Transcript GENETIC ALGORITHMS Tanmay, Abhijit, Ameya, Saurabh Inspiration - Evolution • Natural Selection: – – • “Survival of the Fittest” favourable traits become common and unfavourable traits become uncommon in successive.

GENETIC ALGORITHMS
Tanmay, Abhijit, Ameya, Saurabh
Inspiration - Evolution
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Natural Selection:
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“Survival of the Fittest”
favourable traits become common and
unfavourable traits become uncommon in
successive generations
Sexual Reproduction:
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Chromosomal crossover and genetic
recombination
population is genetically variable
adaptive evolution is facilitated
unfavourable mutations are eliminated
Overview
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
Inspiration
The basic algorithm
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
Why Genetic Algorithms work ?
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Schemas
Hyper-planes
Schema Theorem
Strengths and Weakness
Applications
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
Encoding
Selection
Crossover
Mutation
TSP
Conclusion
THE BASIC ALGORITHM
Ameya Muley
Encoding of Solution Space
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
Represent solution space by strings of fixed length
B
over some alphabet
A
C
TSP:

ordering of points
A

D
B
E
C
B
E
D
A
C
1
0
1
1
0
Knapsack:

inclusion in knapsack
0
0
1
0
1
E
D
Selection
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Fitness function:
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f(x), x is a chromosome in the solution space
f(x) may be:
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an well-defined objective function to be optimised
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a heuristic
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e.g. TSP and knapsack
e.g. N-Queens
Probability distribution for selection:
P X  xi  
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
f ( xi )
M
j 1
f (x j )
Fitness proportional selection
Operators-Crossover and Mutation
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Crossover:
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Applied with high probability
Position for crossover on the two parent chromosomes randomly selected
Offspring share characteristics of well-performing parents
Combinations of well-performing characteristics generated
Mutation:
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Applied with low probability
Bit for mutation randomly selected
New characteristics introduced into the population
Prevents algorithm from getting trapped into a local optimum
The Basic Algorithm
1.
2.
3.
4.
Fix population size M
Randomly generate M strings in the solution space
Observe the fitness of each chromosome
Repeat:
Select two fittest strings to reproduce
Apply crossover with high probability to produce offspring
Apply mutation to parent or offspring with low probability
Observe the fitness of each new string
Replace weakest strings of the population with the offspring
1.
2.
3.
4.
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until
i.
ii.
iii.
fixed number of iterations completed, OR
average/best fitness above a threshold, OR
average/best fitness value unchanged for a fixed number of
consecutive iterations
Example
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Problem specification:
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string of length 4
two 0’s and two 1’s
0’s to the right of the 1’s
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f(x) = number of bits that match the ones in the solution
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0 0 1 1
Solution space: 0,14
Fitness function (heuristic):
Initialization (M = 4):
1 0 0 0
f ( A)  1
0 1 0 0
f ( B)  1
0 1 0 1
f (C )  2
0 0 1 0
f ( D)  3
f av  1.75
0 1 0 1
0 1 0 0
f (X ) 1
0 0 1 0
0 0 1 1
f (Y )  4
0 1 0 0
0 1 1 0
f (Z )  2
Example (contd.)
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After iteration 1:
0 1 0 1
f ( A)  2
0 1 1 0
f ( B)  2
0 0 1 0
f (C )  3
0 0 1 1
f ( D)  4
f av  2.75
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After iteration 2:
0 1 0 1
f ( A)  2
0 0 0 1
f ( B)  3
0 0 1 0
f (C )  3
0 0 1 1
f ( D)  4
f av  3
0 1 0 1
0 1 1 0
f (X )  2
0 0 1 0
0 0 0 1
f (Y )  3
WHY GENETIC ALGORITHMS WORK?
Tanmay Khirwadkar
Schemas

Population
 Strings
 E.g.

over alphabet {0,1} of length L
s  10010
Schema
A
schema is a subset of the space of all possible
individuals for which all the genes match the template
for schema H.
 Strings over alphabet {0,1,*} of length L
 E.g. H  [1* *10]  {10010,10110,11010,11110}
Hyper-plane model
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Search space
A
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hyper-cube in L dimensional space
Individuals
 Vertices
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of hyper-cube
Schemas
 Hyper-planes
formed by vertices
0**
Sampling Hyper-planes
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Look for hyper-planes (schemas) with good fitness
value instead of vertices (individuals) to reduce
search space
Each vertex
of 3L hyper-planes
 Samples hyper-planes
 Member
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Average Fitness of a hyper-plane can be estimated
by sampling fitness of members in population
Selection retains hyper-planes with good estimated
fitness values and discards others
Schema Theorem
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Schema Order O(H)
Schema order, O(.) , is the number of non ‘*’ genes in
schema H.
 E.g. O(1**1*) = 2
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Schema Defining Length δ(H)
Schema Defining Length, δ(H), is the distance between first
and last non ‘*’ gene in schema H
 E.g. δ(1**1*) = 4 – 1 = 3
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Schemas with short defining length, low order with
fitness above average population are favored by GAs
Formal Statement
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Selection probability
m( H , t ) f ( H , t )
E (m( H , t  1)) 
f (H , t)
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Crossover probability
P(hcrossover )  pc L(H1)
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Mutation probability
P(hmutation)  ( H ) pm
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Expected number of members of a schema
m( H , t ) f ( H , t )
 (H )
E (m( H , t  1) 
f (H , t)
(1  pc
L 1
 pm( H ))
Why crossover and mutation?
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Crossover
 Produces
new solutions while ‘remembering’ the
characteristics of old solutions
 Partially preserves distribution of strings across schemas
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Mutation
 Randomly
generates new solutions which cannot be
produced from existing population
 Avoids local optimum
STRENGTHS AND WEAKNESS
Abhijit Bhole
Area of application
GAs can be used when:
 Non-analytical problems.
 Non-linear models.
 Uncertainty.
 Large state spaces.
Non-analytical problems
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Fitness functions may not be expressed analytically
always.
Domain specific knowledge may not be computable
from fitness function.
Scarce domain knowledge to guide the search.
Non-linear models
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Solutions depend on starting values.
Non – linear models may converge to local
optimum.
Impose conditions on fitness functions such as
convexity, etc.
May require the problem to be approximated to fit
the non-linear model.
Uncertainty
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Noisy / approximated fitness functions.
Changing parameters.
Changing fitness functions.
Why do GAs work? Because uncertainty is common
in nature.
Large state spaces
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Heuristics focus only on the immediate area of initial
solutions.
State-explosion problem: number of states huge or
even infinite! Too large to be handled.
State space may not be completely understood.
Characteristics of GAs
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Simple, Powerful, Adaptive, Parallel
Guarantee global optimum solutions.
Give solutions of un-approximated form of
problem.
Finer granularity of search spaces.
When not to use GA!
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Constrained mathematical optimization problems
especially when there are few solutions.
Constraints are difficult to incorporate into a GA.
Guided domain search is possible and efficient.
PRACTICAL EXAMPLE - TSP
Saurabh Chakradeo
TSP Description
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Problem Statement: Given a complete weighted
undirected graph, find the shortest Hamiltonian
cycle. (n nodes)
The size of the solution space in (n-1)!/2
Dynamic Programming gives us a solution in time
O(n22n)
TSP is NP Complete
TSP Encoding
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Binary representation
 Tour
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1-3-2 is represented as ( 00 10 01 )
Path representation
 Natural
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Adjacency representation
 Tour
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–(132)
1-3-2 is represented as ( 3 1 2 )
Ordinal representation
A
reference list is used. Let that be ( 1 2 3 ).
 Tour 1-3-2 is represented as ( 1 2 1 )
TSP – Crossover operator
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Order Based crossover (OX2)
 Selects
at random several positions in the parent tour
 Imposes the order of nodes in selected positions of one
parent on the other parent
 Parents: (1 2 3 4 5 6 7 8) and (2 4 6 8 7 5 3 1)
 Selected positions, 2nd , 3rd and 6th
 Impose order on (2 4 6 8 7 5 3 1) &(1 2 3 4 5 6 7 8)
 Children are (2 4 3 8 7 5 6 1) and (1 2 3 4 6 5 7 8)
TSP – Mutation Operators
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Exchange Mutation Operator (EM)
 Randomly
select two nodes and interchange their
positions.
 ( 1 2 3 4 5 6 ) can become ( 1 2 6 4 5 3 )
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Displacement Mutation Operator (DM)
 Select
a random sub-tour, remove and insert it in a
different location.
 ( 1 2 [3 4 5] 6 ) becomes ( 1 2 6 3 4 5 )
Conclusions
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Plethora of applications
 Molecular
biology, scheduling, cryptography,
parameter optimization
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General algorithmic model applicable to a large
variety of classes of problems
Another in the list of algorithms inspired by
biological processes – scope for more parallels?
Philosophical Implication:
 Are
humans actually moving towards their global
optimum?
References
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Adaptation in Natural and Artificial Systems, John H.
Holland, MIT Press, 1992.
Goldberg, D. E. 1989 Genetic Algorithms in Search,
Optimization and Machine Learning. 1st. AddisonWesley Longman Publishing Co., Inc.
Genetic Algorithms for the Travelling Salesman
Problem: A Review of Representations and
Operators, P. Larranaga et al., University of Basque,
Spain. Artificial Intelligence Review, Volume 13,
Number 2 / April, 1999