Evolutionary Computation (EC) eie426-ec-200809.ppt 2017/4/14 EIE426-AICV

Contents Basic Concepts of EC Genetic Algorithms An Example Chromosome Representation Stopping Criteria Initial Population Selection Mechanisms Crossover and Mutation Fitness Functions Another Example Application: Routing Optimization Advantages and Disadvantages of EC 2017/4/14 EIE426-AICV

Evolution and Search Evolution - search through the enormous genetic parameter space for the best genetic make-up. Borrow ideas from nature to help us solve problems that have equally large search spaces or similarly changing environment. 2017/4/14 EIE426-AICV

Natural Evolution and Evolutionary Computation Individual Fitness Environment Evolutionary Computing Candidate Solution Quality Problem 2017/4/14 EIE426-AICV

Different ECs Several classes of EC algorithms have been developed: - Genetic algorithms (GA’s): model genetic evolution - Genetic programming: based on GA’s, but individuals are programs (represented as trees) - Evolutionary programming: from the simulation of adaptive behavior in evolution (phenotype evolution) - Evolution strategies: model the strategic parameters that control variation in evolution, i.e., the evolution of evolution - Culture evolution: models the evolution of culture of a population and how the culture influences the evolution of individuals. - Co-evolution: individuals evolve through cooperation, or in competition with one other. 2017/4/14 EIE426-AICV

Basic Concepts Chromosome: individual Population: many individuals Gene: each characteristics of chromosome (one parameter) Allele: the value of a gene Crossover: generate offspring by combining parts of the parents. Mutation: introduce new genetic material into an existing individual. Fitness: the survival strength of an individual Culling (removing) and elitism (copying) 2017/4/14 EIE426-AICV

Evolutionary Computation Selection Parents Recombination (Crossover) Population Mutation The evolutionary cycle Replacement Offspring 2017/4/14 EIE426-AICV

Genetic Algorithms The GA was the first EC paradigm developed and applied (Holland 1975). The features of the original GA’s: A bit string representation Proportional selection Cross-over as the primary method to produce new individuals. Several changes have been made: Different representation schemes Different selection methods Different GA operators (cross-over, mutation and elitism) 2017/4/14 EIE426-AICV

Random Search The GA is a search procedure. Random search is possibly the simplest search procedure. Its training time may be very long before an acceptable solution is obtained. Procedure: Start from an initial search point or a set of initial points. Random perturbations to the points Repeat until an acceptable solution is reached or a maximum number of iterations is exceeded. 2017/4/14 EIE426-AICV

General Genetic Algorithm Let g = 0. Initialize the initial generation Cg . While no stopping criterion is satisfied (a) Evaluate the fitness of each individual in Cg . (b) g  g+1. (c) Select parents from Cg-1. (d) Recombine selected parents through cross-over to form offspring Og (with a probability pc). (e) Mutate offspring in Og (with a probability pm). (f) Select the new generation Cg from (the previous generation Cg-1, e.g., the best individuals are copied) and the offspring Og. g: generation Note: The things in () might or might not be carried out. 2017/4/14 EIE426-AICV

An Example 2017/4/14 EIE426-AICV

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Use a genetic algorithm to solve the problem: Coding (chromosome representation of a solution) Generation of initial population (solutions) Fitness calculation Genetic operation 2017/4/14 EIE426-AICV

Coding (chromosome representation): Use a binary string to represent x. If the solution is to be precise to 10-6, then the interval (2-(-1)) = 3 should be divided into 3× 106. At least 22 bits should be used because 2017/4/14 EIE426-AICV

Decoding 2017/4/14 EIE426-AICV

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Generation of initial population A set of N 22-bit binary strings can be randomly generated as the initial population. 2017/4/14 EIE426-AICV

Fitness calculation Since f(x) > 0 in the interval, we can directly use f(x) as a fitness function: f(s) = f(x) e.g., s1 = <1000101110110101000111>, f(s1)=2.586345 s2 = <0000001110000000010000>, f(s2)=1.078878 s3 = <1110000000111111000101>, f(s3)= 3.250650 2017/4/14 EIE426-AICV

(1) Selection: based on the fitness of individuals Genetic operation (1) Selection: based on the fitness of individuals e.g., roulette wheel selection (fitness proportionate selection) (2) Crossover (with a probability pc) e.g., s2 = <00000 | 01110000000010000>, f(s2)=1.078878 s3 = <11100 | 00000111111000101>, f(s3)= 3.250650 After the crossover operation: s’2 = <00000 | 00000111111000101>, f(s’2)=1.940865 s’3 = <11100 | 01110000000010000 >, f(s’3)= 3.459245 2017/4/14 EIE426-AICV

(3) Mutation (with a probability pm) e.g., s3 = <1110000000111111000101> f(s3)= 3.250650 After the mutation operation: s’3 = <1110100000111111000101 > f(s’3)= 0.917743 or s3 = <1110000000111111000101> s”3 = <1110000001111111000101 > f(s”3)= 3.343555 2017/4/14 EIE426-AICV

Simulation results: N = 50, pc = 0.25, pm = 0.01, at 89 generations, the best individual was obtained: smax = <1101001111110011001111> xmax = 1.850 549 f(xmax) = 3.850 274 2017/4/14 EIE426-AICV

The chromosome of the best individual The best individual at each iteration (up to 150 generations) Generation The chromosome of the best individual x fitness 1 1000111000010110001111 1.831 624 3.534 806 11 0110101011100111001111 1.854 860 3.833 286 17 1110101011111101001111 1.847 536 3.842 004 54 1000110110100011001111 1.848 699 3.847 155 71 0100110110001011001111 1.850 897 3.850 162 89 1101001111110011001111 1.850549 3.850274 150 2017/4/14 EIE426-AICV

Summary on Basic Concepts Evolution is an optimization process, where the aim is to improve the ability of individuals to survive. An evolutionary algorithm (EA) is a stochastic search for an optimal solution to a given problem. Evolution - search through the enormous genetic parameter space for the best genetic make-up. Borrow ideas from nature to help us solve problems that have an equally large search spaces or similarly changing environment. 2017/4/14 EIE426-AICV

Genotype: describes the genetic composition of an individual Phenotype: the expressed behavioral traits of an individual in a specific environment. Selection: use the fitness evaluations to decide which are the best parents to reproduce. Crossover: generate offspring by combining parts of the parents Mutation: introduce new genetic material into an existing individual. Coding: phenotype  genotype Decoding: genotype  phenotype 2017/4/14 EIE426-AICV

Simple Genetic Algorithm (SGA) Representation Binary strings Recombination (crossover) N-point (commonly used 1-point and 2-point) or uniform; pc typically in range (0.6, 0.9) Mutation Bitwise bit-flipping with fixed probability pm (typically between 1/pop_size and 1/ chromosome_length) Parent selection Fitness-Proportionate Survivor selection All children replace parents Speciality Emphasis on crossover 2017/4/14 EIE426-AICV

Chromosome Representation Genotype space = {0,1}I I-bit binary strings Phenotype space Coding (encoding or (representation) 10010001 10010010 010001001 011101001 Decoding (inverse representation) 2017/4/14 EIE426-AICV

Binary-valued variables: no extra coding required Nominal-valued variables D-bit with 2D discrete nominal values e.g., four colors: red (00), blue (01), green (10), yellow (11) Continuous-valued variables 2017/4/14 EIE426-AICV

Other Representations Gray coding of integers (still binary chromosomes) Gray coding is a mapping that means that small changes in the genotype cause small changes in the phenotype (unlike binary coding, e.g., 0111(7) and 1000 (8)). It is generally accepted that it is better to encode numerical variables directly as Integers Floating point variables 2017/4/14 EIE426-AICV

Stopping Criteria The maximum number of generation is exceeded An acceptable best fit individual has evolved The average and/or maximum fitness value do not change significantly over the past g generations. 2017/4/14 EIE426-AICV

Initial Population The standard way of generating the initial population is to choose gene values randomly from the allowed set of values. The goal of random selection is to ensure that the initial population is a uniform representation of the entire search space. A large population covers a larger area of the search space, and may require less generations to converge. In the case of a small population, the EA can be forced to explore a large search space by increasing the rate of mutation. 2017/4/14 EIE426-AICV

Selection Mechanisms Selection operators Random selection Proportional selection: roulette wheel selection Tournament selection Rank-based selection 2017/4/14 EIE426-AICV

Random Selection Individuals are selected randomly with no reference to fitness at all. All the individuals, good or bad, have an equal chance of being selected. 2017/4/14 EIE426-AICV

Proportional Selection: Roulette Wheel Selection Individual Chromosome Fitness fi Selection probability, Pi Accumulated probability 1 0001100000 8 0.086 957 2 0101111001 5 0.054 348 0.141 304 3 0000000101 0.021 739 0.163 043 4 1001110100 10 0.108 696 0.271 739 1010101010 7 0.076 087 0.347 826 6 1110010110 12 0.130 435 0.478 261 1001011011 0.532 609 1100000001 19 0.206 522 0.739 130 9 0.847 826 0001010011 14 0.152 174 1.000 000 2017/4/14 EIE426-AICV

Roulette Wheel Selection It can be visualized as the spinning of the wheel and testing which slide ends up at the top. Fitness values are usually normalized to [0,1]. 2017/4/14 EIE426-AICV

Assume that the following random number sequence is generated: 0.070 221 0.545 929 0.784 567 0.446 930 0.507 893 0.291 198 0.176 340 0.272 901 0.371 435 0.854 641 Compared the random number sequence with the accumulated probabilities, we select the individuals: 1, 8, 9, 6, 7, 5, 8, 4, 6, 10. Individuals 2 and 3 were removed and replaced with individuals 8 and 6. The individuals with high fitness tends to survive but those with low fitness may be removed. 2017/4/14 EIE426-AICV

The pseudocode: 2017/4/14 EIE426-AICV

Tournament Selection A group of k individuals is randomly selected. The individual with the best fitness is selected from the group. The advantage: the worse individuals of the population will not be selected and the best individuals will not dominate in the reproduction process. For crossover, two tournaments are held to select each of two parents. It is possible that (1) A parent can be selected to reproduce more than once; and (2) One individual can combine with itself to reproduce offspring. 2017/4/14 EIE426-AICV

Rank-Based Selection The rank ordering of the fitness values is used to determine the probability of selection and not the fitness values itself. Non-deterministic linear sampling Individuals are sorted in decreasing fitness value. (1) Let n = random(random(N)) where N is the total number of individuals and random(N) return a number between 1 and N. (2) Return n as the selected individual 2017/4/14 EIE426-AICV

Elitism Elitism involves the selection of a set of individuals from the current generation to survive to the next generation. The number of individuals to survive to the next generation, without being mutated, is referred to as the generation gap. Generation gap = k k best individuals or k individuals selected using any selection operator 2017/4/14 EIE426-AICV

Crossover Crossover 2017/4/14 EIE426-AICV

Uniform Crossover A mask (vector) of length I (I-bit binary string) is created at random for each pair of individuals selected for reproduction. A bit with value of 1 indicates that the corresponding allele (bit) has to be swapped. 2017/4/14 EIE426-AICV

Parent1 Parent 2 Mask Offspring 1 Offspring 2 1 2017/4/14 EIE426-AICV

One-point Crossover 2017/4/14 EIE426-AICV

Parent1 Parent 2 Mask Offspring 1 Offspring 2 1 2017/4/14 EIE426-AICV

Two-point Crossover 2017/4/14 EIE426-AICV

Parent1 Parent 2 Mask Offspring 1 Offspring 2 1 2017/4/14 EIE426-AICV

Arithmetic Crossover Arithmetic crossover can be used in the case of continuous-valued genes. 2017/4/14 EIE426-AICV

Mutation Alter each gene independently with a probability pm (the mutation rate) Real-valued representations, mutation occurs by adding a random value (usually sampled from a Gaussian distribution ) to allele. The variance is usually a function of the fitness of the individual. Individuals with a good fitness value will be mutated less, while a bad fitness value will lead to large mutations. 2017/4/14 EIE426-AICV

1 1 Mutation (fox) 2017/4/14 EIE426-AICV

The Evolution Mechanism Increasing diversity by using genetic operators mutation crossover Decreasing diversity by selection of parents things to kill 2017/4/14 EIE426-AICV

Fitness Functions Common fitness functions: Use the objective function f(x) directly (1) For a maximization problem, Fit(f(x)) = f(x) e.g., (2) For a minimization problem, Fit(f(x)) = -f(x) e.g., solution for x2 + x = 2, to minimize f(x) = x2 + x - 2 2017/4/14 EIE426-AICV

Clipping 2017/4/14 EIE426-AICV

Mapping 2017/4/14 EIE426-AICV

Linear transformation of fitness functions 2017/4/14 EIE426-AICV

Fig. ft1 Fig. ft2 2017/4/14 EIE426-AICV

Genetic algorithms: case study To find the maximum of the “peak” function of two variables x and y: -3 ≤ x, y ≤ 3 2017/4/14 EIE426-AICV

Chromosome representation 2017/4/14 EIE426-AICV

Initial population 2017/4/14 EIE426-AICV

The first generation 2017/4/14 EIE426-AICV

Local maximum 2017/4/14 EIE426-AICV

Global maximum 2017/4/14 EIE426-AICV

Evolutionary Computation: Applications Robotics Control Design Scheduling/Routing/Resource Allocation Machine Learning Pattern Recognition Market forecasting Data Mining Game Playing - Robocode - Backgammon - Chess 2017/4/14 EIE557-CI&IA

Robocode An Open Source educational game started by Mathew Nelson (originally provided by IBM). Currently contributions are being made by various people; officially Flemming N. Larsen is working on Robocode to keep it current and fix the bugs. The game is designed to help people learn to program in Java and enjoy the experience. Genetic Programming (GP) Robocode 2017/4/14 EIE426-AICV

Application: Routing Optimization The problem: Given a network of M switches, an origin and a destination switch, the objective is to find the best route to connect a call between the origin and destination switches (Sevenster and Engelbrecht 1996). PSTN: Public Switch Telephone Network 2017/4/14 EIE557-CI&IA

Chromosome representation: - variable length - each gene representing one switch - integer values for switch numbers - the first gene and last gene representing the origin and last switches, respectively Examples: (1 3 6 10) (1 5 2 5 10) = (1 5 2 10) Duplicate switches are ignored 2017/4/14 EIE557-CI&IA

Initialization of population: - randomly generated with the restriction that the first gene represents the origin switch and the last gene the destination switch 2017/4/14 EIE557-CI&IA

Fitness function: a multi-criteria objective function was applied. 2017/4/14 EIE557-CI&IA

Selection: any selection operator Crossover: any crossover operator Mutation: replacing selected genes with a uniformly random selected switch in the range [1, M]. 2017/4/14 EIE557-CI&IA

Real World EC Tends include: More complex representations and operators Use of problem specific knowledge for seeding the initial population and creating heuristic operators Hybridisation with other methods 2017/4/14 EIE557-CI&IA

Advantages of EC Handles huge search spaces Balances exploration and exploitation Easy to try - not knowledge intensive Easy to combine with other methods Provides many alternative solutions Can continually evolve solutions to fit with a continually changing problem 2017/4/14 EIE557-CI&IA

Disadvantages of EC No guarantee for optimal solution within finite time Weak theoretical basis May need extensive parameter tuning Often computationally expensive, i.e., slow 2017/4/14 EIE557-CI&IA