MITESD_77S10_lec11

MITESD_77S10_lec11 - Multidisciplinary System Design...

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Unformatted text preview: Multidisciplinary System Design Optimization A Basic Introduction to Genetic Algorithms Lecture 11 Prof. Olivier de Weck 1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Heuristic Search Techniques Main Motivation for Heuristic Techniques: (1) To deal with local optima and not get trapped in them J x (2) To allow optimization for systems, where the design variables are not only continuous, but discrete (categorical), integer or even Boolean xi R xi ={1,2,3,4,5}, xi ={„A‟,‟B‟,‟C‟} xi ={true, false} These techniques do not guarantee that global optimum can be found. Generally Karush-Kuhn-Tucker conditions do not apply. 2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Principal Heuristic Algorithms • Genetic Algorithms (Holland – 1975) – Inspired by genetics and natural selection – max fitness • Simulated Annealing (Kirkpatrick – 1983) – Inspired by statistical mechanics– min energy • Particle Swarm Optimization (Eberhart Kennedy - 1995) – Inspired by the social behavior of swarms of insects or flocks of birds – max “food” These techniques all use a combination of randomness and heuristic “rules” to guide the search for global maxima or minima 3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Today: Genetic Algorithms • • • • • • 4 Genetics and Natural Selection A simple genetic algorithm (SGA) “The Genetic Algorithm Game” Encoding - Decoding (Representation) Fitness Function - Selection Crossover – Insertion - Termination © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Premise of GAs • Natural Selection is a very successful organizing principle for optimizing individuals and populations of individuals • If we can mimic natural selection, then we will be able to optimize more successfully • A possible design of a system – as represented by its design vector x - can be considered as an individual who is fighting for survival within a larger population. • Only the fittest survive – Fitness is assessed via objective function J. 5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics MATLAB® GA demo (“peaks”) • Maximize “peaks” function • Population size: 40 • Generations: 20 • Mutation Rate: 0.002 -Observe convergence -Notice “mutants” -Compare to gradient search 6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Natural Selection Charles Darwin (1809-1882) Controversial and very influential book (1859) On the origin of species by means of natural selection, or the preservation of favored races in the struggle for life Observations: • Species are continually developing • Homo sapiens sapiens and apes have common ancestors • Variations between species are enormous • Huge potential for production of offspring, but only a small/moderate percentage survives to adulthood Evolution = natural selection of inheritable variations 7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Inheritance of Characteristics Gregor Mendel (1822-1884) Investigated the inheritance of characteristics (“traits”) Conducted extensive experiments with pea plants Examined hybrids from different strains of plant Tall Tall Tall Short Short Short Short Tall Tall Image by MIT OpenCourseWare. Character (gene) for tallness is dominant Character (gene) for shortness is recessive 8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics GA Terminology chromosome population gene individuals selection crossover insertion mutation genetic operators Generation n 9 Generation n+1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Chromosomes Chromosome (string) alleles gene 0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1 Each chromosome represents a solution, often using strings of 0‟s and 1‟s. Each bit typically corresponds to a gene. This is called binary encoding. The values for a given gene are the alleles. A chromosome in isolation is meaningless need decoding of the chromosome into phenotypic values 10 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics GA over several generations Initialize Population (initialization) next generation Select individual for mating (selection) Mate individuals and produce children (crossover) Mutate children (mutation) Insert children into population (insertion) n Are stopping criteria satisfied ? y Finish 11 Ref: Goldberg (1989) © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics “The GA Game” Population size: N=40 Mean Fitness: F=6.075 Ca. 15 minutes Generation 1: (Fitness F = total number of 1‟s in chromosome) 1 1 1 5 8 9 6 3 3 3 0 0 40 1 2 3 20 40 54 42 24 27 30 0 0 6.075 GA Game Initial Population Number of Individuals 1 2 3 4 5 6 7 8 9 10 11 12 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 Fitness Value 0 <= F <= 12 12 Goal: Maximize Number of “1”s © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Creating a GA on Computer (1) define the representation (encoding-decoding) (2) define “fitness” function F - incorporate feasibility (constraints) and objectives (3) define the genetic operators - initialization, selection, crossover, mutation, insertion (4) execute initial algorithm run - monitor average population fitness - identify best individual (5) tune algorithm - adjust selection, insertion strategy, mutation rate 13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Encoding - Decoding phenotype genotype coded domain Biology UGCAACCGU (“DNA” blocks) Design 10010011110 (chromosome) decision domain expression sequencing “blue eye” decoding H encoding Genetic Code: (U,C,G,A are the four bases of the nucleotide building blocks of messenger-RNA): Uracil-CytosinAdenin-Guanin - A triplet leads to a particular aminoacid (for protein synthesis) e.g. UGG-tryptophane 14 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Radius R=2.57 [m] Decoding x1 x2 xn 0 1 0 1 1 1 1 0 1 0 0 1 ….. 0 1 Radius (genotype) Height Material E.g. binary encoding of integers: 10100011 (1*27+0*26+1*25+0*24+0*23+0*22+1*21+1*20) 128 + 0 + 32 + 0 + 0 + 0 + 2 + 1 = 163 Coding and decoding MATLAB® functions available: decode.m, encode.m 15 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Binary Encoding Issues Number of bits dedicated to a particular design variable is very important. Number of bits needed: Resolution depends on: - upper and lower bounds xLB, xUB - number of bits x nbits xUB xLB x ln 2 xLB xUB x=(xUB- xLB)/2nbits Example [G]=encode(137.56,50,150,8) G = 1 1 0 1 1 [X]=decode(G,50,150,8); X = 137.4510 So Loss in precision !!! 16 ln 1 1 1 x= (150-50)/28 = 0.39 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Other Encoding Schemes Not all GA chromosomes are binary strings Can use a different ALPHABET for GA coding Most common is binary alphabet {0,1} can also have The set of symbols is the “alphabet” - ternary: {0,1,2} {A,B,C} - quaternary: {0,1,2,3} {T,G,C,A} => biology - integer: {1,2,….13,….} -real valued: {3.456 7.889 9.112} -Hexadecimal {1,2,..,A,B,C,D,E,F} Used for Traveling Salesman Problem 17 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics A representation for the fire station location problem 1 2 4 3 7 6 5 9 8 11 10 12 13 14 Image by MIT OpenCourseWare. 10101000010010 18 “1” represents a fire station © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Fitness and Selection Probability Typically, selection is the most important and most computationally expensive step of a GA. x1 xn 01001110101 F 19 f (J ) Al-7075 decode F drives probability of being selected P(selected ) 1.227 yes F no Map raw objective to Fitness Evaluate objective simcode function J f x © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Fitness Function • Choosing the right fitness function is very important, but also quite difficult • GAs do not have explicit “constraints” • Constraints can be handled in different ways: – via the fitness function – penalty for violation – via the selection operator (“reject constraint violators”) – implicitly via representation/coding e.g. only allow representations of the TSP that correspond to a valid tour – Implement a repair capability for infeasible individuals Choosing the right fitness function: an important genetic algorithm design Issue 20 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Maximization vs. Minimization There are many ways to convert a minimization problem to a maximization problem and vice-versa: • N-obj • 1/obj • -obj 21 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Selection by Ranking • Goal is to select parents for crossover • Should create a bias towards more fitness • Must preserve diversity in the population (1) Selection according to RANKING Example: Let D j P 1 j select the kth most fit member of a population 1 to be a parent with probability Pk D1 k Better ranking has a higher probability of being chosen, e.g. 1st 1, 2nd 1/2, 3rd 1/3 ... 22 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Selection by Fitness (2) Proportional to FITNESS Value Scheme Example: Let F j P Fitness j select the kth most fit member of a population to be a parent with probability Pk Fitness(k ) F 1 Probability of being selected for crossover is directly proportional to raw fitness score. 23 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Roulette Wheel Selection Roulette Wheel Selection Probabilistically select individuals based on some measure of their performance. 1 2 6 3 5 4 Sum Sum of individual‟s selection probabilities 3rd individual in current population mapped to interval [0,Sum] Selection: generate random number in [0,Sum] Repeat process until desired # of individuals selected Basically: stochastic sampling with replacement (SSR) 24 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Tournament Selection Dominant performer placed in intermediate population of survivors 2 members of current population chosen randomly n Population Filled ? y Crossover and Mutation form new population Old Population Fitness 101010110111 8 100100001100 4 001000111110 6 25 Survivors Fitness 101010110111 8 001000111110 6 101010110111 8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Crossover 0 1 0 1 1 1 1 0 1 1 1 1 ….. 1 1 P1 1 1 1 0 0 1 0 0 0 1 0 1 ….. 0 0 P2 crossover O1 ? O2 ? Question: How can we operate on parents P1 and P2 to create offspring O1 and O2 (same length, only 1‟s and 0‟s)? 26 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Crossover in Biology P1 P2 Child a b cd Crossover produces either of these results for each chromosome 27 This is where the word crossover comes from ac ac OR ad OR bc OR bd © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Crossover Operator (I) Crossover (mating) is taking 2 solutions, and creating 1 or 2 more Classical: single point crossover crossover point P1 0 1 1 0 1 0 1 1 1 1 O1 P2 1 0 0 1 1 1 0 0 0 1 O2 The parents 28 The children (“offspring”) © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Crossover Operator (II) More on 1-point crossover …. P1 0 1 1 0 1 P2 1 0 0 1 1 i=3 0 1 1 1 1 C1 1 0 0 0 1 C2 l=length of chromosome i=3 l=5 A crossover bit “i” is chosen (deliberately or randomly), splitting the chromosomes in half. Child C1 is the 1st half of P1 and the 2nd half of P2 Child C2 is the 1st half of P2 and the 2nd half of P1 29 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Crossover Operator (III) • One can also do a 2-point crossover or a multi-point crossover • The essential aspect is to create at least one child (solution/design) from two (or more) parent (solutions/designs) • there are many solutions to do this Some crossover operations: - single point, versus multiple point crossover - path re-linking 30 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Path Relinking • Given Parents P1 and P2 • Create a sequence of children – The first child is a neighbor of P1 – Each child is a neighbor of the previous child – The last child is a neighbor of P2 C2 C1 P1 31 ... Cn P2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Example: Path Relinking Parents P1 1 0 0 1 0 0 1 and 0 0 1 0 1 0 0 P2 Children 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 32 Create a path of children, then select the best one. Good approach, but solutions tend to be interpolations of initial population. © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Some Insertion Strategies • Can replace an entire population at a time (go from generation k to k+1 with no survivors) - select N/2 pairs of parents - create N children, replace all parents - polygamy is generally allowed • N = # of members in population if steady state Can select two parents at a time - create one child - eliminate one member of population (weakest?) • “Elitist” strategy - small number of fittest individuals survive unchanged • “Hall-of-fame” - remember best past individuals, but don‟t use them for progeny 33 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Initialization Somehow we need to create an initial population of solutions to start the GA. How can this be done? • Random initial population, one of many options • Use random number generator to create initial population (caution with seeds !) • Typically use uniform probability density functions (pdf‟s) • Typical goal: Select an initial population that has both quality and diversity Example: Nind - size of binary population Lind - Individual chromosome length round(rand(1,6)) >> 1 1 1 1 0 0 Need to generate Nind x Lind random numbers from {0,1} Rule of thumb: Population Size at Least Nind ~ 4 Lind 34 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics GA Convergence Typical Results Average Fitness global optimum (unknown) Converged too fast (mutation rate too small?) generation Average performance of individuals in a population is expected to increase, as good individuals are preserved and bred and less fit individuals die out. 35 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics GA Stopping Criteria Some options: • • • 36 X number of generations completed - typically O(100) Mean deviation in performance of individuals in the population falls below a threshold J<x (genetic diversity has become small) Stagnation - no or marginal improvement from one generation to the next: (Jn+1-Jn)<X © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics GAs versus other methods Differ from traditional search/optimization methods: • • GAs use probabilistic transition rules, not deterministic ones • GAs work on an encoding of the design variable set rather than on the variables themselves • 37 GAs search a population of points in parallel, not only a single point GAs do not require derivative information or other auxiliary knowledge - only the objective function and corresponding fitness levels influence search © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Next Lecture • • • • Speciality GA‟s Particle Swarm Optimization (PSO) Tabu Search (TS) Selection of Optimization Algorithms – Which algorithm is most suited to my problem? • Design Optimization Applications 38 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Book References Holland J., “Adaptation in Natural and Artificial Systems”, University of Michigan Press, 1975 Goldberg, D.E.,” Genetic Algorithms in Search, Optimization and Machine Learning”, Addison Wesley, 1989 39 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics MIT OpenCourseWare http://ocw.mit.edu ESD.77 / 16.888 Multidisciplinary System Design Optimization Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
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This note was uploaded on 11/08/2011 for the course AERO 16.851 taught by Professor Ldavidmiller during the Fall '03 term at MIT.

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