Iterative Improvement Algorithms

1
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

2
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

3
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

4
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

5
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

6
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_A
  2_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

7
Reproduction Methods

• Mutation
• 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
• Example
• Inversion
• Reverse the chromsome

8
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

9
Another ExampleTraveling Sales Person Problem

• How to represent a chromosome?
• What effects does this have on crossover and
  mutation?

10
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

11
Order Crossover

• p1 (1 9 2 4 6 5 7 8 3)
• p2 (4 5 9 1 8 7 6 2 3)
• 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)

12
Mutation Inversion

• What can go wrong with mutation?
• What is wrong with inversion?

13
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)