Logistics Assignments Crossover and Mutation Checkpoint 1 -- Problem Graded -- comments on mycourses Checkpoint 2 --Framework Mostly all graded -- comments on mycourses Checkpoint 3 -- Genotype / Phenotype Due last Wednesday Grading not started Checkpoint 4 -- Selection / Fitness Due October 19 (Friday) Final exam date Project Presentation 12 Projects Final exam date has been announced: Presentations: Dates: Week 10 Friday, November 16th Monday, November 5 Wednesday, November 7 2:45pm - 4:45pm 15 minutes / presentation 70-1445 Schedule now on Web Please send me choice of time/day Code, Report, and Grad Survey However… DUE FRIDAY, NOV 9th Plan for this week Evolutionary Algorithms Today: An EA uses some mechanisms inspired by biological evolution: reproduction, mutation, recombination, Crossover and Mutation natural selection and survival of the fittest. Wednesday: Candidate solutions to the optimization problem play the role of individuals in a population, and the cost Guest Speaker: Peter Anderson function determines the environment within which GAs and permutations. the solutions "live". Evolution of the population then takes place after the repeated application of the above operators. Questions before we start 1
Evolutionary Computation process Evolutionary Algorithms To use evolutionary algorithms your must: Initialize Define your problem population Define your genotype Select individuals for crossover (based on fitness function Identify your phenotype Crossover Define the genotype -> phenotype translation Mutation Define crossover and mutation operators Insert new offspring Define fitness into population Determine selection criteria Are stopping criteria satisfied? Set population parameters Finish Reproduction Fitness Generation k+1 Generation k fitness Individual Phenotype Genotype Reproduction output parameters problem Fitness Reproduction Exploration vs. Exploitation Means by which new individuals are produced Exploration Crossover random variation and selection to determine strategies that fit the environment Combination of 2 parents Best of both parents diversity Mutation Mutation modification of a single individual Exploitation Allows for random search through search space. focused repetition of the fittest behavior in the stage of retention. Genetic operators are applied on the Selective pressure genotype. Crossover 2
Exploration vs Exploitation Crossover Challenge is to maintain balance. Combination of the best of both parents. Too much exploration: random search Building blocks Too much exploitation Get stuck local optima Crossover on Strings / Arrays One-Point Crossover Common mechanisms: Crossover point on the parent string is selected. One-Point Crossover Two-Point Crossover All data beyond that point is swapped between the two parents Cut and Splice Uniform and Half-Uniform Crossover Arithmetic Heuristic Two-Point Crossover Cut and Splice Two points are selected on the parent Like one-point crossover, except each strings. parent has a different cut point Everything between the two points is Can result in variable length children. swapped between the parents 3
Arithmetic (Numerical arrays) Uniform Crossover and Half Uniform Crossover uniform crossover scheme (UX) linearly combines two parent chromosome vectors to produce two new offspring individual genes are compared between according to the following equations: two parents. The gene values are swapped with a fixed Offspring1 = a * Parent1 + (1- a) * Parent2 probability, typically 0.5. Offspring2 = (1 – a) * Parent1 + a * Parent2 half uniform crossover scheme (HUX) exactly half of the nonmatching genes are a = randomly determined constant. swapped. Heuristic (Numerical Arrays) GPs: Crossover and Mutation uses the fitness values of the two parent crossover mutation chromosomes to determine the direction of the search. The offspring are created according to the following equations: Before crossover Before mutation After crossover After mutation Offspring1 = BestParent + r * (BestParent – WorstParent) Offspring2 = BestParent Note that operation maintain valid where r is a random number between 0 and individuals. 1. Mutation Standard Mutation Random modification of a single Bit String individual Flip a bit Explore new areas of search space Array Avoids getting stuck in local Modify gene by random amount minima/maxima. Trees Replace branch with random subtree. 4
Traveling Salesman Problem Bad genomes and reproduction Dealing with bad genomes Instance: Total Rejection N cities with distances between pairs of Genetic Repair cities fix in genetic mapping Genetic operators Said another way: Fitness Penalties. Complete graph with n vertices such that all edges are labeled with a cost value Break Example of a permutation problem Traveling Salesman Problem Traveling Salesman Problem Output: Solution: Distance traveled to complete the tour. Tour of the cities such that each city is visited once. Said another way: Complete treatment A permutation of the cities. From [Larranaga, et. al. 1999] Ordered list of the cities Recall: Is this a large search space? Phenotype == ordered tour of the cities n cities = n! permutation TSP - Path Representation TSP - Path Representation Tour is represented as an ordered list Most intuitive and common genotype. (or array) of the cities. But it has it’s problems: Order in array == order of visitation. If city i is the jth element of the array, city 3 2 4 1 7 5 8 6 3 2 4 1 7 5 8 6 i is the jth city to be visited. Eg. 8 7 6 5 4 3 2 1 3 2 4 1 7 5 8 6 3 2 4 1 4 3 2 1 3 2 4 1 3 5 8 6 Tour: 3-2-4-1-7-5-8-6 Not valid tours!!! 5
TSP - Path Representation TSP - Path Representation GeneRepair [Mitchell, et.al.] Genetic Mapping responsible for doing the repair on a “bad genome” Keep a corrective template with a valid tour. Identify duplicate cities Most approaches that use the path Use template to replace duplicate cities. representation use designer crossover/mutation operators Assure valid offspring. Designer crossover operators Partially Mapped Crossover (PMX) Portion of one parent is mapped to a portion Designed for a particular application. of another parent Problem domain constraints Remaining info is exchanged Phenotype constraints How it works: Still operates on genotype. Choose two random cut points Define mapping Goal: Prevent bad genomes Copy mapping section (between cut points) to offspring Fill in remainder of offspring using mapping Partially Mapped Crossover (PMX) Partially Mapped Crossover (PMX) 1 2 3 4 5 6 7 8 4 2 3 1 6 8 7 5 3 7 5 1 6 8 2 4 3 7 8 4 5 6 2 1 parents offspring 6
Cycle Crossover (CX) Cycle Crossover (CX) Creates an offspring where every position is occupied by a corresponding element from one of the parents 1 2 3 4 5 6 7 8 1 2 6 4 7 5 3 8 2 4 6 8 7 5 3 1 parents offspring Cycle Crossover (CX) Order Crossover (OX) Absolute position of (on average) half Observes that order is important, not elements of both parents preserved. necessarily position. How it works: Better results for TSP than PMX. Choose 2 cut points Copy between cut points to offspring Starting from 2nd cut point in one parent, fill missing cities in order they appear in other parent. Order Crossover (OX) Edge Recombination Crossover (ER) Creates a path (offspring) that is similar to a set of existing paths (parents) by looking at the edges rather than the 1 2 3 4 5 6 7 8 4 5 6 8 7 1 2 3 vertices. 2 4 6 8 7 5 3 1 8 7 3 4 5 1 2 6 parents offspring 7
Edge Recombination Crossover (ER) Edge Recombination Crossover (ER) Algorithm Edge Map Let K be the empty list For each node, gives list of other nodes to Let N be the first node of a random parent. which it has an edge in either parent While Length(K) < Length(Parent): K := K + N (append N to K) CABDEF ABCEFD Remove N from all neighbor lists A: B C D B: A C D If N's neighbor list is non-empty then C: A B E F let N* be the neighbor of N with the fewest neighbors D: A B E F in its list (or a random one, should there be multiple) else let N* be a randomly chosen node that is not in K E: C D F N := N* F: C D E Mutation Edge Recombination Crossover (ER) A: B C D Similar problem for standard mutation B: A C D C: A B E F D: A B E F 3 2 4 1 7 5 8 6 E: C D F F: C D E 3 2 4 3 7 5 8 6 K = {} K = ABDF K = A K = ABDFC Not a valid tour K = ABDFCE K = AB K = ABD Displacement Mutation Exchange Mutation Aka Cut Mutation Randomly selects two cities and swaps How it works: Select a subtour at random 3 2 4 1 7 5 8 6 Insert it in a random place 3 2 4 1 7 5 8 6 3 2 7 1 4 5 8 6 3 7 5 2 4 1 8 6 8
Recommend
More recommend
