POPULATION BASED METAHEURISTICS

1
POPULATION BASEDMETAHEURISTICS

• Jacques A. Ferland
• Department of Informatique and Recherche
  Opérationnelle
• Université de Montréal
• ferland_at_iro.umontreal.ca
• http//www.iro.umontreal.ca/ferland/
• Spring School in Optimization
• Ho Chi Minh
  
  March 2007

2
Population Based Techniques (PBT)

• A Population Based Technique (PBT) can be seen as
  an iterative process starting with an initial
  population P0 including N feasible solutions. x
  is the best solution of the population P0 .
• At each iteration ( generation)
• - During a collective process, the solutions
  of the current population are compared and
  combined to generate new offspring-solutions
  inheriting the prevaling characteristics of the
  parent-solutions.
• - Then the new offspring-solutions evolve
  individually according to an individual process.
• - The best solution x is updated.
• - Finally a new population of size N is
  generated.
• The procedure continues until some stopping
  criterion is satisfied.

3
POPULATION BASED

• Genetic Algorithm
• Ant Colony
• Particle Swarm
• http//www.iro.umontreal.ca/ferland/

4
Initial Population
5
Problem used to illustrate

• General problem
• min f(x)
• 
  x ? X
• Assignment type problem Assignment of resources
  j to activities i
• min f(x)
• 
  Subject to ?1 j m xij 1 1 i n
• 
  xij 0 or 1 1 i n,
  1 j m

6
Problem Formulation

• Assignment type problem
• min f(x)
• Subject to ?1 j m xij 1 1
  i n
• xij 0 or 1
  1 i n, 1 j m
• Graph coloring problem Graph G (V,E).
• V i 1 i n E (i, l) (i,
  l) edge of G.
• Set of colors j 1 j m
• min ? 1 j m ? (i, l) ? E xij
  xlj
• Subject to ?1 j m xij 1 1
  i n
• xij 0 or 1 1 i n,
  1 j m

7
Heuristic Constructive Techniques

• Values of the variables are determined
  sequentially
• at each iteration, a variable is
  selected,
• and its value is determined

8
Heuristic Constructive Techniques

• Values of the variables are determined
  sequentially
• at each iteration, a variable is
  selected,
• and its value is determined
• The value of each variable is never modifided
  once it is determined

9
Heuristic Constructive Techniques

• Values of the variables are determined
  sequentially
• at each iteration, a variable is
  selected,
• and its value is determined
• The value of each variable is never modifided
  once it is determined

10
Greedy method

• Next variable to be fixed and its value are
  selected to optimize the objective function given
  the values of the variables already fixed

11
Greedy method

• Next variable to be fixed and its value are
  selected to optimize the objective function given
  the values of the variables already fixed

• Graph coloring problem
• Vertices are ordered in
• decreasing order of their degree
• Vertices selected in that order
• For each vertex, select a color in order to
  reduce the number of pairs of adjacent vertices
  already colored with the same color

12
GRASPGreedy Randomized Adaptive Search Procedure

• Next variable to be fixed is selected randomly
  among those inducing the smallest increase.

13
GRASPGreedy Randomized Adaptive Search Procedure

• Next variable to be fixed is selected randomly
  among those inducing the smallest increase.
• Referring to the general problem,
• i) let J j
  xj is not fixed yet
• and dj be the increase induces by the
  best value that xj can take ( j ? J )

14
GRASPGreedy Randomized Adaptive Search Procedure

• Next variable to be fixed is selected randomly
  among those inducing the smallest increase.
• Referring to the general problem,
• i) let J j
  xj is not fixed yet
• and dj be the increase induces by the
  best value that xj can take ( j ? J )
• ii) Denote d min
  j ? J dj
• and
  a ? 0, 1 .

15
GRASPGreedy Randomized Adaptive Search Procedure

• Next variable to be fixed is selected randomly
  among those inducing the smallest increase.
• Referring to the general problem,
• i) let J j
  xj is not fixed yet
• and dj be the increase induces by the
  best value that xj can take ( j ? J )
• ii) Denote d min
  j ? J dj
• and
  a ? 0, 1 .
• iii) Select randomly j ? j ? J
  dj ( 1 / a ) d
• and fix the value of xj

16
Genetic Algorithm
17
Encoding the solution

• The phenotype form of the solution x ? Rn is
  encoded (represented) as a genotype form vector z
  ? Rm (or chromozome) where m may be different
  from n.
• For example in the assignment type problem
• let x be the following solution for each
  1 i n,
• 
  xij(i) 1
• 
  xij 0 for all other j
• x ? Rnxm can be encoded as z ? Rn where
• zi j(i)
  i 1, 2, , n
• i.e., zi is the index of the resource
  j(i) assigned to activity i

18
Genetic Algorithm (GA)

• An initial population of size N is generated
• At each iteration (generation) three different
  operators are first applied to generate a set of
  new (offspring) solutions using the N solutions
  of the current population
• 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

19
Two variants of GA

• At each iteration of the Classical genetic
  algorithm
• - N 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 mutation operator is applied
• according to some probability to
• each offspring-solution.
• - The population of the next iteration
• includes the offspring-solutions

• At each iteration of the Steady-state population
  genetic algorithm
• - An even number of parent-solutions
• are selected and paired two by two
• - 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.
• -The population of the next iteration
• includes the N best solutions among the
• current population and the offspring-
• solutions

20
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

21
Random selection operator

• Select randomly each parent-solution from the
  current (entire) population
• Properties
• Very straightforward
• Promotes diversity of the population
  generated

22
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 a in the interval
• 
  0, ?1k N ( 1 / f( zk) )
• iii) Let t be the smallest index such that
• 
  ?1k t (1 / f( zk ) ) a
• iv) Then zt is selected
• 1 / f( z1 ) 1 / f( z2 )
  1 / f( z3)
  
  1 / f( zN)
• 
  
• t
• a
• The chance of selecting zk increases with its
  fittness 1 / f( zk)

For the problem Min f (x) where x is
encoded as z 1/f (z) measures the fittness
of the solution z
23
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 ? P f(z)
• iii) Then z is selected as a
  parent-solution

24
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.

25
Diversity preserving selection operator
26
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 offspring-solutions according to
  different crossover (recombination) operators.

27
One point crossover

• The one point crossover generates two
  offspring-solutions from the two parent-solutions
• z1 z11, z21, ,
  zm1
• z2 z12, z22, ,
  zm2
• as follows
• i) Select randomly a position (index) ?,
  0 ? m.
• ii) Then the offspring-solutions are
  specified as follows
• oz1 z11, z21, , z?1,
  z?12, , zm2
• oz2 z12, z22, , z?2,
  z?11, , zm1
• The first ? 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)

28
Two points crossover

• The two points crossover generates two
  offspring-solutions from the two parent-solutions
• z1 z11, z21, ,
  zm1
• z2 z12, z22, ,
  zm2
• as follows
• i) Select randomly two positions
  (indices) µ,?, 1 µ ? m.
• ii) Then the offspring-solutions are
  specified as follows
• oz1 z11, , zµ-11, zµ2, ,
  z?2, z?11, , zm1
• oz2 z12, , zµ-12, zµ1, ,
  z?1, z?12, , zm2
• The offspring oz1 (offspring oz2) has
  components µ, µ1, , ? of parent 2 (parent 1),
  and the rest of the components are the
  corresponding ones of parent 1 (parent 2)

29
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, , zm1 ,
  z2 z12, z22, , zm2
• 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,, zm-11, zm1
• parent 2 z12, z22, z32,
  z42,, zm-12, zm2
• Vector of bits 0 , 1 , 1 , 0 , ,
  1 , 0
• Offspring oz1 z11, z22, z32, z41,,
  zm-12, zm1
• Offspring oz2 z12, z21, z31, z42,,
  zm-11, zm2

The ith component of oz1 (oz2) is the ith
component of 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)

30
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.

31
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.

32
Mutation operator

• Mutation operator is an individual process to
  modify offspring-solutions
• In traditional implementations of Genetic
  Algorithm the mutation operator is used to modify
  arbitrarely each componenet zi with a small
  probability
• For i 1 to m
• Generate a
  random number ß ? 0, 1
• If ß lt ßmax
  then select randomly a new value for zi
• where ßmax 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