G5BAIM Artificial Intelligence Methods

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G5BAIMArtificial Intelligence Methods
Andrew Parkes (ajp)

• DRAFT Examples of neighbourhoods

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NOTE (Feb 15th 2007)

• This list is currently partial!
• As the course proceeds I will add more

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AIM

• In practical implementations of optimization, it
  is crucial to design and use good neighbourhoods
• These slides just give some ideas of the
  neighbourhoods that might be used in the standard
  problem domains
• these are used for local search algorithms, e.g.
  hillclimbing, simulated annealing, tabu search,
• It is not intended to be a complete list! Just to
  give some basic ideas

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Function Optimisation

• Variables x1,,xn
• Real-valued (not necessarily integers)
• Objective value f(x1,,xn)
• where f is some supplied function
• Task
• Find values of x1,,xn that maximise f
• subject to
• constraints on the x1,,xn

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Function Optimisation

• Typical Neighbourhoods
• a ball or radius e around the current point
• x x0 lt e
• or a cube
• x(i) x0(i) lt e forall i

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Exercise

• What might happen if e is made large?

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Discrete Optimisation Binary Variables

• Variables are binary (either 0 or 1 true or
  false)
• Subset Sum
• (Binary) Integer Programming
• (Propositional) Satisfiability

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Exercise

• If given a new problem and decide want to use
  local search then the first thing is to imagine
  and then implement a variety of neighbourhoods to
  try out.
• What neighbourhood(s) might you use for problems
  with binary variables, e.g.
• subset sum
• BIP
• SAT
• Neighbourhoods form the basis of many search
  algorithms, e.g. simulated annealing and tabu
  search

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Binary Variables Neighbourhoods

• Simple Flip
• swap 0 for 1, or vice versa
• for just one variable
• In subset sum this corresponds to moving one
  number either into or out of the subset
• In SAT this corresponds to flipping the truth
  value of one of the boolean variables
• Larger neighbourhoods were considered but did not
  become popular

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Travelling Salesman Problem (TSP)
(some edge lengths missing for clarity)

• Tour permutation of cities e.g. ABCFED(A)
• Length is sum of edges e.g. 211413
• Task Find the shortest tour of the cities

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TSP 2-OPT

• To be completed later

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Graph Colouring Nbds

• Coursework ONE !!

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Expectations

• Become familiar with some neighbourhood on the
  standard problems
• be able to discuss them in the lectures
• be able to discuss and use them in the coursework
  and exam
• be able to identify when they get stuck in local
  minima