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:
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 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

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

10. TSP Chromosome: Ordering of city numbers
( )
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 | | 8 3)
p2 = (4 5 9 | | 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

11. Order Crossover New p1 will look like:
c1 = (x x x | | x x)
To fill in c1, first produce ordered list of cities from p2, starting after cut, eliminating cities in c1
Drop them into c1 in order
c1 = ( )
Do similarly in reverse to obtain
c2 = ( )

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

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