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**What is an Evolutionary Algorithm?**

Chapter 2

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**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

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**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

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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

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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

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General Scheme of EAs

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**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.

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**Pseudo-code for typical EA**

1 – Select parents – stochastic 5 – Survival selection – deterministic

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**What are the different types of EAs (1)**

By Christoph Eick – COSC 6367 evolutionary programming

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**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

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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

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**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

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**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)

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**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

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**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

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**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

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**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

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**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)

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**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

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**Initialisation using domain-specific knowledge**

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**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

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**The 8 queens problem: representation**

1 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

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**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.

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**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.

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**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

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**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

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**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

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**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.

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**8 Queens Problem: summary**

Note that is is only one possible set of choices of operators and parameters

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**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.

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**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.

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**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.

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**Knapsack Problem: summary**

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**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.

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**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; }

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**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

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**Example with two traits**

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**Typical run: progression of fitness**

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

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**Hypervolume comparison in MOEA problems**

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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