1. What is an Evolutionary Algorithm?
Chapter 2

2. Contents Examples : eight queens / knapsack Typical behaviours of EAs
Recap of Evolutionary Metaphor
Basic scheme of an EA
Basic Components:
Representation / Evaluation / Population / Parent Selection / Recombination / Mutation / Survivor Selection / Termination
Examples : eight queens / knapsack
Typical behaviours of EAs
EC in context of global optimisation

3. The Main Evolutionary Computing Metaphor
Environment
Individual
Fitness
PROBLEM SOLVING
Problem
Candidate Solution
Quality
The fundamental methaphor of Evolutionary Computing relates the powerful natural evolution to a particularly style of problem solving – that of trial-and-error or a.k.a. Generate and test. Learning doesn't happen from failure itself but rather from analyzing the failure, making a change, and then trying again.
In the macroscopic view of evolution, natural selection plays a central role. Given an environment that can host a only a limited number of individuals, and the basic instincts of individuals to reproduce, selection becomes inevitable if the population size is not to grow exponentially. Natural selection favours those individuals that are adapted or fit to the environmental condition best (survival of the fittest).
Fitness  chances for survival and reproduction
Quality  chance for seeding new solutions 3

4. Recap of EC metaphor
A population of individuals exists in an environment with limited resources
Competition for those resources causes selection of those fitter individuals that are better adapted to the environment
These individuals act as seeds for the generation of new individuals through recombination and mutation
The new individuals have their fitness evaluated and compete (possibly also with parents) for survival.
Over time Natural selection causes a rise in the fitness of the population

5. Recap 2:
EAs fall into the category of “generate and test” algorithms. Learning doesn't happen from failure itself but rather from analyzing the failure, making a change, and then trying again.
They are stochastic, population-based algorithms
Variation operators (recombination and mutation) create the necessary diversity and thereby facilitate novelty
Selection reduces diversity and acts as a force pushing quality

6. General Scheme of EAs

7. The generic structure of EAs
The algorithms usually start by creating an initial population of one or more individuals. These first individuals are created randomly or by another process. These individuals are then evaluated. If the individuals satisfy the stop conditions (they are good enough, the allocated time has elapsed, etc.) the results are returned.
If the stop condition has not been met the individuals are modified (all of them or only a part of them) using different operators and a new population is generated. The new population is again evaluated and the algorithm continues.
Some of the evolutionary algorithms use all the operators some of them use only a part. This is where usually the difference between algorithms appears.

8. Pseudo-code for typical EA
1 – Select parents – stochastic
5 – Survival selection – deterministic

9. What are the different types of EAs (1)
By Christoph Eick – COSC 6367 evolutionary programming

10. What are the different types of EAs (2)
Historically different flavours of EAs have been associated with different representations
Binary strings : Genetic Algorithms
Real-valued vectors : Evolution Strategies
Finite state Machines: Evolutionary Programming
LISP trees: Genetic Programming
These differences are largely irrelevant, best strategy
choose representation to suit problem
choose variation operators to suit representation
Selection operators only use fitness and so are independent of representation

11. Representations
Candidate solutions (individuals) exist in phenotype space
They are encoded in chromosomes, which exist in genotype space
Encoding : phenotype=> genotype (not necessarily one to one)
Decoding : genotype=> phenotype (must be one to one)
Chromosomes contain genes, which are in (usually fixed) positions called loci (sing. locus) and have a value (allele)
In order to find the global optimum, every feasible solution must be represented in genotype space

12. Evaluation (Fitness) Function
Represents the requirements that the population should adapt to
a.k.a. quality function or objective function
Assigns a single real-valued fitness to each phenotype which forms the basis for selection
So the more discrimination (different values) the better
Typically we talk about fitness being maximised
Some problems may be best posed as minimisation problems, but conversion is trivial

13. Population Holds (representations of) possible solutions
Usually has a fixed size and is a multiset of genotypes
Some sophisticated EAs also assert a spatial structure on the population e.g., a grid.
Selection operators usually take whole population into account i.e., reproductive probabilities are relative to current generation
Diversity of a population refers to the number of different fitnesses / phenotypes / genotypes present (note not the same thing)

14. Parent Selection Mechanism
Assigns variable probabilities of individuals acting as parents depending on their fitnesses
Selection acts as a force pushing quality
Usually probabilistic
high quality solutions more likely to become parents than low quality
but not guaranteed
even worst in current population usually has non-zero probability of becoming a parent
This stochastic nature can aid escape from local optima

15. Variation Operators Role is to generate new candidate solutions
Work on the individual level and creates diversity and therefore facilitate novelty
Usually divided into two types according to their arity (number of inputs):
Arity 1 : mutation operators
Arity >1 : Recombination operators
Arity = 2 typically called crossover
There has been much debate about relative importance of recombination and mutation
Nowadays most EAs use both
Choice of particular variation operators is representation dependant

16. Mutation Acts on one genotype and delivers another
Element of randomness is essential and differentiates it from other unary heuristic operators
Importance ascribed depends on representation and dialect:
Binary GAs – background operator responsible for preserving and introducing diversity
EP for FSM’s/ continuous variables – only search operator
GP – hardly used
May guarantee connectedness of search space and hence convergence proofs

17. Recombination Merges information from parents into offspring
Choice of what information to merge is stochastic
Most offspring may be worse, or the same as the parents
Hope is that some are better by combining elements of genotypes that lead to good traits
Principle has been used for millennia by breeders of plants and livestock

18. Survivor Selection a.k.a. replacement
Most EAs use fixed population size so need a way of going from (parents + offspring) to next generation
Often deterministic
Fitness based : e.g., rank parents+offspring and take best
Age based: make as many offspring as parents and delete all parents
Sometimes do combination (elitism)

19. Initialisation / Termination
Initialisation usually done at random,
Need to ensure even spread and mixture of possible allele values
Can include existing solutions, or use problem-specific heuristics, to “seed” the population
Termination condition checked every generation
Reaching some (known/hoped for) fitness (the discovery of an optimal or near optimal solution)
Reaching some maximum allowed number of generations
Reaching some minimum level of diversity
Reaching some specified number of generations without fitness improvement (the EC detects the problem has no feasible solution)
Convergence on a single solution or set of similar solutions
After a user-specified threshold has been reached

20. Initialisation using domain-specific knowledge

21. Example: the 8 queens problem
For a certain application, explain how could be the Genotype and Phenotype?
Place 8 queens on an 8x8 chessboard in
such a way that they cannot check each other

22. The 8 queens problem: representation1 2 3 4 5 6 7 8
Genotype:
a permutation of
the numbers 1 - 8
Phenotype:
a board configuration
The Phenotype space is the set
of all such configurations.
Obvious mapping

23. EA FOR 8-QUEENS PROBLEM Solution Representation
A chromosome is a permutation of the number 1,…,8 and a given g = < i1,…, i8> denotes the board configuration where the k-th column contains exactly one queen placed on the ik th row.
Example: The permutation g = < 1,2,3,4,5,6,7,8> represents a board where the queens are placed along the main diagonal.
The solution space is now the set of all permutations of 1,…,8.

24. Solution Representation (cont.)
By using such chromosome, we restrict the search to board configurations where horizontal constraint violation (two queens on the same column) and vertical constraint violation (two queens on the same row) do not occur.
The representation guarantees “half” number of the constraints and what remains to be minimized is the number of diagonal constraint violations.

25. 8 Queens Problem: Fitness evaluation
Penalty of one queen:
the number of queens she can check.
Penalty of a configuration:
the sum of the penalties of all queens.
Note: penalty is to be minimized
Fitness of a configuration:
inverse penalty to be maximized

26. The 8 queens problem: Mutation
Small variation in one permutation, e.g.:
swapping values of two randomly chosen positions,
Alegerea genei / genelor asupra carora fac mutatia poate fi aleatoare dar poate fi stabilita si prin selectie determinista – care regine dau cele mai multe conflicte, intrucat celelalte poate ar fi bine sa nu le mut intrucat au putine conflicte. 1 2 3 4 5 6 7 8

27. The 8 queens problem: Recombination
Combining two permutations into two new permutations:
choose random crossover point
copy first parts into children
create second part by inserting values from other parent:
in the order they appear there
beginning after crossover point
skipping values already in child 8 7 6 4 2 5 3 1

28. The 8 queens problem: Selection
Parent selection:
Pick 5 parents and take best two to undergo crossover
Survivor selection (replacement)
When inserting a new child into the population, choose an existing member to replace by:
sorting the whole population by decreasing fitness
enumerating this list from high to low
replacing the first with a fitness lower than the given child
Usually survivor selection: (replace worst) after merging the population and offsprings, then ranks them according to fitness and deletes the worst two.

29. 8 Queens Problem: summary
Note that is is only one possible
set of choices of operators and parameters

30. Example: 0-1-KNAPSACK problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.
It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most useful items.

31. 0-1 Knapsack problem with Greedy Algorithm
05/25/11
0-1 Knapsack problem with Greedy Algorithm
A greedy algorithm obtains an optimal solution to a problem by making a sequence of choices. At each decision point, the algorithm chooses the locally optimal solution. In other words, when we are considering which choice to make, we make the choice that looks best in the current situation, without considering results from subproblems.
This is called the 0-1 knapsack problem because each item must be taken in its entirely. If we use a greedy strategy to solve this problem, we would take the objects in order of their benefit/weight value.

32. The Greedy method does not work for the 0-1 Knapsack Problem.
05/25/11
The Greedy method does not work for the 0-1 Knapsack Problem.

33. Knapsack Problem: summary

34. Knapsack Problem: Implementation question
From the viewpoint of programming language, how can you create/represent individuals and the population to manipulate them?
i.e. an individual could be a bit, byte, int, char, ..., and the population is a vector, matrix, structure, etc.? Justify your answer.

35. Knapsack Problem: individual / population
public void initPopulation() { population = new string[Configuration.population_size]; for (int i = 0; i < Configuration.population_size; i++) { string currentGenome = getRandomBitStream(); population[i] = currentGenome; } } public string getRandomBitStream() { string genome = ""; for (int i = 0; i < itemsNr; i++) { string randomGene = ""; randomGene = random_gene.Next(2).ToString(); genome += randomGene; } return genome; }

36. Typical behaviour of an EA
Phases in optimising on a 1-dimensional fitness landscape
Early phase:
quasi-random population distribution
Mid-phase:
population arranged around/on hills
Late phase:
population concentrated on high hills

37. Example with two traits

38. Typical run: progression of fitness
Best fitness in population
Time (number of generations)
Typical run of an EA shows so-called “anytime behavior”

39. Hypervolume comparison in MOEA problems

40. Hypervolume definition
Hypervolume by Coello et al. (2002), a.k.a. hyperarea, computes the volume enclosed by the current
Pareto front approximation and the axes, for a maximization problem. If it is computed at each generation, the evolution of this volume shows if the algorithm makes progress. Also if multiple algorithms are compared it can
show the quality of the results of one algorithm over the other. For a minimization problem a point has to be
established, called hypervolume reference point, which will replace the origin in the hypervolume computation. The
hypervolume reference point (HV) is set to the coordinates of the maximum values of the objectives (see Fig). If
multiple algorithms are compared the maximum values area enclosed between the axes and the HV remains fixed.
have to be searched from all the dierent runs so that the

41. Are long runs beneficial?
Best fitness in population
Time (number of generations)
Progress in 1st half
Progress in 2nd half
Answer:
- it depends how much you want the last bit of progress
- it may be better to do more shorter runs

42. Is it worth expending effort on smart initialisation?
T: time needed to reach level F after random initialisation T
Time (number of generations)
Best fitness in population
F: fitness after smart initialisation F
Answer : it depends:
- possibly, if good solutions/methods exist.
- care is needed, see chapter on hybridisation

43. Evolutionary Algorithms in Context
There are many views on the use of EAs as robust problem solving tools
For most problems a problem-specific tool may:
perform better than a generic search algorithm on most instances,
have limited utility,
not do well on all instances
Goal is to provide robust tools that provide:
evenly good performance
over a range of problems and instances

44. EAs as problem solvers: Goldberg’s 1989 view
Special, problem tailored method
Evolutionary algorithm
Performance of methods on problems
Random search
Scale of “all” problems

45. EAs and domain knowledge
Trend in the 90’s:
adding problem specific knowledge to EAs
(special variation operators, repair, etc)
Result: EA performance curve “deformation”:
better on problems of the given type
worse on problems different from given type
amount of added knowledge is variable
Recent theory suggests the search for an “all-purpose” algorithm may be fruitless

46. Michalewicz’ 1996 view EA 4 EA 2 EA 3 EA 1 P Scale of “all” problems
Performance of methods on problems
EA 1 P
Scale of “all” problems

47. EC and Global Optimisation
Global Optimisation: search for finding best solution x* out of some fixed set S
Deterministic approaches
e.g. box decomposition (branch and bound etc)
Guarantee to find x* , but may run in super-polynomial time
Heuristic Approaches (generate and test)
rules for deciding which x  S to generate next
no guarantees that best solutions found are globally optimal

49. EC and Neighbourhood Search
Many heuristics impose a neighbourhood structure on S
Such heuristics may guarantee that best point found is locally optimal e.g. Hill-Climbers:
But problems often exhibit many local optima
Often very quick to identify good solutions
EAs are distinguished by:
Use of population,
Use of multiple, stochastic search operators
Especially variation operators with arity >1
Stochastic selection