Logistics. Final exam date. Project Presentation. Plan for this week. Evolutionary Algorithms. Crossover and Mutation

Transcription

1 Logistics Crossover and Mutation Assignments Checkpoint -- Problem Graded -- comments on mycourses Checkpoint --Framework Mostly all graded -- comments on mycourses Checkpoint -- Genotype / Phenotype Due last Wednesday Grading not started Checkpoint -- Selection / Fitness Due October 9 (Friday) Final exam date Project Presentation Final exam date has been announced: Friday, November th :pm - :pm 0- However Projects Presentations: Dates: Week 0 Monday, November Wednesday, November minutes / presentation Schedule now on Web Please send me choice of time/day Code, Report, and Grad Survey DUE FRIDAY, NOV 9th Plan for this week Today: Crossover and Mutation Wednesday: Guest Speaker: Peter Anderson GAs and permutations. Questions before we start Evolutionary Algorithms An EA uses some mechanisms inspired by biological evolution: reproduction, mutation, recombination, natural selection and survival of the fittest. Candidate solutions to the optimization problem play the role of individuals in a population, and the cost function determines the environment within which the solutions "live". Evolution of the population then takes place after the repeated application of the above operators.

2 Evolutionary Computation process Evolutionary Algorithms Initialize population Select individuals for crossover (based on fitness function Crossover Mutation Insert new offspring into population Are stopping criteria satisfied? Finish To use evolutionary algorithms your must: Define your problem Define your genotype Identify your phenotype Define the genotype -> phenotype translation Define crossover and mutation operators Define fitness Determine selection criteria Set population parameters Reproduction Generation k Generation k+ Fitness Individual fitness Genotype Phenotype Reproduction parameters problem output Fitness Reproduction Exploration vs. Exploitation Means by which new individuals are produced Crossover Combination of parents Best of both parents Mutation modification of a single individual Allows for random search through search space. Genetic operators are applied on the genotype. Exploration random variation and selection to determine strategies that fit the environment diversity Mutation Exploitation focused repetition of the fittest behavior in the stage of retention. Selective pressure Crossover

3 Exploration vs Exploitation Challenge is to maintain balance. Too much exploration: random search Too much exploitation Get stuck local optima Crossover Combination of the best of both parents. Building blocks Crossover on Strings / Arrays Common mechanisms: One-Point Crossover Two-Point Crossover Cut and Splice Uniform and Half-Uniform Crossover Arithmetic Heuristic One-Point Crossover Crossover point on the parent string is selected. All data beyond that point is swapped between the two parents Two-Point Crossover Two points are selected on the parent strings. Everything between the two points is swapped between the parents Cut and Splice Like one-point crossover, except each parent has a different cut point Can result in variable length children.

4 Uniform Crossover and Half Uniform Crossover uniform crossover scheme (UX) individual genes are compared between two parents. The gene values are swapped with a fixed probability, typically 0.. half uniform crossover scheme (HUX) exactly half of the nonmatching genes are swapped. Arithmetic (Numerical arrays) linearly combines two parent chromosome vectors to produce two new offspring according to the following equations: Offspring = a * Parent + (- a) * Parent Offspring = ( a) * Parent + a * Parent a = randomly determined constant. Heuristic (Numerical Arrays) GPs: Crossover and Mutation uses the fitness values of the two parent chromosomes to determine the direction of the search. The offspring are created according to the following equations: Offspring = BestParent + r * (BestParent WorstParent) Offspring = BestParent Before crossover After crossover crossover mutation Before mutation After mutation where r is a random number between 0 and. Note that operation maintain valid individuals. Mutation Random modification of a single individual Explore new areas of search space Avoids getting stuck in local minima/maxima. Standard Mutation Bit String Flip a bit Array Modify gene by random amount Trees Replace branch with random subtree.

5 Bad genomes and reproduction Dealing with bad genomes Total Rejection Genetic Repair fix in genetic mapping Genetic operators Fitness Penalties. Break Traveling Salesman Problem Instance: N cities with distances between pairs of cities Said another way: Complete graph with n vertices such that all edges are labeled with a cost value Example of a permutation problem Traveling Salesman Problem Traveling Salesman Problem Solution: Tour of the cities such that each city is visited once. Output: Distance traveled to complete the tour. Said another way: A permutation of the cities. Ordered list of the cities Is this a large search space? n cities = n! permutation Complete treatment From [Larranaga, et. al. 999] Recall: Phenotype == ordered tour of the cities TSP - Path Representation Tour is represented as an ordered list (or array) of the cities. Order in array == order of visitation. If city i is the jth element of the array, city i is the jth city to be visited. Eg. Tour: TSP - Path Representation Most intuitive and common genotype. But it has it s problems: Not valid tours!!!

6 TSP - Path Representation GeneRepair [Mitchell, et.al.] Keep a corrective template with a valid tour. Identify duplicate cities Use template to replace duplicate cities. TSP - Path Representation Genetic Mapping responsible for doing the repair on a bad genome Most approaches that use the path representation use designer crossover/mutation operators Assure valid offspring. Designer crossover operators Partially Mapped Crossover (PMX) Designed for a particular application. Problem domain constraints Phenotype constraints Still operates on genotype. Goal: Prevent bad genomes Portion of one parent is mapped to a portion of another parent Remaining info is exchanged How it works: Choose two random cut points Define mapping Copy mapping section (between cut points) to offspring Fill in remainder of offspring using mapping Partially Mapped Crossover (PMX) Partially Mapped Crossover (PMX) parents offspring

7 Cycle Crossover (CX) Cycle Crossover (CX) Creates an offspring where every position is occupied by a corresponding element from one of the parents parents offspring Cycle Crossover (CX) Absolute position of (on average) half elements of both parents preserved. Better results for TSP than PMX. Order Crossover (OX) Observes that order is important, not necessarily position. How it works: Choose cut points Copy between cut points to offspring Starting from nd 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 vertices. parents offspring

8 Edge Recombination Crossover (ER) Edge Recombination Crossover (ER) Edge Map For each node, gives list of other nodes to which it has an edge in either parent CABDEF ABCEFD A: B C D B: A C D C: A B E F D: A B E F E: C D F F: C D E Algorithm Let K be the empty list Let N be the first node of a random parent. While Length(K) < Length(Parent): K := K + N (append N to K) Remove N from all neighbor lists If N's neighbor list is non-empty then let N* be the neighbor of N with the fewest neighbors in its list (or a random one, should there be multiple) else let N* be a randomly chosen node that is not in K N := N* Edge Recombination Crossover (ER) Mutation K = {} K = A K = AB K = ABD K = ABDF K = ABDFC K = ABDFCE A: B C D B: A C D C: A B E F D: A B E F E: C D F F: C D E Similar problem for standard mutation Not a valid tour Displacement Mutation Aka Cut Mutation How it works: Select a subtour at random Insert it in a random place Exchange Mutation Randomly selects two cities and swaps

9 Insertion Mutation Randomly choose a city Remove it and insert in a random place Like displacement with a subtour of Simple Inversion Mutation Choose two cut points Reverse subtour between cuts Scramble Mutation Choose a random subtour and scramble TSP Seen enough? For complete list see [Larranaga 999] Questions? Take Home Messages Reproduction Types Crossover Mutation Designer Operators for Given Problem Next time APPLICATION: Peter Anderson: GAs and permutations. CP: Crossover / Mutation 9