate step.

For each fixed pair ¯ γ and β we have

α¯ γβto obtain a new

X

α∈Ω

ΛC

α¯ γβ= 1.

The dependence on both, ¯ γ and β, can be explained by

the fact that not each crossover operator creates the new

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Page 3

population α by solely utilizing the mating pool ¯ γ, which

is only the case for the Generation Replacement Model. In

the Steady-State Replacement Model it is also possible that

other states βi from the current population β are kept.

An exemplary procedure could work as follows: After get-

ting a mating pool ¯ γ of m states m new states are created

by recombination from ¯ γ. In addition to the current n states

then there are n + m states available and n states are kept

for the new generation α applying some standard procedure

(e.g. keep the best n of all n+m states). In the special case

of generation replacement we have n = m and β is replaced

completely by the states from recombination.

Remark 1. Think of the recombination itself that states

from the mating pool ¯ γ are taken one after another. Each

possible tuple of states for one crossover operation is chosen

with the same probability, i.e. probability is uniform dis-

tributed among all possible tuples of desired size in ¯ γ. Most

commonly pairs are chosen and for each pair a split posi-

tion for one point crossover or more than one split position

for multi point crossover procedures are determined - again

uniform distributed (the proof is general enough that also

other distributions or procedures are possible here).

Combining selection and crossover leads to a transition

probability Γt

αβfrom one population β to the next popula-

tion α.

In summary the dynamics of GAs can be described as a

Markovian random walk in state space. For the develop-

ment of the probability pt

αto be in state α (which means to

have a certain population in the GA) the master equation

pt

α=

X

β∈Ω

Γt

αβ· pt−1

β

(2)

is applicable. Here Γt

αβis defined to be

Γt

αβ=

X

¯ γ∈¯Ω

ΛC

α¯ γβΛS

¯ γβ. =

X

¯ γ∈¯Ω

ΛC

α¯ γβ

m

Y

i=1

di,t. (3)

In the next step the dependence of the performance of

the GA on the probability distributions d1,t,d2,t,...,dm,t

over the ranks in the population is investigated and which

choice is an optimal one for these distributions considering

an optimization run with S steps.

Most commonly one of the following objectives is used [11]

(here slightly adapted in the notation for GAs):

(O1) The mean of the fitness of the best individual in the

final population should be as large as possible.

(O2) The probability of having a final population contain-

ing a member of optimal fitness should be as large as

possible.

To optimize according to (O1) one chooses

g1(α) = f(α) = max{f(αi) | i = 1,2,...,n}

which means essentially that the quality of a population is

assumed to be equivalent to the quality of the best individual

in the population, and to optimize according to (O2) one

chooses

g2(α)=

1

0

| if α contains a state with fitness fmax

| otherwise,

i.e.

values different from zero. Other objectives are possible and

the only important fact for the proof is that they are linear

in the final state probabilities, as we will see.

The optimization process consists of a finite number of S

steps (t := 1,2,...,S). Note, that Γt

for i fixed. The arguments below apply in general to any

objective function which is linear in the final state probabil-

ities pS

αas e.g. (O1) and (O2). The state probabilities at

time t are considered as vector ptand the linear objective

function with values g(α) for each state α as vector g. If

(·)trdenotes the transpose, the measure of performance is

equivalent to

only optimal states with fitness fmax have objective

αβis linear in di,t(k)

g(pS) = gtr· pS=

X

α∈Ω

g(α) · pS

α−→ max .(4)

4.SETUP OF A VECTOR SPACE

In the following the distributions di,t(k),k = 1,2,...,n,

are considered to be n dimensional vectors di,twith entries

di,t

k

∈ [0,1]. Consider without loss of generality n − 1 of

these distributions di,t,i ∈ 1,2,...,n to be fixed. Only one

remaining distribution denoted by dr,tis open to optimize.

The question is then how to choose dr,tto maximize the

objective function. As a consequence of the assumptions

(A2) and (A3), the region F of feasible vectors dr,tis de-

fined by the n + 1 linear inequations in (A2) and one linear

equation in (A3) where the first inequation 1 ≥ dr,t

from the others. Of the remaining n inequations n−1 must

be set to equations to find extreme points (vertices) in the

region F. Letting V denote the set of extreme points of

F, the elements of V are exactly those vectors dr,twhich

have the initial sequence of i entries equal to 1/i followed

by a sequence of n − i entries equal to zeros. Explicitly,

V = {v1,v2,...,vn}, where v1 = (1,0,0,...,0)tr, v2 =

(1/2,1/2,0,0,...,0)tr, vi = (1/i,1/i,...,1/i,0,0,...,0)tr,

and vn = (1/n,1/n,...,1/n)tr. Note that the elements of

V are linearly independent. Then F is exactly the convex

hull C(V ) of V , which is a simplex.

This equivalence of F and C(V ) can be shown by standard

calculations, see e.g. [13] as reference.

1

follows

5.PROOF

Now the Bellman principle of dynamic programming is

applied, starting with the last step in the optimization pro-

cess. The output of the last step is pSand used to determine

the optimality criterion (4). In the last step S one has to

solve the optimization problem (4) for the given input pS−1.

Using (2) one gets

g(pS) =

X

α,β∈Ω

g(α) · ΓS

αβ· pS−1

β

−→ max

with ΓS

function on a simplex has to be found. To do so choose the

distribution dr,S, which selects one of the m individuals for

crossover, equal to one of the vertices vi ∈ V . The corre-

sponding transition probabilities are denoted by ΓS, because

then all m distributions are fixed in this stage. Considering

now the step before, i.e. step S − 1, one gets

αβgiven by (3). Hence, the maximum of a linear

gtr· pS= (gtr· ΓS) ·

0

@

X

α,β∈Ω

ΓS−1

αβ · pS−2

β

1

A−→ max .

2183

Page 4

Defining gS−1= gtr·ΓSas new objective function the same

arguments can be applied to choose dr,S−1. The resulting

matrix is then denoted with ΓS−1where the optimal transi-

tion probabilities are again found by taking dr,S−1to be an

element of V . For all other steps the same argument holds

as well, i.e. dr,t,t = 1,2,...,S, are all elements of the vertex

set V .

Because an arbitrary distribution dr,thas been chosen for

the proof to be variable, the same arguments hold for all

distributions dr,t,r = 1,2,...,n as well. Hence, the proof

shows that a rectangular distribution over the fittest indi-

viduals in each generation in the iterations t = 1,2,...,S

in GAs gives the best implementation of the selection step

for each individual used for the crossover step in iteration t.

For the mutation operator the same as for crossover holds,

because this is an operator with equivalent characteristics

with regard to our proof but only one input state.

6. CONCLUSIONS

In this paper the problem of selecting individuals from the

population of a GA for crossover based on a fitness function

has been considered. The master equation was the means of

choice to describe the corresponding dynamics as a random

walk in state space and some straightforward assumptions

on the probability distributions for selecting the individuals

in a certain generation have been formulated.

The goal was to find transition probabilities assuring the

optimum control of the evolutionary development in the GA.

A rectangular distribution of selection probabilities is prov-

ably optimal, provided the performance is measured by a lin-

ear function in the state probabilities, which includes many

reasonable choices as for instance maximizing the mean fit-

ness of the best individual in the final population.

The proof above is based on the fundamental theorem of

linear programming, which states that a linear function de-

fined on a simplex assumes its minimum at a vertex. The

proof does not state that all optimal crossover selection

strategies in GAs are rectangular. Other strategies may do

equally well, but not better. If there exists an optimal strat-

egy other than Threshold Selecting, it follows that an edge

or a face of the described simplex does equally well. Thus,

it seems unlikely that a strictly monotonic distribution can

be optimal [11], which would imply that all the vertices in

V do equally well.

As presented the proof can be applied for any crossover

procedure in GAs with independent probability distributions

for the selection of the crossover individuals and both for the

Generation Replacement Model, where the mating pool has

size n for populations of size n, and also for the Steady-

State Replacement Model, where only some individuals are

replaced [21].

Currently the knowledge that best performance can be

achieved using Threshold Selecting is only of limited use,

since the cutoff ranks γ to be used are not known a priori.

Therefore it would be interesting to perform numerical ex-

periments comparing different possible distributions empiri-

cally. Further, it is reasonable to introduce a schedule on the

cutoff rank γ, narrowing the rectangular distribution during

the optimization process and thus increasing the evolution-

ary pressure gradually. Moreover, it would be interesting

to obtain also theoretical progress concerning the choice of

one of the possible rectangular distributions or to reduce the

choice to a certain assortment.

Our proof was based on the assumption that the objective

measuring the performance of the GA is a linear function of

the state probabilities. While this includes very common

measures, it does not include them all.

As an example, the best-so-far fitness over individuals of

all previous generations as a measure is beyond the scope of

the proof presented here. From a practical point of view this

can be fixed easily, adding one individual to the population

and adapting the crossover operator in a way to keep the

best individual in each iteration, if not one with better or

equal fitness is found.

Clearly this adaption is possible for each given crossover

operator C to obtain an operator C′. Applying C′the ob-

jective value of the individual with best fitness in the final

population is equivalent to the best-so-far fitness but the dis-

tributions are as well rectangular because the proof applies

as before.

Further, the proof above had to assume a finite state

space. The exploration of continuous state spaces would be

interesting as well, but considering the discrete arithmetic

of digital computers any state space in practice is effectively

finite [13].

Finally, the arguments presented here establish the struc-

ture of a provably optimal strategy which could be applied

to study also other heuristic approaches to global optimiza-

tion.

7. ACKNOWLEDGMENTS

The authors would like to thank the German Research

Foundation (DFG) for partially funding this research.

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