Searching and optimization

1
Chapter 13

• Searching and optimization

2
Topics

• Applications and techniques
• Branch-and-bound search
• Genetic algorithms
• Successive refinement
• Hill climbing

3
Applications and techniques

• Characterized by looking for a solution to a
  problem that has many potential solutions
• An exhaustive search is not feasible for many
  problems
• Directed search is used instead
• May not necessarily search for the optimal (best)
  solution
• Non-optimal solutions may be acceptable

4
Applications and techniques

• Examples of problems
• Travelling salesperson problem
• 0/1 knapsack problem
• n-queens problem
• 8-puzzle
• 15-puzzle
• All of these have an extremely large number of
  permutations
• Searches can be very time-consuming

5
Applications and techniques

• Examples of real world applications
• Financial forecasting
• Airline fleet and crew assignment
• VLSI chip layout
• Examples of search and optimization techniques
• Branch-and-bound search
• Dynamic programming
• Hill climbing
• Simulated annealing
• Genetic algorithms

6
Branch-and-bound search

• Uses a state space tree
• Root node is the starting point
• Transition from one level to the next represents
  a choice being made towards the solution
• Can be considered as dividing the problem into
  sub-problems
• Each node represents a sub-problem
• How the tree is constructed depends on the
  problem
• Called a dynamic tree
• If the choice at each node just consists of
  selecting either true or false, a binary tree is
  created
• Called a static tree

7
Branch-and-bound search

• The state space tree can be explored using
  various methods
• Depth-first search
• Breath-first search
• Best-first search
• Searching all nodes is expensive
• Consider only exploring paths that will likely
  lead to the best solution (i.e., best-first
  search)
• Avoiding unpromising paths prunes the state space
  tree

8
Branch-and-bound search

• A bounding (or cut-off) function may be used to
  determine whether or not a given path is
  promising
• When a node is reached, the bounding function
  produces an upper or lower bound
• This can be compared to the value of the best
  path that has been found so far
• Determine if it is worth it to continue searching
  along this path or not

9
Branch-and-bound search

• Terminology
• Live node A node that has been reached, but not
  all of its children have been explored
• E-node (Expanded node.) A live node in which
  its children are currently being explored
• Dead node A node where all of its children have
  been explored
• As the search proceeds, a list of live nodes is
  maintained in a queue
• Called an open list
• A list of dead nodes may also be maintained
• Called a closed list
• Used to check for duplicates

10
Branch-and-bound search

• Parallelization
• Simplest approach...allocate a chuck of the state
  space tree to each processor and let them search
  their portion independently
• Possible to parallelize the evaluation of the
  bounding function
• However, there are some issues

11
Branch-and-bound search

• The size of the state space tree may not be known
  in advance
• Issue of load balancing
• The current lower or upper bound (generated by
  the bounding function) should be known by all
  processors
• Must be updated so that all processors can prune
  their state space tree quickly

12
Branch-and-bound search

• Another approach is to allow the open list queue
  to be accessed concurrently (i.e., in shared
  memory)
• When a processor chooses an item from the queue,
  that item must be locked so that another
  processor doesnt choose it as well
• However, this will limit the potential speedup

13
Branch-and-bound search

• Speedup anomalies
• Acceleration anomaly Super-linear speedup can
  be achieved if one processor finds a solution
  extremely quickly
• Deceleration anomaly A solution may be
  positioned such that it cannot be found in 1/p of
  the sequential time
• Detrimental anomaly There is evidence that
  shows that the speedup factor could be less than
  1

14
Genetic algorithms

• Attempts to simulate the natural evolution of a
  population of individuals
• Chromosomes contain information that
  characterizes a living being
• Evolution works at the chromosome level
• The chromosomes of offspring are generated from
  the chromosomes of the offsprings parents
• This blending is called crossover
• Sometimes, there may be a random change an
  individuals chromosome pattern
• This is called mutation

15
Genetic algorithms

• With a low mutation rate, there is a tendency for
  fit individuals to pass on their traits to the
  next generation
• Unfit individuals tend to die out
• With a high mutation rate, individuals of the
  next generation are very randomly different from
  the individuals of the previous generation
• Can cause instability or non-convergence

16
Genetic algorithms

• Basically, you have a population of solutions.
  Crossover and mutation occur. A new generation
  of solutions is produced and this process repeats
• Algorithm
• Create an initial population of solutions
  (individuals)
• Evaluate each solution to determine how fit
  they are
• Characterize the solutions from most fit to least
  fit
• Select a subset of the population (favouring more
  fit solutions)
• Use the subset to produce a new generation of
  offspring (using crossover)
• Apply random mutations to some of offspring
• Repeat this process until some termination
  condition is satisfied

17
Genetic algorithms

• Representation of an individual (their
  chromosomes)
• Commonly, binary strings (e.g., 000101011110)
• More recently, floating points, gray codes, and
  integer strings
• Bits in the binary string have some sort of
  meaning
• For example, the first bit might represent the
  sign of a number and the remaining bits might
  represent the number itself
• An initial population is generated using a random
  number generator
• Individuals are evaluated according to some
  function
• Could produce a number, with higher numbers
  representing that an individual is very fit

18
Genetic algorithms

• Constraints might be placed on a solution
• For example, if an individual falls outside of a
  range of values, it could be discarded or
  repaired by changing some bits
• The number of individuals in a population
  affects
• How quickly a good solution will be found
• How much computation needs to be done per
  generation

19
Genetic algorithms

• Selection process
• In nature, there is a bias towards the most fit
  individuals
• However, there may be problems where there are
  many local optimum solutions
• Just selecting the most fit individuals could
  gather around a single local optimum rather than
  the global optimum
• One approach is tournament selection
• A set of individuals from the population is
  selected
• The most fit individual wins the tournament is
  selected as a parent to generate offspring
• Many of these tournaments are played until the
  whole population has participated

20
Genetic algorithms

• The individuals chosen by the selection process
  are paired up as parents and produce offspring,
  using crossover and mutation
• Single-point crossover
• Given two parents, A and B, their chromosomes are
  split into two at some boundary
• The rightmost portions of the parents
  chromosomes are then swapped to generate two
  children
• Other types of crossover Multi-point crossover,
  uniform crossover

21
Genetic algorithms

• Mutation
• Generally, the mutation rate is kept small
• Mutation occurs by simply changing one or more
  bits in a childs chromosome
• Other variations that can be incorporated into
  the algorithm
• Carry over some of the most fit individuals from
  the previous generation into the next generation
• Randomly generate new individuals for the next
  generation
• Vary the population size from one generation to
  the next

22
Genetic algorithms

• Possible termination conditions
• Stop after a number of generations
• Cannot tell if you have an optimal solution or if
  the population has moved far away from it
• Consider the degree of improvement from one
  generation to the next
• Consider the similarity of all the individuals in
  the population

23
Genetic algorithms

• Two approaches to parallelization
• Let each processor control its own population and
  allow some migration to occur between populations
• Let each processor do a portion of each step of
  the algorithm on a shared population

24
Genetic algorithms

• Isolated subpopulations
• After a certain number of generations, the
  processors share their best individuals
• Immigrants must be selected, send, received, then
  integrated into the population
• Sending and receiving is done easily with message
  passing
• There are a couple models of migration to
  consider
• The island models allow individuals to move to
  any other subpopulation
• Models nature better, but requires more
  communication
• The stepping-stone model only allows moving
  between neighbouring populations

25
Genetic algorithms

• Common population
• The steps of the algorithm can easily be
  parallelized, but under a message passing system,
  there would be a huge amount of communication
• This is more feasible with shared memory

26
Successive refinement

• Related to solving problems like finding the
  maximum value of a function
• Consider a graph of a function
• A grid is placed with some amount of spacing
  between grid lines
• At each of these grid lines, a point is examined
• The k best points are kept
• A finer grid spacing is then used
• Around each of the chosen best points, points on
  the grid lines are examined
• The best points are kept and the process repeats
  until some condition is met

27
Successive refinement

• This is simple, but requires more computations
  than genetic algorithms
• Though, still much better than an exhaustive
  search
• Parallelization is straightforward
• Divide up the points to be examined amongst all
  the processors
• Each processor sends its best points to the
  master processor
• The master processor determines which of those
  points are the best, then distributes them to the
  slave processors
• The points to be examined are divided up again
  and the process repeats

28
Hill climbing

• Consider a blindfolded hiker placed in a terrain
  having many local maximums
• The hikers task is to find the global maximum
• Simple algorithm the hiker uses
• Dont go downhill
• If the hiker always moves uphill, it will always
  reach the top of a maximum
• However, if there are many local maximums, its
  unlikely the hiker will stop on the global maximum

29
Hill climbing

• So...randomly place thousands of blindfolded
  hikers on the terrain
• Still no guarantee, but this increases the
  chances that one of them will stop on the global
  maximum
• Hill climbing is a Monte Carlo search technique
• Again, this relates to problems like finding the
  global maximum of a function

30
Hill climbing

• Parallelization
• The area can be divided up amongst all the
  processors
• Each processors generates a random number of
  starting points and moves all of those points
• Each processor reports the maximum they found to
  the master processor
• The master processor returns the largest maximum
  from the findings

31
Hill climbing

• Some hikers might take longer than others to
  reach a maximum
• A tiny hill versus a steep mountain
• Some processors might finish earlier than others
• A work-pool approach considers this
• Divide the area up into even smaller segments
• As processors finish a segment, they report their
  maximum to the master processor and choose
  another segment from the work-pool