Iterative Improvement Algorithms For some problems, path to solution is irrelevant: just want solution Start with initial state, and change it iteratively to improve it (find a “best”, or “minimum” value Examples: Finding the optimal way of assigning dates within n people (match.com problem) Traveling salesperson problem Knapsack problem

Calculus approach If you know the function, can take derivative: solve derivative = 0 Example: Given fencing of length 100 feet, find dimensions to get maximum area

Hill-climbing search (or gradient descent) Example: Given 20 people, find the optimal way to distribute $1000 to them to maximize # of dates they can get Too much money to one person: that person might run out of time Money to other people might not be worthwhile Goal: Maximize # of dates For a given allocation of money, try it, then measure # of dates in 1 day period Start at a guess, then start hill climbing from there

Hill-climbing in general Move in direction of increasing value Useful when path to solution is irrelevant Drawbacks: Local maxima Plateaux Can get around this some with random-restart hill climbing

Simulated Annealing Technique inspired by engineering practice of cooling liquid At each iteration make a random move If position is better than current, do it Over time, slowly drop “temperature” T If position is worse, do it with probability P P becomes smaller as T drops P = exp(change in value / T) Eventually, algorithm reverts to hill climbing Popular in VLSI layout

Genetic Algorithms (Evolutionary Computing) Genetic Algorithms used to try to “evolve” the solution to a problem Generate prototype solutions called chromosomes (individuals) Knapsack problem as example: http://home.ksp.or.jp/csd/english/ga/gatrial/Ch9_A2_4.html All individuals form the population Generate new individuals by reproduction Use fitness function to evaluate individuals Survival of the fittest: population has a fixed size Individuals with higher fitness are more likely to reproduce

Reproduction Methods Mutation Cross-over Inversion Alter a single gene in the chromosome randomly to create a new chromosome Example Cross-over Pick a random location within chromosome New chromosome receives first set of genes from parent 1, second set from parent 2 Inversion Reverse the chromsome

Interpretation Genetic algorithms try to solve a hill climbing problem Method is parallelizable The trick is in how you represent the chromosome Tries to avoid local maxima by keeping many chromosomes at a time

Another Example: Traveling Sales Person Problem How to represent a chromosome? What effects does this have on crossover and mutation?

TSP Chromosome: Ordering of city numbers (1 9 2 4 6 5 7 8 3) What can go wrong with crossover? To fix, use order crossover technique Take two chromosomes, and take two random locations to cut p1 = (1 9 2 | 4 6 5 7 | 8 3) p2 = (4 5 9 | 1 8 7 6 | 2 3) Goal: preserve as much as possible of the orderings in the chromosomes Crossover: might end up visiting the same city twice, or not at all

Order Crossover New p1 will look like: c1 = (x x x | 4 6 5 7 | x x) To fill in c1, first produce ordered list of cities from p2, starting after cut, eliminating cities in c1 2 3 9 1 8 Drop them into c1 in order c1 = (2 3 9 4 6 5 7 1 8) Do similarly in reverse to obtain c2 = (3 9 2 1 8 7 6 4 5)

Mutation & Inversion What can go wrong with mutation? What is wrong with inversion? Mutation: same problem as crossover Inversion: same solution

Mutation & Inversion Redefine mutation as picking two random spots in path, and swapping p1 = (1 9 2 4 6 5 7 8 3) c1 = (1 9 8 4 6 5 7 2 3) Redefine inversion as picking a random middle section and reversing: p1 = (1 9 2 | 4 6 5 7 8 | 3) c1 = (1 9 2 | 8 7 5 6 4 | 3)