Download 1. Metaheuristic_Pop-Based_McGill

METAHEURISTIC
Jacques A. Ferland
Department of Informatique and Recherche Opérationnelle
Université de Montréal
[email protected]
March 2015
Overview
• Heuristic Constructive Techniques:
Generate a good solution
• Local (Neighborhood) Search Methods (LSM):
Iterative procedure improving a solution
• Population-based Methods:
Population of solutions evolving to mimic a natural process
Population-based Methods
•
•
•
•
•
•
•
Genetic Algorithm (GA)
Hybrid GA-LSM
Genetic Programming (GP)
Adaptive Genetic Programming (AGP)
Scatter Search (SS)
Ant Colony Algorithm (ACA)
Particle Swarm Optimiser (PSO)
Genetic Algorithm (GA)
• Population based algorithm
• At each generation three different operators are first applied to generate a
set of new (offspring) solutions using the N solutions of the current
population P:
selection operator: selecting from the current population parent-solutions
that reproduce themselves
crossover (reproduction) operator: producing offspring-solutions from each
pair of parent-solutions
mutation operator: modifying (improving) individual offspring-solution
• A fourth operator (culling operator) is applied to determine a new
population of size N by selecting among the solutions of the current
population and the offspring-solutions according to some strategy
Two variants of GA
•
At each generation of the Classical genetic
algorithm:
•
At each generation of the Steady-state
population genetic algorithm:
- N parent solutions are selected and
paired two by two
- An even number of parent-solutions
are selected and paired two by two
- A crossover operator is applied to
each pair of parent-solutions according to some
probability to generate two offspring-solutions.
Otherwise the two parent-solutions become their
own offspring-solutions
- A crossover operator is applied
to each pair of parent-solutions to
generate two offspring-solutions.
- A mutation operator is applied
according to some probability to
each offspring-solution.
- A mutation operator is applied
according to some probability to
each offspring-solution.
- The population of the next generation
includes the offspring-solutions
-The population of the next generation
includes the N best solutions among the
current population and the offspringsolutions
Problem used to illustrate
• General problem
min f  x 
xX
• Assignment type problem: Assignment of resources j to activities i
f  x
min
m
s.t.
x
j 1
ij
1
xij  0 or 1
i  1,
,n
i  1,
, n ; j  1,
,m
Encoding the solution
• The phenotype form of the solution x є ℝp is encoded (represented) as a
genotype form vector z є ℝn (or chromozome) where m may be different
Assigning resources j to activities i
from n.
min f  x 
m
Subject to  x  1
• For example in the assignment type problem:
x  0 or 1
let x be the following solution: for each 1≤ i ≤ n,
xij(i) = 1
xij = 0 for all other j
j 1
ij
x є ℝnxm can be encoded as z є ℝn where
zi = j(i)
i = 1, 2, …, n
i.e., zi is the index of the resource j(i) assigned to activity i
ij
i  1,
,n
i  1,
,n
j  1,
,m
Genetic Algorithm (GA)
• Population based algorithm
• At each generation three different operators are first applied to generate a
set of new (offspring) solutions using the N solutions of the current
population P:
selection operator: selecting from the current population parent-solutions
that reproduce themselves
crossover (reproduction) operator: producing offspring-solutions from each
pair of parent-solutions
mutation operator: modifying (improving) individual offspring-solution
• A fourth operator (culling operator) is applied to determine a new
population of size N by selecting among the solutions of the current
population and the offspring-solutions according to some strategy
Selection operator
• This operator is used to select an even number (2, or 4, or …, or N) of
parent-solutions.
• Each parent-solution is selected from the current population according to
some strategy or selection operator.
• Note that the same solution can be selected more than once.
• The parent-solutions are paired two by two to reproduce themselves.
• Selection operators:
Random selection operator
Proportional (or roulette whell) selection operator
Tournament selection operator
Diversity preserving selection operator
Random selection operator
• Select randomly each parent-solution from the current (entire) population
• Properties:
Very straightforward
Promotes diversity of the population generated
Proportional (Roulette whell) selection
operator
• Each parent-solution is selected as follows:
i) Consider any ordering of the solutions z1, z2, …, zN in P
ii) Select a random number α in the interval
For the problem
min f (x)
[0, ∑1≤k≤ N ( 1 / f( zk) )]
where x is encoded as z
iii) Let τ be the smallest index such that
1/f (z) measures
∑1≤k≤ τ (1 / f( zk ) ) ≥ α
the fittness of the
τ
iv) Then z is selected
solution z
1 / f( z1 )
|
|
1 / f( z2 )
1 / f( z3)
|
|
τ
1 / f( zN)
…
|
|
α
The chance of selecting zk increases with its fittness 1 / f( zk)
Tournament selection operator
• Each parent-solution is selected as the best solution in a subset of randomly
chosen solutions in P:
i) Select randomly N’ solutions one by one from P (i.e., the same solution
can be selected more than once) to generate the subset P’
ii) Let z’ be the best solution in the subset P’:
z’ = argmin z є P’ f(z)
iii) Then z’ is selected as a parent-solution
Elitist selection
• The main drawback of using elitist selection operator like Roulette whell
and Tournament selection operators is premature converge of the algorithm
to a population of almost identical solutions far from being optimal.
• Other selection operators have been proposed where the degree of elitism is
in some sense proportional to the diversity of the population.
Genetic Algorithm (GA)
• Population based algorithm
• At each generation three different operators are first applied to generate a
set of new (offspring) solutions using the N solutions of the current
population P:
selection operator: selecting from the current population parent-solutions
that reproduce themselves
crossover (reproduction) operator: producing offspring-solutions from each
pair of parent-solutions
mutation operator: modifying (improving) individual offspring-solution
• A fourth operator (culling operator) is applied to determine a new
population of size N by selecting among the solutions of the current
population and the offspring-solutions according to some strategy
Crossover (recombination) operators
• Crossover operator is used to generate new solutions including
interesting components contained in different solutions of the
current population.
• The objective is to guide the search toward promissing regions
of the feasible domain X while maintaining some level of
diversity in the population.
• Pairs of parent-solutions are combined to generate offspringsolutions according to different crossover (recombination)
operators.
One point crossover
• The one point crossover generates two offspring-solutions from the two
parent-solutions
z1 = [ z11, z21, …, zn1]
z2 = [ z12, z22, …, zn2]
as follows:
i) Select randomly a position (index) ρ, 1 ≤ ρ ≤ n.
ii) Then the offspring-solutions are specified as follows:
oz1 = [ z11, z21, …, zρ1, zρ+12, …, zn2]
oz2 = [ z12, z22, …, zρ2, zρ+11, …, zn1]
Hence the first ρ components of offspring oz1 (offspring oz2) are the
corresponding ones of parent 1 (parent 2), and the rest of the components
are the corresponding ones of parent 2 (parent 1)
Two points crossover
• The two points crossover generates two offspring-solutions from the two
parent-solutions
z1 = [ z11, z21, …, zn1]
z2 = [ z12, z22, …, zn2]
as follows:
i) Select randomly two positions (indices) μ,ν, 1 ≤ μ ≤ ν ≤ n.
ii) Then the offspring-soltions are specified as follows:
oz1 = [ z11, …, zμ-11, zμ2, …, zν2, zν+11, …, zn1]
oz2 = [ z12, …, zμ-12, zμ1, …, zν1, zν+12, …, zn2]
Hence the offspring oz1 (offspring oz2) has components μ, μ+1, …, ν of
parent 2 (parent 1), and the rest of the components are the corresponding
ones of parent 1 (parent 2)
Uniform crossover
• The uniform crossover requires a vector of bits (0 or 1) of dimension n to
generate two offspring-solutions from the two parent-solutions
z1 = [ z11, z21, …, zn1] , z2 = [ z12, z22, …, zn2] :
i) Generate randomly a vector of n bits, for example [0, 1, 1, 0, …, 1, 0]
ii) Then the offspring-solutions are specified as follows:
parent 1: [ z11, z21, z31, z41,…, zn-11, zn1]
parent 2: [ z12, z22, z32, z42,…, zn-12, zn2]
Vector of bits: [ 0 , 1 , 1 , 0 , …, 1 , 0 ]
Offspring oz1 : [ z11, z22, z32, z41,…, zn-12, zn1]
Offspring oz2: [ z12, z21, z31, z42,…, zn-11, zn2]
Uniform crossover
• The uniform crossover requires a vector of bits (0 or 1) of dimension m to
generate two offspring-solutions from the two parent-solutions
z1 = [ z11, z21, …, zn1] , z2 = [ z12, z22, …, zn2] :
i) Generate randomly a vector of bits, for example [0, 1, 1, 0, …, 1, 0]
ii) Then the offspring-solutions are specified as follows:
parent 1: [ z11, z21, z31, z41,…, zn-11, zn1]
Hence the ith component of oz1
parent 2: [ z12, z22, z32, z42,…, zn-12, zn2]
(oz2) is the ith component of
Vector of bits: [ 0 , 1 , 1 , 0 , …, 1 , 0 ]
Offspring oz1 : [ z11, z22, z32, z41,…, zn-12, zn1]
Offspring oz2: [ z12, z21, z31, z42,…, zn-11, zn2]
parent 1 (parent 2) if the ith
component of the vector of bits
is 0, otherwise, it is equal to the
ith component of parent 2
(parent 1)
Path relinking
Let d  z1 , z 2  the Hamming
distance between z1 and z 2 :
Denote by N  z1  the set of
d  z1 , z 2   number of components of
feasible solution obtained
by modifying slightly z1.
z1 and z 2 that are different.
•
The path relinking procedure generates a sequence of feasible solutions along a path linking two
solutions xz1 and xz22
Path relinking procedure:
1. ps  z and ps  z
1
1
2
2
Let also


N  ps1 , ps 2  = z  N  ps1  : d  z, ps 2   d  ps1 , ps 2 
    
2. Use the roulette whell selection operator to select z  N p s1 , p s 2
If f ( z )  f ( z1 ) and f ( z )  f ( z 2 ), stop with z
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d  ps1 , ps 2   0, then stop with z; otherwise repeat 2
1. ps1  z1 and ps 2  z 2

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z1
z2

1. ps1  z1 and ps 2  z 2

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
ps1  z1
2
ps 2  z

1. ps1  z1 and ps 2  z 2
Let also

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 


N  ps1 , ps 2  = z  N  ps1  : d  z, ps 2   d  ps1 , ps 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
ps1  z1
2
ps 2  z

1. ps1  z1 and ps 2  z 2

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z
ps  z
1
1
2
ps 2  z

1. ps1  z1 and ps 2  z 2

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z
1
ps 2
2
ps1  z

1. ps1  z1 and ps 2  z 2
Let also

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 


N  ps1 , ps 2  = z  N  ps1  : d  z, ps 2   d  ps1 , ps 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z
1
ps 2
2
ps1  z

1. ps1  z1 and ps 2  z 2

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z
1
ps 2
2
ps1  z
z

1. ps1  z1 and ps 2  z 2

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z
1
ps1
z2
ps 2

1. ps1  z1 and ps 2  z 2
Let also

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 


N  ps1 , ps 2  = z  N  ps1  : d  z, ps 2   d  ps1 , ps 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z
1
ps1
z2
ps 2

1. ps1  z1 and ps 2  z 2

2. Use the roulette whell selection operator to select z  N p  s1  , p  s 2 
If f ( z )  f ( z ) and f ( z )  f ( z ), stop with z
1
2
3. Otherwise ps1  ps 2 , and ps 2  z
4. If d ( ps1 , ps 2 )  0, then stop with z; otherwise repeat 2
z
1
ps1
z
z2
ps 2

Ad hoc crossover operator
The preceding crossover operators are sometimes too general
to be efficient. Hence, whenever possible, we should rely on
the structure of the problem to specify ad hoc problem
dependent crossover operator in order to improve the
efficiency of the algorithm.
Recovery procedure
Furthermore, whenever the structure of the problem is such
that the offspring-solutions are not necessarily feasible, then an
auxiliary procedure is required to recover feasibility. Such a
procedure is used to transform the offspring-solution into a
feasible solution in its neighborhood.
Genetic Algorithm (GA)
• Population based algorithm
• At each generation three different operators are first applied to generate a
set of new (offspring) solutions using the N solutions of the current
population P:
selection operator: selecting from the current population parent-solutions
that reproduce themselves
crossover (reproduction) operator: producing offspring-solutions from each
pair of parent-solutions
mutation operator: modifying (improving) individual offspring-solution
• A fourth operator (culling operator) is applied to determine a new
population of size N by selecting among the solutions of the current
population and the offspring-solutions according to some strategy
Mutation operator
• Mutation operator is an individual process to modify offspring-solutions
• In traditional variants of Genetic Algorithm the mutation operator is used to
modify arbitrarely each componenet zi with a small probability:
For i = 1 to n
Generate a random number β  [0, 1]
If β < βmax then select randomly a new value for zi
where βmax is small enough in order to modify zi with a small probability
• Mutation operator simulates random events perturbating the natural
evolution process
• Mutation operator not essential, but the randomness that it introduces in the
process, promotes diversity in the current population and may prevent
premature convergence to a bad local minimum
Hybrid GA - LSM
Genetic
Algorithm
(GA)
Hybrid
GA - LSM
• Population based algorithm
• At each generation three different operators are first applied to generate a
set of new (offspring) solutions using the N solutions of the current
population P:
selection operator: selecting from the current population parent-solutions
that reproduce themselves
crossover (reproduction) operator: producing offspring-solutions from each
pair of parent-solutions
mutation operator: modifying
(improving)
individual
offspring-solution
improve each
offspring-solution
using
a LSM
• A fourth operator (culling operator) is applied to determine a new
population of size N by selecting among the solutions of the current
population and the offspring-solutions according to some strategy
HybridGA
Methods
Hybrid
- LSM
• This is a good strategy since it is well known that in general, Genetic
Algorithms GA(and population based algorithms in general) are very time
consuming and generate worse solutions than LSM
• Strength of hybrid methods comes from combining complementary search
strategy to take advantage of their respective strength. For instance,
- Intensify the search in a promissing region with the LSM
- Diversify the search through the selection operator, crossover operator
of the GA
Genetic Programming GP 
Genetic
GeneticProgramming
Algorithm (GA)
GP 
• Population based algorithm
At each generation, use the solutions in the current population P to
• generate
At eachoffspring
generation
three different
operators
are firstof
applied
to generate
a :
- solutions
to create
the population
the next
generation
set of new (offspring) solutions using the N solutions of the current
population
P:
Repeat
the following
process until N  offspring-solutions are generated:
Select randomly a generation strategy gs  GS :
selection operator: selecting from the current population parent-solutions
Select parent-solu
tions in P themselves
for applying the generation strategy
that reproduce
gs (reproduction)
GS :
crossover
operator: producing offspring-solutions from each
pair of parent-solutions
Generate offspring-solutions
to be included in the intermediary
population
mutation
operator:Pmodifying (improving) individual offspring-solution
fourthoperator
operator(culling
(culling
operatoris) applied
is applied
to determine
• A
A fourth
operator)
to determine
a newa new
population of
selecting
among
the solutions
of the
current to
population
ofsize
sizeNNbyby
selecting
solutions
in P P
according
population
and
some
strateg
y the offspring-solutions according to some strategy
Genetic
GeneticProgramming
Algorithm (GA)
GP 
• Population based algorithm
At each generation, use the solutions in the current population P to
• generate
At eachoffspring
generation
three different
operators
are firstofapplied
to generate
a :
- solutions
to create
the population
the next
generation
set of new (offspring) solutions using the N solutions of the current
population
P:
Repeat
the following
process until N  offspring-solutions are generated:
Select randomly a generation strategy gs  GS :
selection operator: selecting from the current population parent-solutions
Select parent-solu
tions in P themselves
for applying the generation strategy
that reproduce
gs (reproduction)
GS :
crossover
operator: producing offspring-solutions from each
pair of parent-solutions
Generate offspring-solutions
to be included in the intermediary
population
mutation
operator:Pmodifying (improving) individual offspring-solution
culling ope
rator) isisapplied
to to
determine
a new
• A fourth operator
operator((culling
operator)
applied
determine
a new
population of
in Psolutions
P according
population
ofsize
sizeNNby
byselecting
selectingsolutions
among the
of the to
current
some
strategand
y the offspring-solutions according to some strategy
population
Generation strategies GS
Transfer some of the best parent-solutions of P in P
Some iterations of a local search method on a parent-solution
Crossover on a pair of parent-solutions
Crossover on a pair of parent-solutions + mutation on offsring-solutions
Mutation on a parent-solution
Descent method
One point crossover
Tabu Search
Two points crossover
Simulated annealing
Uniform crossover
Large Multiple-Neighborhood Search
Ad hoc crossover
Adaptive Large neighborhood Seach
Path relinking
Genetic
GeneticProgramming
Algorithm (GA)
GP 
• Population based algorithm
At each generation, use the solutions in the current population P to
• generate
At eachoffspring
generation
three different
operators
are firstofapplied
to generate
a :
- solutions
to create
the population
the next
generation
set of new (offspring) solutions using the N solutions of the current
population
P:
Repeat
the following
process until N  offspring-solutions are generated:
Select randomly a generation strategy gs  GS :
selection operator: selecting from the current population parent-solutions
Select parent-solu
tions in P themselves
for applying the generation strategy
that reproduce
gs (reproduction)
GS :
crossover
operator: producing offspring-solutions from each
pair of parent-solutions
Generate offspring-solutions
to be included in the intermediary
population
mutation
operator:Pmodifying (improving) individual offspring-solution
culling ope
rator) isisapplied
to to
determine
a new
• A fourth operator
operator((culling
operator)
applied
determine
a new
population of
in Psolutions
P according
population
ofsize
sizeNNby
byselecting
selectingsolutions
among the
of the to
current
some
strategand
y the offspring-solutions according to some strategy
population
Parent - solution selection
Selection operators (like in GA):
Random selection operator
Proportional (or roulette whell) selection operator
Tournament selection operator
Diversity preserving selection operator
Genetic
GeneticProgramming
Algorithm (GA)
GP 
• Population based algorithm
At each generation, use the solutions in the current population P to
• generate
At eachoffspring
generation
three different
operators
are firstofapplied
to generate
a :
- solutions
to create
the population
the next
generation
set of new (offspring) solutions using the N solutions of the current
population
P:
Repeat
the following
process until N  offspring-solutions are generated:
Select randomly a generation strategy gs  GS :
selection operator: selecting from the current population parent-solutions
Select parent-solu
tions in P themselves
for applying the generation strategy
that reproduce
gs (reproduction)
GS :
crossover
operator: producing offspring-solutions from each
pair of parent-solutions
Generate offspring-solutions
to be included in the intermediary
population
mutation
operator:Pmodifying (improving) individual offspring-solution
culling ope
rator) isisapplied
to to
determine
a new
• A fourth operator
operator((culling
operator)
applied
determine
a new
population of
in Psolutions
P according
population
ofsize
sizeNNby
byselecting
selectingsolutions
among the
of the to
current
some
strategand
y the offspring-solutions according to some strategy
population
Generation strategies GS
Transfer some of the best parent-solutions of P in P
Some iterations of a local search method on a parent-solution
Crossover on a pair of parent-solutions
Crossover on a pair of parent-solutions + mutation on offsring-solutions
Mutation on a parent-solution
Descent method
Tabu Search
Genetic

One point crossover
Two) points crossover
Algorithm (GA
Simulated annealing
Uniform crossover
Large Multiple-Neighborhood Search
Ad hoc crossover
Adaptive Large neighborhood Seach
Path relinking
Adaptive Genetic Programming  AGP 
Adaptive Genetic
min(GA)
g  AGP
Genetic
Programming
GP 
GeneticProgram
Algorithm
• Population based algorithm
At each generation, use the solutions in the current population P to
• generate
At eachoffspring
generation
three different
operators
are firstofapplied
to generate
a P :
- solutions
to create
the population
the next
generation
set of new (offspring) solutions using the N solutions of the current
population
P:
Repeat
the following
process until N  offspring-solutions are generated:
Select ran
domly a generation strateg
strategy
randomly
y gs  GS :according to its score  gs :
selection operator: selecting from the current population parent-solutions
Select parent-solu
tions in P themselves
for
applying
the
Adaptive
layer
: generation strategy
that reproduce
gs (reproduction)
GS :
crossover
operator: producing offspring-solutions
 j from each
Probabilistic choices on
to specify
pair of parent-solutions
Generate offspring-solutions
to be included in the intermediary

 i
population
mutation
operator:Pmodifying (improving) individual offspring-solution
the intervals in the roulette wheel selection
culling ope
rator) isisapplied
to to
determine
a new
• A fourth operator
operator((culling
operator)
applied
determine
a new
population of
in Psolutions
P according
population
ofsize
sizeNNby
byselecting
selectingsolutions
among the
of the to
current
some
strategand
y the offspring-solutions according to some strategy
population
Adaptive Genetic
min(GA)
g  AGP
Genetic
Programming
GP 
GeneticProgram
Algorithm
• Population based algorithm
At each generation, use the solutions in the current population P to
• generate
At eachoffspring
generation
three different
operators
are firstofapplied
to generate
a P :
- solutions
to create
the population
the next
generation
set of new (offspring) solutions using the N solutions of the current
population
P:
Repeat
the following
process until N  offspring-solutions are generated:
Select ran
domly a generation strateg
strategy
randomly
y gs  GS :according to its score  gs :
selection operator: selecting from the current population parent-solutions
Select parent-solu
tions in P themselves
for applying the generation strategy
that reproduce
gs (reproduction)
GS :
Score:
crossover
operator: producing offspring-solutions
from each
pair of parent-solutions
Generate offspring-solutions
to be included
the intermediary
Pastinperformance
to contribute
population
mutation
operator:Pmodifying (improving) individual
during offspring-solution
the solution process
culling ope
rator) isisapplied
to to
determine
a new
• A fourth operator
operator((culling
operator)
applied
determine
a new
population of
in Psolutions
P according
population
ofsize
sizeNNby
byselecting
selectingsolutions
among the
of the to
current
some
strategand
y the offspring-solutions according to some strategy
population
Adaptive Genetic
ing
Genetic
Programming
GP 
 AGP
GeneticProgramm
Algorithm
(GA)
• Population based algorithm
At each generation, use the solutions in the current population P to
• generate
At eachoffspring
generation
three different
operators
are firstofapplied
to generate
a P :
- solutions
to create
the population
the next
generation
set of new (offspring) solutions using the N solutions of the current
population
P:
Repeat
the following
process until N  offspring-solutions are generated:
Score update:Select ran
domly a generation strateg
strategy
randomly
y gs  GS :according to its score  gs :
selection
operator: selecting from the current population parent-solutions
New best
solution
Select parent-solu
tions in P themselves
for applying the generation strategy
that reproduce
Better solution
gs than
(reproduction)
GSparent-solutions
:
crossover
operator: producing offspring-solutions from each
Not previously
visited solutions
pair of parent-solutions
Generate
offspring-solutions
to be included in the intermediary
Worsemutation
solution
than
paren
ions (improving) individual offspring-solution
modifying
population
Pt-solut
operator:
Update the score  gs
Sc
update praocess:
culling ope
rator) isisapplied
toore
determine
new
• A fourth operator
operator((culling
operator)
applied
to
determine
a new
population of
in P
P according
solutions
at
population
ofsize
sizeNNby
byselecting
selectingsolutions
among the
ofobtained
the to
current
i : the score
some
strategand
y the offspring-solutions according to some strategy
population
the current iteration
Then
 i :  i  1     i