Note to 4th year students

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Note to 4th year students

• students interested in doing masters degree and
  those who intend to apply for OGS/NSERC
  scholarships should complete the required forms
  and apply before October 19. The forms can be
  obtained through the research office website.

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Optimization Problems

• searching a space when paths dont matter

3
Local search algorithms

• If paths dont matter, algorithms are able to
  jump from state to state (ie not follow edges)
• Example
• the n queens problem
• place n chess queens, ngt3, on an n x n chess
  board so no queen threatens another according to
  chess rules

4
Optimisation problems

• find maximum value of a function over a parameter
  state space
• e.g., one-dimensional max f(x) over x

f(x)
x
5
Optimisation problems

• e.g., 2-dimensional max f(x,y) over x,y

y
f(x,y)
x
6
Algorithms

• hill-climbing (greedy best-first dfs)
• weaknesses
• local optima
• ridges
• plateaux
• variations on hill-climbing to avoid the traps

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Hill climbing
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Getting outside the local area

• variations on hill-climbing
• choosing a successor that may not be the optimal
  (escape by path)
• random restart (escape by jump)
• simulated annealing

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Getting outside the local area

• choosing a successor that may not be the optimal
  (escape by path)

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Getting outside the local area

• random restart

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Getting outside the local area

• simulated annealing
• random move is generated, probability of moving
  is based on change in value

f(x)
x
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Simulated annealing

• Probability of move function
• Probability of making bad move decreases with time

probability of moving

-
13
Avoiding paths altogether

• genetic algorithms
• pick set of states randomly
• order states by fitness
• create new set of states by combining state
  variables of most fit
• make a few random changes to state variables
• go to 2

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Genetic algorithm example

• Guessing a 32 bit sequence
• fitness function number of matching bits
• (hill-climbing would be better!)

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Genetic algorithm example

• population size 4
• first generation random fitness
• 0110 1010 1011 0110 0110 1010 1010 1110 14
• 1100 1101 0110 0101 1101 0010 0000 1010 15
• 1101 0110 1011 1010 1001 1010 1010 1110 19
• 0010 1101 1000 0111 0010 0110 1001 1001 13
• order 3,2,1,4
• cross 3 x 2 and 3 x 1 for next generation

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Genetic algorithm example

• crossing 3 x 2
• pick random cut point after 9th
• 1100 1101 0110 0101 1101 0010 0000 1010
• 1101 0110 1011 1010 1001 1010 1010 1110
• recombine crossed pieces
• 1100 1101 0011 1010 1001 1010 1010 1110 21
• 1101 0110 1110 0101 1101 0010 0000 1010 13
• crossing 3 x 1 (cut after 21)
• 1101 0110 1011 1010 1001 1010 1010 1110 18
• 0110 1010 1011 0110 0110 1010 1010 1110 15
• (4 potential new sequences)

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Genetic algorithm example

• next generation
• 1100 1101 0110 0101 1101 0010 0000 1010 15
• 1101 0110 1011 1010 1001 1010 1010 1110 19
• 1100 1101 0011 1010 1001 1010 1010 1110 21
• 1101 0110 1011 1010 1001 1010 1010 1110 18
• (2 best new combinations replace 2 worst from
  original population)
• repeat