MEMETIC ALGORITHMS

1
MEMETIC ALGORITHMS

• Pablo Moscato
• DENSIS FEEC - UNICAMP
• moscato_at_densis.fee.unicamp.br
• www.densis.fee.unicamp.br/moscato
• Memetic Algorithms Home Page
• www.densis.fee.unicamp.br/moscato/memetic_home.ht
  ml

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GAs and MAs

• For GAs population ? group of living
  organisms
• For MAs population ? group with knowledge of
  solution methods for the problem... (and the
  class !)
• Knowledge means... proper and rational reuse of
• Heuristics (constructive and iterative
  improvement)
• Approximation Algorithms - PTAS - FPTAS
• Exact methods (BnB - truncated BnB - Branch and
  Cut)
• Properties of the best or optimal solution(s)
• Transformations and reductions...
• Recombination has a more general character than
  GAs

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Basic Memetic Algorithm
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Cultural Evolution

• Shared characteristics are not inherited due to
  simple processes of recombination of previous
  solutions
• Examples
• Formula 1 (technological evolution)
• Martial Arts
• The use of totally random mutation has a
  significantly less important rôle.
• Heavy use of historical information and an
  external logic to speed-up the process.

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Memetic Algorithms for combinatorial optimization
problems

• Meme word introduced by Richard Dawkins in
  his best-seller book The Selfish Gene (76).
• unit of imitation, analogous rôle of gene but
  
• in the field of cultural evolution.
• Memetic Algorithms (MAs, 89) caracterization
  of evolutionary algorithms that can hardly fit
  the GAs methaphor - no, or small, relation with
  biology
• hybrid GAs ?
  MAs
• Scatter Search (Glover, 77) ? MAs

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Comparative Glossary

• In GAs we generally define
• Individual (genotype, cromossome, struture,
  string, etc.)
• chain of genes that codifies for a given
  configuration/solution of the problem at hand.
• coding (several ways, i.e. general rules, kind
  of art )
• In MAs prefer to use the word Agent
• we code not only for a confg./solution., we also
  include resolution methods for the problem (and
  problem class) we wish to solve.
• Use of multiple representations (cultural
  evolution).

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Multiple representations - Example

• Number Partitioning
• exemple partition of 15, 12, 10, 9, 4 in two
  sets
• 15, 10 and 12, 4, 9.
• Binary Direct Representation s (1, -1, 1, -1,
  -1).
• Ternary Direct s(TD) (1, ?, 1, ?, -1)
• with greedy decoding ( s (TD,g) -gt s).
• Binary direct ? Ternary direct.
• decoding known as growth function, since it
  maps from genotype s(TD) to phenotype s.

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Multiple representations (continuation)

• Genotype a configuration in search space.
• Phenotype ? feasible solution (not always the
  case...).
• Diferent decoders give birth do diferent
  representations (since the same genotype can give
  diferent phenotypes)
• Generally, decoding/growth function is a
  deterministic process (though randomization may
  be useful for sampling, specially in problems
  beyond NP).
• A phenotype can be the image of several
  genotypes.
• GAs are generally more related with a single type
  of decoding (MAs have implicit more freedom).

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Other types of multiple representations

• Sometimes the best representation is not the
  naive one for a given problem
• Exemple 1 binary matrix representation
• xij 1 if the integers ai and aj are in
  diferent partitions xij 0, otherwise.
• To discuss advantages and disadvantages...
• Exemple 2 tree representation
• Fact Every tree has a 2-coloring ? we can
• then represent two partitions...

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Tree representation for number partitioning

• In the previous example...
• phenotype partition in two groups
• 15, 10 and 12, 4, 9.
• Genotype is a tree

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Chain or path representation

• Exemple 3 let the phenotype be the partition
• 125, 365, 40, 120, 237, 43, 158, 101
• It can be represented by the genotype
• 125 -- 237 -- 365 -- 43 -- 40 -- 158 -- 120 --
  101
• cost 650 - 539 111
• Local search if we pick a subpath that starts
  and ends with
• a diferent colour (for instance 237 -- 365 -- 43
  -- 40)
• and we change switch the endpoints we get
• 125 -- 40 -- 43 -- 365 -- 237 -- 158 -- 120 --
  101
• cost 525 - 664 139

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Mutation and Representation

• For GAs we say
• mutation
• small random perturbation that modifies a
  chromosome
• For MAs
• totally random mutation is allowed...
• Sometimes is needed but is not very efficient.
  
• small... it depends on the representation.
• Mutation is important
• for some specific type of non-structured
  instances or when recombination is not working
  well.
• when diversity crisis occurs

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Invariant glossary of GAs and MAs

• Population
• a set of individuals (resp. agents) that
  represent the associated configurations in the
  search space.
• Generation
• arbitrary separation of different populations as
  time evolves
• Fitness function (we prefer guiding function)
• it can be a multiobjetive function, even in the
  case of a single objective function naturally
  associated with the problem.

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Invariant glossary (continuation)

• Reproductive operator (recombination/crossover)
• In MAs we classify
• crossover ? recombination
• crossover
• only between two genotypes
• low complexity - maximum O(n log n)
• recombination
• it can be blind ou myopic or have
• complete knowledge

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Representation - Good and desired features

• Complete representation and valid representation.
• Complete all feasible solution of the problem is
  represented by at least one configuration in
  configuration space.
• Valid at least one of the optimal solutions of
  the problem is represented by one configuration
  in the search space. Complete -gt Valid, the
  inverse is not true.
• Mutation small structural changes should have as
  a consequence small variations of the objective
  function and in the guiding function as well.
• Homework Study several natural mutations for the
  representations we discussed for the number
  partitioning problem. Worst-case vs. Average-case
  analysis ?

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Representation - Good problems ?

• Some problems apparently have some naturally good
  representations for GAs o MAs.
• Problems in the DLS class (discrete linear
  subset).
• Problems with low epistasis.
• DLS (Definition) Let N1, ..., n and let F be
  a set of subsets of N, F ? 2N with the property
  that no set in F is properly contained in another
  one. The problem DLS (F, w) is the combinatorial
  optimization problem with feasible set F and with
  cost function
• cost(f) ? wi f?F, ? i ? f
• where w (w1, ..., wn) is a weight vector
  (given) in Rn.

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Problems in DLS, are they all NP-Hard ?

• By the previous definition, the answer is No.
  Examples...
• The Min TSP is a DLS problem. Let n be the
  number of edges of a complete graph K_n, then F
  exactly contains the set of integers (i.e., the
  integers that represent edges) corresponding to
  tours in K_n. The vector of costs w is a 1-dim
  array representation of the distance matrix
  between cities.
• The Minimum Spanning Tree, is a polynomial-time
  problem that also belongs to DLS.
• Shortest-Path, Min-cost flow, Maximum weight set
  in a matroid, Weighted matroid intersection, and
  weighted 3-matroid intersection also belong to
  DLS.

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The traveling salesman problem (from Chvatals
page)

• 1800-1900 first descriptions of the problem
• 1920-1930 problem becomes well defined
• 1940-50 it starts to be recognized as hard
• 1954 an instance with 42 cities is solved to
  optimality.

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120 cities of West Germany (1977)
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532 cities from the United States (1987) att532
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Optimal solution for instance pcb3038
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Largest instance solved to optimality (1998)
usa13509
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Moscato and Tinetti ( 92, MA for the TSP)
solution pocket/fitness solution current/fitness

• PocketPropagation()
• Agents use three different heuristics (2-opt,
  one-city insertion, e two-city insertion ).
• Solutions get modified when they go up the
  hierarchy
• example agents 10 -gt 3 -gt 1...
• Transputers (constraint and inspiration)

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Strategic Edge Crossover (Norman Moscato, 92)

• based on Edge Recombination (Whitley, 91)
• generalized to the ATSP, called SAX
• (Buriol, França, Moscato, 99)
• selection of two tours A and B
• EdgeMap (auxiliary data structure)
• build strings with edges from A and B.
• Karps patching heuristic !!
• Faster patching methods can be used.
• Very good results for ATSP instances !!

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DPX para o TSP (Merz Freisleben, 96)

• Objetive Recombination with a good synergy with
  a powerful local search scheme (based on a
  version of Lin-Kernighan)
• Problem Lin-Kernighans local minima have many
  edges in common. Recombination operators that do
  not take this into account would suffer great
  diversity losses.
• Solution Forbid edges in (A-B) ? (B-A) !
• while including all edges in (A ? B)
• Property d(y,A) d(A,B) (suggested the name)
• Result Excellent ! (best 2-parent MA ever
  built !)

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MFNNRER (Holstein Moscato, 98)

• MFNNRER Multiple Fragment with Nearest
  Neighbour Repair Edge Recombination
• 1) Find all edges in SA ? B and include them in
  the child
• 2) Order edges in L(A ? B) - S (A-B) ? (B-A)
  in incresing length order (breaking ties at
  random).
• 3) Starting with the first edge in L, add it to
  the child
• if a) no city should have three edges
  incident to it
• b) no subtour should be formed
• 4) Run a Nearest Neighbour heuristic having as
  input the cities that only have 0 or 1 edges
  incident in the strings formed on step 3.

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Why so many recombination operators ?

• Actually, they all reflect an underlying, more
  general strategy
• 1) Find some kind of similarity between two
  solutions A and B, call the set SA ? B and
  insert it in the child.
• 2) Define two sets S(in) and S(out) of desired
  and non-desired features (one or both can be
  empty).
• 3) Trying to keep all elements of S in the new
  solution, try to find another feasible solution
  that maximizes (respectively, minimize) the
  elements from S(in) (respectively, from S(out)).

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Why so many recombination operators ?
(continuation)

• In the previous examples
• MFNNRER was used in an MA with a very simple
  local search scheme (2-opt).
• SAX is being used in an MA with a local search
  restricted to a subset of the 3-opt neighborhood.
  
• SAX e MFNNRER try to give priority to edges
  present in both parent solutions.
• Fact two 2-opt tours have on average 66 of
  the edges in common (intensification is
  needed).
• DPX explicitly forbids edges which are not
  common (diversification is needed due to LK
  neighborhood used).

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New combinatorial optimization problems !!

• Recombination can be regarded as a heuristic way
  of solving new types of combinatorial
  optimization problems !!!
• THIRD (BETTER-THAN-WORST) HAMILTONIAN CYCLE
• Instance Graph G(V,E,W), and two hamiltonian
  cycles C1 and C2 of G, such that C1?C2, and
  w.l.o.g. Long(C1) ? Long(C2).
• Question ? another C, Hamiltonian cycle of
  G,
• such that C? C1, C2 and Long(C) lt Long(C2)
  ?
• Unknown computational complexity !!!
  (NP-complete ?)
• This means.... Less chances of being fired !
• (not only MAs help us to address NP-hard
  problems...
• ...they help us to create new ones
  !!)

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A chance to revisit old friends...

• Recombination would also help us to revisit some
  old combinatorial optimization problems.
• RESTRICTED HAMILTONIAN CYCLE (RHC)
• Instance Graph G(V,E) and a Hamiltonian path P
  of G.
• Question ? a Hamiltonian cycle in G ?
• Computational complexity known !!!
• NP-complete... (see Papadimitriou Steiglitz,
  Combinatorial Optimization, Chapter 19, pp.
  477-480).
• This means.... May be there is a way of reducing
  RHC to our previous problem and prove it
  NP-Complete. (???)

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Recombination Procedure (SAX)
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Local Search Algorithm Recursive Arc Insertion
(RAI)(Buriol, França, and Moscato)

• Recursive application of 3-changes
• Key idea divide the tour in a subtour and a path
  string and insure feasibility back as soon as a
  decreasing tour can be constructed.

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Recursive Arc Insertion - Step-by-Step
i
i
i
j
j
i
i
i
j
j
j
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Computational Results with RAI and four
recombinations
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Computational results with other methods for the
ATSP
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Structured vs. non-sturctured populations

• 30 runs on each of the 27 instances of the ATSP
  available from the TSPLIB which were solved to
  optimality.

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MAs with Multiple Representations

• Example let us suppose we have a phenotype
• 125, 365, 40, 120, 237, 43, 158, 101
• it can be represented by the genotype
• 125 -- 237 -- 365 -- 43 -- 40 -- 158 -- 120 --
  101
• cost 650 - 539 111
• we can get the subpath 237 -- 365 -- 43 -- 40
• and make a 2-opt move we get
• 125 -- 40 -- 43 -- 365 -- 237 -- 158 -- 120 --
  101
• cost 525 - 664 139
• ... Lesson we can use 2-opt-like algorithms for
  this problem...

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Future issues - gt MemePool as a generic framework

• MAs with multiple representations
• Frameworks for MAs to exploit code reuse
• MemePool Project
• Polynomial Merger Algorithms
• Problems outside NP
• Linear Programming ?
• Maximum Cardinality Matching ?
• Problems in PSPACE ?
• GAs vs MAs
• Complete Memetic Algorithms

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Representação e Ótimos Locais

• Analise o problema e a representação
• max f(x) (x-14)2,
• s.a. x ? 0,31 e inteiro
• representação ? binaria, cada inteiro como um
  cromossomo de 5 bits. O vetor (0,0,0,0,0)
  representa 0 e (1,1,1,1,1) -gt representa 31.
• população inicial ? aleatória
• tamanho da população e tipo ? 5, fixa, sem
  estrutura.
• função de fitness ? f(x)

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Exemplo

• População inicial gerada
• x f(x)(x-14)2
• A1 ? 1 0 0 1 1 19 25
• A2 ? 0 0 1 0 1 5 81
• A3 ? 1 1 0 1 0 26 144
• A4 ? 1 0 1 0 1 21 49
• A5 ? 0 1 1 1 0 14 0
• Escolha dos pais com maior fitness ? A3 , A2
• geração de 1 filho, que substituirá o indivíduo
  com menor fitness A5

Pai 1
Pai 2
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Exemplo

• x f(x)(x-14)2
• Pai1 A2 ? 0 0 1 0 1 5 81
• Pai2 A3 ? 1 1 0 1 0 26 144
• Filho ? 0 0 0 1 0 2 144
• mutação ? 0 0 0 1 0 2 144
• x f(x)
• A1 ? 1 0 0 1 1 19 25
• A2 ? 0 0 1 0 1 5 81
• A3 ? 1 1 0 1 0 26 144
• A4 ? 1 0 1 0 1 21 49
• A5 ? 0 0 0 1 0 2 144

Pai 2
Pai 1
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Exemplo

• x f(x)
• Pai1 A5 ? 0 0 0 1 0 2 144
• Pai2 A3 ? 1 1 0 1 0 26 144
• Filho ? 0 0 0 1 0 2 144
• mutação ? 0 0 0 0 0 0 196
• x f(x)
• A1 ? 0 0 0 0 0 0 196
• A2 ? 0 0 1 0 1 5 81
• A3 ? 1 1 0 1 0 26 144
• A4 ? 1 0 1 0 1 21 49
• A5 ? 0 0 0 1 0 2 144

Pai 1
Pai 2
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Exemplo

• x f(x)
• Pai1 A1 ? 0 0 0 0 0 0 196
• Pai2 A5 ? 0 0 0 1 0 2 144
• Filho ? 0 0 0 0 0
• mutação ? 0 0 0 0 1 1 169
• x f(x)
• A1 ? 0 0 0 0 0 0 196
• A2 ? 0 0 1 0 1 5 81
• A3 ? 1 1 0 1 0 26 144
• A4 ? 0 0 0 0 1 1 169
• A5 ? 0 0 0 1 0 2 144
• e assim continua...

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Representação binaria e código de Gray

• x f(x)(x-7)2 binaria Gray
• 0 49 0000 0000
• 1 36 0001 0001
• 2 25 0010 0011
• 3 16 0011 0010
• 4 9 0100 0110
• 5 4 0101 0111
• 6 1 0110 0101
• 7 0 0111 0100
• 8 1 1000 1100
• 9 4 1001 1101
• 10 9 1010 1111
• 11 16 1011 1110
• 12 25 1100 1010
• 13 36 1101 1011
• 14 49 1110 1001
• 15 64 1111 1000

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Representação, código de Gray e fitness
landscape

• f(x)(x-7)2 delta_f binaria diff
• 49 1564-49 0000 4
• 36 1349-36 0001 1
• 25 1125-36 0010 2
• 16 9 0011 1
• 9 7 0100 3
• 4 5 0101 1
• 1 3 0110 2
• 0 1 0111 1
• 1 1 1000 4
• 4 3 1001 1
• 9 5 1010 2
• 16 7 1011 1
• 25 9 1100 3
• 36 11 1101 1
• 49 13 1110 2
• 64 15 1111 1

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Representação, código de Gray e fitness
landscape

• delta_f diff
• 15 4 60 suma total
• 13 1 13 representação binaria
• 11 2 22
• 9 1 9 601322... 240
• 7 3 21
• 5 1 5 suma total
• 3 2 6 representação de Gray
• 1 1 1
• 1 4 4 151311... 128
• 3 1 3
• 5 2 10
• 7 1 7 240/128 1.875
• 9 3 27 (métrica de comparação ?)
• 11 1 11
• 13 2 26
• 15 1 15

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Representação - Hipercubo binario
7
15
3
6
14
11
2
4
5
13
12
0
1
8
9
10
48
Representação - Código de Gray
5
10
2
4
11
13
3
7
6
9
8
0
1
15
14
12
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Representação, código de Gray e fitness
landscape

• Problema cómo avaliar uma métrica dada ?
• 1) Análise estatística
• 2) Análise asintótico... (?)
• 3) Análise do pior caso... pode ser feito, mas é
  pouco relevante para entender a dinámica do
  algoritmo evolutivo.
• Existe alguma métrica geralmente aceita para
  poder analisar diferentes representações ?
• Não, mais geralmente se traduz em principios,
  e.g.
• building block hypotheses, baixa epistasis,
  etc.
• Ideia básica configurações vizinhas (para a
  mutação) tem que ter valores semelhantes da
  função de fitness (correlação local).

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Análise estatística e representaçãoes boas

• Moscato (89) On evolution, search..., discute
  a correlação de ótimos locais como essencial
  para os MAs. Dependencia não só do problema mais
  da instancia !
• Radcliffe The Algebra of Genetic Algorithms,
  (preprint, 92) discute uma formalização dos GAs
  em termos de formae. Inspiração em Algebra
  Linear.
• Hoffman (Tese, TU Münich, 93), usa formae
  theory para analizar varios operadores de
  recombinação para o problema do caixeiro
  viajante.
• Moscato (93) paper em Annals of Operations
  Research, vol. 41.
• a) representações com ótimos locales são as
  veces naturais para alguns problemas. Se não,
  o problema é cómo desenhar elas para MAs
  eficentes.

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Análise estatística e representaçãoes boas

• Moscato (93) paper em Annals of Operations
  Research, vol. 41.
• b) fitness distance analisis para o problema
  do perceptron binario.
• c) Tabu Search Muito bom para diversificação
  (sempre que não seja excessiva) da população.
• Radcliffe Surry (94) dois trabalhos Formal
  Memetic Algorithms, e Fitness variance of
  formae and performance prediction.
• a) MAs são muito melhores que GAs para alguns
  problemas (e.g. TSP)
• b) Como Hoffmann, análise estatística e
  possivel com ajuda da simulação computacional.

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População inicial - GAs e MAs

• Aleatória ou com uma heurística de construção
• válido para GAs e MAs
• perturbação para gerar diferentes indivíduos.
• perturbação anterior nos dados de entrada da
  heurística -gt construção de diferentes soluções.
• nos MAs
• Efeitos transitórios...
• Causa ótimos locais dos processos de busca podem
  ser bons, mas muito distantes entre eles.

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Estrutura da População

• Idéia básica
• Aproveitar o isolamento por distância entre
  indivíduos para diminuir o problema de perda
  de diversidade.
• Conceito importante
• deriva genética (random genetic drift)
• migração
• Tarefa veja o que acontece se num GA que você
  tenha já implementado para um problema X a
  função de fitness é constante para todo
  fenotipo...

54
Politicas de reprodução em MAs

• Escolha dos pais totalmente aleatória.
• Aleatória mais relacionada com o valor de fitness
• Otimalidade de Pareto.
• Crise de diversidade modifica a seleção.
• quando há estrutura, migração pode ser uma boa
  alternativa (ver por exemplo os artigo de Martina
  Gorges-Schleuter, Mülhenbein, Markus Schwem).

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Exemplo de uma estrutura de população
solução pocket/fitness solução current/fitness

• 13 agentes ? 2 cromossomos ? Pocket e Current.
• para todo agente, pocket é sempre melhor que
  current.
• agentes 2, 3, e 4 são a sua vez lideres e
  subordinados.
• se pocket subordinado melhor que pocket do líder
  então troca de solução
• pocket de 1 terá a melhor solução

56
Moscato - Tinetti (92, MA para o TSP)
solução pocket/fitness solução current/fitness

• PocketPropagation(), e um processo de migração.
• Agentes usavam tres diferentes heurísticas
  (2-opt, one-city insertion, e two-city insertion
  ).
• Soluções modificavam-se ao subir na hierarquia
• exemplo agentes 10 -gt 3 -gt 1...
• Transputers (necessidade e inspiração)

57
Recombinação com personalidade ou
behaviourBerretta Moscato, in New Ideas in
Optimization, Corne, Glover, and Dorigo (eds.),
McGraw-Hill, 99

• R e um operador de recombinação parameterizado
• conciliador R(A 0 0 1 0 1 , B 1 0 1 1 0 )
  -gt x 0 1 x x -gt yc
• rebelde R(A 0 0 1 0 1, B 1 0 1 1 0 ) -gt
  1 x x 1 0 -gt yr
• bajulador R(A 0 0 1 0 1, B 1 0 1 1 0 ) -gt
  0 x x 0 1 -gt yb
• Propiedade d(yc,A) ? d(A,B) para tudo yc
• Notar que para yr 1 1 0 1 0 d(yr,A) 5,
  então em geral o comportamento rebelde e
  bajulador não satisfazem a propriedade acima.
• Em geral GAs só tem tipo conciliador -gt perda de
  diversidade

58
Recombinação com behaviours para o TSP ?

• R e um operador de recombinação parameterizado
• conciliador input (A, B) -gt arcos em A ? B
  -gt yc
• rebelde input (A,B) -gt arcos em B-A -gt
  yr
• bajulador input (A,B) -gt arcos em A-B -gt
  yb
• Propriedade d(yc,A) ? d(A,B) para tudo yc
  ainda é válida ?
• Caso 1 d(A,B) n - A ? B
• onde n é o número de
  cidades.
• Caso 2 d(A,B) n - de corners comuns
  

59
DPX para o TSP (Merz Freisleben, 96)

• Objetivo Recombinação que colabore com uma busca
  local poderosa (baseada no algoritmo de
  Lin-Kernighan)
• Problema Mínimos locais do algoritmo de
  Lin-Kernighan têm muitas arestas semelhantes.
  Recombinações que não tenham em conta esse fato
  dão MAs com grandes perdas de diversidade
  (performance ruim).
• Solução Proibir arestas em (A-B) ? (B-A) !
• Incluir todas as arestas em (A ? B)
• Propriedade d(y,A) d(A,B)
• Resultado Excelente! (com estruturas de dados
  adicionais)

60
Strategic Edge Crossover

• Introduzido por Norman Moscato, 92
• baseado no Edge Recombination (Whitley, 91)
• Composto por quatro fases
• seleção de dois tours A e B para recombinação
• construção do EdgeMap
• geração das strings provenientes de arcos de A e
  B
• união destas para obter um novo tour factível.
• Karps patching heuristic !!

61
SAX, o seu ArcMap (Buriol, França, Moscato)
62
CreateString e PatchString

• Strings provenientes de arcos de A e B geradas
  pelo
• procedimento chamado CreateString.
• Algoritmo PatchString factibiliza a rota.

63
MAs for Single Machine Scheduling (with set-up
times and due dates) Mendes, França, Moscato

• O Problema é definido como
• Input n tarefas tem que ser processadas em uma
  máquina.
• p p1, ..., pn ( vetor de tempos de
  processamento
• para cada tarefa).
• d d1, ..., dn (vetor de datas de entrega
  ).
• Sij, n x n, matriz de inteiros, where Sij is
  the tempo de setup que a maquina precisa para
  começa a processar a tarefa j depois de ter
  completado a tarefa i.
• Task Achar a permutação do conjunto de tarefas
  que minimize a seguinte função objetivo
• onde ck e o chamado completion time para a
  tarefa k.

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Algoritmo Memético baseado em busca local

65
Algoritmo Memético para o SMS - Detalhes de
implementação

• Representação -gt vetor de inteiros 1,...,n
  (permutacões)
• Recombinação -gt OX (Order Crossover)
• Pai A 1 3 2 7 8 6 9 4 5
• Pai B 1 7 8 4 5 6 2 3 9
• Filho inicial 7 8 6 9
  (Pai A)
• 1 7 8 6 9 (Pai B)
• 1 4 7 8 6 9 (Pai B)
• 1 4 5 7 8 6 9 (Pai B)
• 1 4 5 7 8 6 9 2 (Pai B)
• Filho final 1 4 5 7 8 6 9 2 3 (Pai B)

66
Algoritmo Memético para SMS - Detalhes de
implementação

• Mutação -gt Troca de posição de tarefas
• Exemplo
• 1 3 2 7 8 6 9 4 5 ? 1 3 6 7 8 2
  9 4 5
• Original Mutado
• Função de Fitness
• onde Ti e o atraso total para a solução i na
  população.