cis*6080 genetic algorithms instructor:robert collier e-mail:[email protected]
TRANSCRIPT
2Robert Collier
Map Colouring
Colour the regions of a map such that adjacent nodes are assigned different colours.
3Robert Collier
Graph Colouring Example
If the graph is planar, only four colours are necessary.
This graph isn't planar, so it might not be possible (with only four colours).
How many possible solutions (i.e. candidate solutions) exist?
Colour the nodes such that adjacent nodes are assigned different colours.
4Robert Collier
Graph Colouring Example
The graph is made up of 30 nodes and this example is using 4 colours.
What would the solution (if one actually exists) actually look like?
How many possible solutions are there?
At a rate of 1 solution / second, how long will it take to check them all ?
Solution = { Node01 = #FF0000, Node02 = #0000FF, ..., Node 30
= #007F00}
Colours = { #FF0000, #0000FF, #007F00, #FF00FF }
Nodes = { Node01, Node02, Node03, ..., Node 30}
4 4 4 4 4 ... 4 = 430
= 1 152 921 504 606 846 976
1267650600228229401496703205376 / (60 60 24 365.25)
36 533 877 880 years
5Robert Collier
Graph Colouring Example
The left solution violates 21 constraints while the right solution violates only 3.
Therefore we can conclude that the right solution is superior to the left.
Even if there is no perfect solution, there is a solution as good or better than all others.
There may or may not be a perfect solution to this problem, but consider...
6Robert Collier
Definition: The Genetic Algorithm
The Genetic Algorithm is a Population Simulation that uses
the principles of Darwinian Evolution to solve Optimization
Problems.
7Robert Collier
Course Outline
Instructor: Robert Collier
Office Hours: Wednesdays (by appointment)E-mail: [email protected] Website: http://www.uoguelph.ca/~collierr/CIS6080/index.htm
The primary objectives of this course are:
to introduce students to the simple genetic algorithm,
to explore each of the component processes that drive the
mechanism, and
to explore many of the different variations on these component
processes.
The secondary objective of this course is:
to teach students how to conduct a comparative evaluation of two
algorithms.
Course Objectives:
8Robert Collier
This course introduces the student to basic genetic algorithms, which are based on the process of natural evolution. It is explored in terms of its mathematical foundation and applications to optimization in various domains.
Description:
Course Outline
Evaluation:
Assignment 1 15%
Assignment 2 15%
Assignment 3 15%
Final Project 55%
Optional Bonus Assignment 10%
9Robert Collier
Course Overview
Lecture Topics:
Week 1 Foundational Material
Week 2 The Simple Genetic Algorithm
Weeks 3, 4 Component Process: Selection
Weeks 5, 6 Component Process: Variation
Weeks 7, 8 Component Process: Replication
Weeks 9, 10 Experimental Design and Statistical Analysis
Week 11 Genetic Programming, Artificial Life, etc.
Week 12 Learning Classifier Systems
10Robert Collier
Foundational Material
The Genetic Algorithm is a Population Simulation that uses
the principles of Darwinian Evolution to solve Optimization
Problems.
Optimization Problems is the domain of problems to be solved.
Darwinian Evolution provided the inspiration for the technique.
Population Simulations model the environment where evolution was observed.
11Robert Collier
Optimization Problems
The objective is to find the best possible solution from the set of feasible solutions.
This implies: a) that some solutions are better than others
b) there must be a way to quantify relative
superiority
Contrast the set of Optimization Problems with:
Decision Problems
Search Problems
Counting Problems
i.e. Does a particular entity meet certain criteria?
i.e. Where is a particular entity located in the space of all entities?
i.e. How many of a particular entity are contained in the space of entities?
12Robert Collier
Optimization Problems
The objective is to find the best possible solution from the set of feasible solutions.
This implies: a) that some solutions are better than others
b) there must be a way to quantify relative
superiority
Map Colouring
Common Optimization Problems:
13Robert Collier
Windsor
London Hamilton
Toronto
Sudbury
Kingston
OttawaNorth Bay
Optimization Problems
The objective is to find the best possible solution from the set of feasible solutions.
This implies: a) that some solutions are better than others
b) there must be a way to quantify relative
superiorityCommon Optimization Problems:
Traveling Salesman
Windsor
London Hamilton
Toronto
Sudbury
Kingston
OttawaNorth Bay
Windsor
London Hamilton
Toronto
Sudbury
Kingston
OttawaNorth Bay
14Robert Collier
90
100
50
40
30
90
Optimization Problems
The objective is to find the best possible solution from the set of feasible solutions.
This implies: a) that some solutions are better than others
b) there must be a way to quantify relative
superiorityCommon Optimization Problems:
Knapsack Problem
90
100
50
4030
15Robert Collier
Simplistic Approaches
Exhaustive Approach
Every element of the set of feasible solutions can be tested.
F(x) =
x =
Greedy Approach
A candidate solution can often be constructed in stages, and the
component added at each stage is the one offering the most immediate
benefit.
Example: How can the minimum of y = x2 be found (without the
derivative)?
Consider the implications of these approaches?
Why look at neighbours?
Why the superior neighbour?
16Robert Collier
Complex Candidate Solution Spaces
The set of candidate solutions is typically a space of candidate
solutions...
... however, this space can be very complex.
F'(x) =
x =
17Robert Collier
Complex Candidate Solution Spaces
The set of candidate solutions is typically a space of candidate
solutions...
... however, this space can be very complex.
For a two dimensional solution (i.e. an ordered pair (x, y)),
the space is often conceptualized as a landscape.
An intuitive approach directly results from this analogy...
F'(x) =
x =
18Robert Collier
Hill Climbing Approach
solution = selectRandomSolution()best_fitness_so_far = getFitnessOf(solution)other_solutions_remaining = TRUE
WHILE other_solutions_remainingother_solutions_remaining = FALSEother_solutions = getNeighbouringSolutions(solution)FOR EACH neighbouring_solution IN other_solutions
IF getFitnessOf(neighbouring_solution) > best_fitness_so_far
best_fitness_so_far = getFitnessOf(neighbouring_solution)
solution := neighbouring_solutionother_solutions_remaining = TRUE
END IFEND FOR
END WHILE
RETURN solution
19Robert Collier
Simulated Annealing Approach
solution = selectRandomSolution()current_fitness = getFitnessOf(solution)temperature = starting_temperature
WHILE termination_criteria_not_metneighbouring_solution = getRandomNeighbour(solution)IF getFitnessOf(neighbouring_solution) > best_fitness_so_far
current_fitness = getFitnessOf(neighbouring_solution) solution = neighbouring_solution
ELSEIF random < exp(- (current_fitness -
getFitnessOf(neighbouring_solution)) )temperature
current_fitness = getFitnessOf(neighbouring_solution) solution = neighbouring_solution
END IFEND IFreduce(temperature)
END WHILE
RETURN solution
20Robert Collier
Summary of Optimization Problems
Optimization Problems are solved by finding the best possible solution
(i.e. the optima) from the space of candidate solutions.
Some candidate solutions are better than others.
Some candidate solutions are closer together and some are further apart.
Candidate solution spaces are often complex, and hill climbing alone is not sufficient (as they become trapped at local optima)
Processes observed in the natural world (i.e. annealing, evolution) can serve as inspiration for effective optimization
approaches.
21Robert Collier
Foundational Material
The Genetic Algorithm is a Population Simulation that uses
the principles of Darwinian Evolution to solve Optimization
Problems.
Optimization Problems is the domain of problems to be solved.
Darwinian Evolution provided the inspiration for the technique.
Population Simulations model the environment where evolution was observed.
22Robert Collier
Relevant Biology
Chromosome a structure defining an organism's properties
Gene a component of a chromosome
Trait a property encoded by one of more genes
Allele a possible value setting for a trait
Locus a position in a chromosome
Genotype the genetic constitution of an organism
Phenotype the physical constitution of an organism
Important Definitions
23Robert Collier
Context for the Analogous Biology
Solution = { Node01 = #FF0000, Node02 = #0000FF, ..., Node 50
= #007F00}
Solutions to the map colouring problem discussed previously...
Colours = { #FF0000, #0000FF, #007F00, #FF00FF }
Nodes = { Node01, Node02, Node03, ..., Node 50}
How could a chromosome be constructed from this information?
Possible Encodings: String? Character? Integer? Binary? Other?
Integer Valued Chromosome 013213021302130132031230 ... 13203Binary Chromosome010100101111010101111010 ... 01011
Traits? Alleles? Loci?
24Robert Collier
Biomimicry and the Natural World
Leonardo Da Vinci's sketch of a flying machine was inspired by the form of a bird.
Biomimicry uses natural processes as inspiration for solving problems. Why?
... because the natural world has solved many complex problems. How?
... through the process of natural selection.
25Robert Collier
Natural Selection
The process by which traits become more or less common,
according to the survival and/or reproduction of organisms with those
traits.
carbonaria
typica
These moths are an example of a solution to the problem of camouflage.
26Robert Collier
The Evolutionary Mechanism
Jean-Baptiste Lamarck
Philosophie Zoologique, 1809
“as new modifications will necessarily continue to operate, however slowly, not only will there continually be found new species, new genera, and new orders, but each species will vary in some part of its structure and form ... individuals which from special causes are transported into very different situations from those where the others occur, and then constantly submitted to other influences - the former, I say, assume new forms, and then they constitute a new species”
... believed that traits acquired over an organism's life could be transferred to offspring
... believed there were complexifying and adaptive forces at work
27Robert Collier
Instead of Lamarck's complexifying force, genetic variation causes diversity in the chromosomes of living things, and those better suited to their environment are more likely to survive and pass the various traits to their offspring.
Darwin proposed the process of Natural Selection
Instead of Lamarck's complexifying force, genetic variation causes diversity in the chromosomes of living things, and those better suited to their environment are more likely to survive and pass the various traits to their offspring.
The Evolutionary Mechanism
What are the key elements of this process?
Variation
Evaluation
Selection
Replication
28Robert Collier
Summary of Darwinian Evolution
Genetic algorithm terminology borrows heavily from biology.
The natural world has already solved many complex problems.
Natural selection states that variations appear in populations, and those variations that are beneficial are transmitted to
future generations.
The component processes of the evolutionary mechanism are:Evaluation, Selection, Variation, and Replication
Any system with these processes is an evolving system.
29Robert Collier
Foundational Material
The Genetic Algorithm is a Population Simulation that uses
the principles of Darwinian Evolution to solve Optimization
Problems.
Optimization Problems is the domain of problems to be solved.
Darwinian Evolution provided the inspiration for the technique.
Population Simulations model the environment where evolution was observed.
30Robert Collier
Simulation Basics
"The imitation of the operation of a real-world process or system over time."
A population of organisms ...
... in which the likelihood that an organism survives or reproduces is
proportional
the degree to which that organism can solve a particular optimization
problem.
Evaluation
SelectionVariation
Replication
The components of the evolutionary mechanism:
31Robert Collier
Population Simulation
Two Simplifying Assumptions:
1. Each member of the population lives for exactly one
generation.
2. The population size never changes (i.e. complete
replacement)Population at Time t Population at Time t+1
Terminate?
The EvolutionaryMechanism
32Robert Collier
Overview of the Genetic Algorithm
Initialization
Evaluation
Selection
Variation
Replication
Terminate?
Termination
true
false
This graduate level course in genetic algorithms was developed and taught by Robert Collier, MSc,
BSc.
It was most recently offered at the University of Guelph during the
summer 2011 semester.
The remaining slides have been omitted from this
preview.