Problem Difficulties

1
Problem Difficulties

• Search space
• (can the best answer be found in time?)
• 2. Complexity
• (simplification renders the solution useless)
• 3. Objective function
• (noise and time variance)
• 4. Constraints
• (difficult to find one feasible solution,
• harder to find optimum solution)
• People and Politics

2
Algorithmic Approach to Problem Solving

• Three basic concepts are common to every
  algorithmic approach to problem solving
• Representation
• Objective
• Evaluation Function
• Representation encoding of problem solution.
• Objective what is the goal?
• Evaluation (Fitness) Function quality of
  solution.

3
Problem Types
It is important to recognise different problem
types as it may suggest appropriate methods to
find the solution.

• Linear,
• SAT Boolean Satisfiability Problem
• TSP Travelling Salesman Problem
• NLP Non-linear Programming
• Proofs
• Constraint Satisfaction Problems
• Other types and subdivisions exist

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Classical Algorithms

• Two main classes of classical algorithms
• Operate on complete solutions
• Interruptible to give potential solution
• Quality depends on the accuracy of
  procedure-problem
• Requires knowledge of problem
• Each algorithm can only tackle a class of
  problems
• Evaluate partially constructed or approximate
  solutions

Enumeration fails on real-world problems when the
number of alternative solutions is prohibitively
large
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Classical Algorithms

• Two main classes of classical algorithms
• Operate on complete solutions
• Evaluate partially constructed or approximate
  solutions
• Utilised when complete solution methods fail
• Decompose problem into simpler sub-problems and
  then reassemble
• Or solve complex problem in a series of simpler
  stages

Methods include Greedy, Dynamic programming, A,
Branch Bound and Divide Conquer.
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Summary of Traditional Methods

• Some problems can be solved by classical
  algorithms
• e.g. a quadratic evaluation function can be
  solved by gradient minimisation methods.
• Each algorithm is suited to a problem and may
  falter or fail on other problems
• Could use complete solutions
• (time-consuming and may get trapped in local
  optima )
• or partial solutions,
• (may be complex to implement and may not improve
  things)
• Now Consider Modern Heuristics

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Escaping Local Optima

• Tabu search and simulated annealing
• TS usually makes worse moves when stuck,
  whereas SA can at any time
• TS is deterministic, SA is stochastic
• Both work on complete solutions,

But require more parameters than classical
techniques How to choose these parameters? The
more sophisticated the method, the more human
judgment used!
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Contents

• Problems
• Basic Concepts Representation, objective,
  evaluation problem definition, neighbourhoods
  and optima
• Basic Problems Linear, SAT, TSP, NLP
  Constraint satisfaction problems
• Basic Techniques Exhaustive, Local Search
  Simplex method.
• Methods 1 Greedy, A, Branch Bound Dynamic
  programming, and Divide Conquer.
• Methods 2 Simulated Annealing, Tabu Search
• Methods 3 Evolutionary Approaches
• Constraint Handling Techniques
• Hybridise and Tune Practical tips
• Test are you sure that you have the best
  solution?

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Evolutionary Algorithms

• Previous algorithms, e.g. simulated annealing and
  greedy algorithm, all use single solutions.
• Evolutionary algorithms search utilising a
  population of solutions.
• Eight-bit SAT problem x1 x8
• s1 (0,1,0,0,0,1,1,1) 37
• s2 (1,0,0,0,0,1,1,1) 32
• s3 (1,1,1,0,0,1,1,1) 31
• s4 (1,1,0,1,0,1,1,1) 31
• s5 (1,1,0,0,1,1,1,1) 30
• s6 (1,1,0,0,0,0,1,1) 28
• s7 (1,1,0,0,0,1,0,1) 30
• s8 (1,1,0,0,0,1,1,0) 31

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Evolutionary Algorithms

• Competition between solutions is used to improve
  future solutions.
• Random variation is used to search the domain.
• Successive generations lead to the analogy of
  Evolution and Survival of the Fittest
• Biological terms offer illustration, but often
  are not a direct connection!
• Note
• Parallel processing can be used, but this is not
  unique as Hill climbing methods can be made
  parallel.

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Biological Illustration

• Darwinian Learning the fit survive, whether they
  want to or not.
• Baldwin effect That learning and other lifetime
  adaptations can effect the course of evolution
• Lamarckian evolution lifetime characteristics
  can be transferred to offspring.
• Therefore, can break natural laws!
• One or multiple parents
• Age, gender and mating can be controlled.

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Biological Illustration

• A solution is stored as a chromosome,
• With part solutions known as alleles.
• s1 (0,1,0,0,0,1,1,1) ? 01000111
• Phenotype an individuals expressed behaviour.
• Genotype an individuals genetic composition.
• The same genotype may have different phenotypic
  behaviours, e.g.
• 01000111 ? (0,1,0,0,0,1,1,1) in SAT
• 01000111 ? 71 in NLP
• Similarly, same phenotype may have different
  genotypes

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Steps to Evolution

• Select a problem domain then
• Create a population of individuals that represent
  potential solutions
• Evaluate the individuals
• Introduce some selective pressure to promote
  better individuals (or eliminate lesser quality
  individuals)
• Apply some variation operators to generate new
  solutions
• Repeat

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Steps to Evolution

• Procedure evolutionary algorithm
• Begin
• initialise P(t)
• evaluate P(t)
• while (not termination-condition) do
• begin
• select P(t) from P(t - 1)
• alter P(t)
• evaluate P(t)
• end
• End

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Ask the Correct Question

• You are standing in front of two doors. One
  leads to treasure, whilst the other will lead to
  something unpleasant.
• Two guards stand in front of the doors. One
  always tells the truth, whilst the other always
  lies. Naturally, you do not know who is who, but
  they know each other's true nature.
• Both the guards know which way is correct, but
  you can only ask one question to one of them
  before making your choice.
• What will you ask and which door do you go
  through?

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Mating

• Several methods can be applied to select parents
  to produce future offspring
• 1. Best
• Highest scoring individual(s) selected
• 2. Tournament selection
• A random subpopulation selected (20), then best
  selected from these.
• 3. Roulette wheel
• Probabilistically select based on fitness

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Mating

• Several methods can be applied to select parents
  to produce future offspring
• 1. Best
• Highest scoring individual(s) selected
• 2. Tournament selection
• A random subpopulation selected (20), then best
  selected from these.
• 3. Roulette wheel
• Probabilistically select based on fitness

MATE
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Mating

• Offspring could replace all parents!
• Elitism is where some parents survive,
• (? parents, ? offspring), shortened to (?,?)
• e.g. (30, 90) make a population of 120
• Crossover
• j1 00010001 j1 01010001
• j2 01110001 j2 00110001
• Mutation
• j1 10010101 j1 10010001

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Mating

• GA j1 00010001 Crossover
• j2 01110001
• Select crossover point

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Mating

• GA j1 00010001 Crossover
• j2 01110001

21
Mating

• GA j1 00010001 Crossover
• j2 01110001
• Applying crossover

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Mating

• GA j1 01010001
• j2 00110001
• Crossover complete.

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Mating

• GA j1 10010101 Mutation
• Randomly select bit to mutate

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Mating

• GA j1 10010001
• Mutation complete.

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Single-objective
xxx
Population will eventually converge to single
solution, which may be the global optimum
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Multi-objective
x x x x
Multiple solutions can be forced using additional
methods, e.g. fitness sharing, Again no guarantee
of finding optimum cf http//www.ie.ncsu.edu/gaot
/
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Types of Evolutionary Algorithm

• Genetic Algorithms
• Optimise numeric solution of fitness function.
• Evolutionary Systems
• Optimise the solution based on a behavioural
  instead of genetic level.
• Learning Classifier Systems
• Optimise the co-operation of rules for solving
  an input/output fitness function.
• Genetic Programming
• Optimise the interaction of code to solve a
  programming function.
• Existing since 1950s in various forms,
  interchanged methods and no clear boundaries
• EAs population based, random selection and
  random variation.

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Summary

• Understanding of problem domain is essential.
• Selection of representation based on problem
  domain.
• Crossover and mutation are common genetic
  operators.
• Care required in setting up EA
• Optimum, optimal and locally-optimal solutions
  can all be produced.