Local Search and Optimization Presented by Collin Kanaley

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Local Search and OptimizationPresented by
Collin Kanaley
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Local Search Algorithms and Optimization Problems
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Local Search Algorithms

• -Local search algorithms are useful when the path
  to the goal does not matter for example, in the
  eight-queens problem, what matters is the
  configuration of the queens, not the order in
  which they are added to the board.
• -This class of problems includes many important
  applications such as integrated-circuit design,
  factory-floor layout, job-shop scheduling,
  automatic programming, telecommunications network
  optimization, vehicle routing, and portfolio
  management.

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• -Local search algorithms operate using a single
  current state (rather than multiple paths)?
• -Paths followed are typically not retained
• -Local search algorithms have two key advantages
• 1. They use very little memory
• 2. They can often find reasonable solutions in
  large or infinite state spaces for which
  systematic algorithms are not suitable
• -Local search algorithms are also useful for
  solving pure optimization problems, which aim to
  find the best state according to an objective
  function

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• -The state space landscape is useful for
  understanding local search a landscape has both
  location (defined by the state) and elevation
  (defined by the value of the heuristic cost
  function or objective function)?

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State Space Landscape (continued)?

• -If elevation corresponds to cost, then the aim
  is to find the lowest valley, called a global
  minimum
• -If elevation corresponds to an objective
  function, then the aim is to find the highest
  peak, called a global maximum

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• State Space Landscape (continued)?
• In a state space landscape
• -A complete local search algorithm always
  finds a goal if one exists
• -An optimal algorithm always finds a global
  maximum/minimum

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Hill-Climbing Search

• -The hill-climbing search is a loop that
  continually moves uphill in the direction of
  increasing value
• -It terminates when it reaches a peak where no
  neighbour has a higher value
• -It does not maintain a search tree
• -This algorithm does not look beyond the
  immediate neighbours of the current state
• -Hill climbing is sometimes called greedy local
  search because it grabs a good neighbour state
  without thinking ahead about where to go next

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• -While hill-climbing searches often perform quite
  well, they also often get stuck due to
• 1. Local maxima a peak that is higher than
  each of its neighbouring states, but lower
  than the global maximum. Hill-climbing
  algorithms that reach the vicinity of a local
  maximum will be drawn upwards towards the
  peak, but then be stuck with now where else to
  go.
• 2. Ridges a sequence of local maxima
• 3. Plateaux an area of the state space
  landscape where the evaluation function is
  flat

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Hill-climbing variations

• -Stochastic hill climbing chooses at random from
  among the uphill moves the probability of
  selection can vary with the steepness of the
  uphill move
• -First-choice hill climbing implements stochastic
  hill climbing by generating successors randomly
  until one is generated that is better than the
  current state. This is good when a state has
  many (e.g., thousands) of successors
• -Random-restart hill climbing conducts a series
  of hill-climbing searches from randomly generated
  initial states, stopping when a goal is found
• -Simulated annealing combines hill climbing with
  a random walk this yields both efficiency and
  completeness

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Local Beam Search
-Local beam search - this algorithm keeps track
of multiple states as opposed to just one. It
begins with k randomly generated states. At each
step, all the successors of all k states are
generated, and if any is a goal the algorithm
halts otherwise it selects the k best successors
from the complete list and repeats. This is
different from a random-restart search in that
useful information is passed among the k parallel
search threads. Therefore, unfruitful searches
are quickly abandoned and resources are moved to
where the most progress is being made
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Stochastic beam search

• -Stochastic beam search this variant of the
  local beam search chooses k successors at random,
  as opposed to choosing k from a pool of candidate
  successors, with the probability of choosing a
  given successor being an increasing function of
  its value.
• -This helps alleviate the problem local beam
  search algorithms can have when they become too
  concentrated in a small region of the state space

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Genetic Algorithms

• -A genetic algorithm is a variant of stochastic
  beam search in which successor states are
  generated by combining two parent states, rather
  than by just modifying a single state.

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• Genetic Algorithms (continued)?
• -Genetic algorithm's begin with a set of k
  randomly generated states, called the population
• -Each state, or individual, is represented as a
  string over a finite alphabet (most commonly a
  string of 0s and 1s)?
• -Each state is evaluated by the fitness function,
  which rates better states more highly

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Local Search in Continuous Spaces
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Local Search in Continuous Spaces

• None of the previously described algorithms can
  handle continuous state spaces. Introduced here
  are some local search techniques for finding
  optimal solutions in continuous spaces.
  Basically, anything that deals with the real
  world is in such a space.

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• Problems from local maxima, ridges, and plateaux
  are just as prevalent in continuous state spaces
  as they are in local search methods.

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• -One way to avoid continuous problems is to
  simply discretize the neighbourhood of each
  state. This means changing continuous models
  into discrete models.
• -The gradient of the landscape can be used to
  find a maximum
• -When the objective function is not available in
  a differential form at all, an empirical gradient
  can be determined by evaluating the response to
  small increments and decrements in each
  coordinate. (Empirical gradient search is the
  same as steepest-ascent hill climbing in a
  discretized version of the state space.)?

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Newton-Raphson method

• -Newton-Raphson method for may problems this
  algorithm is the most effective this is a
  general technique for finding roots of functions,
  a.k.a. Solving equations of the form g(x)0

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• -The Newton-Raphson method works by computing a
  new estimate for the root x according to Newton's
  formula
• x lt- x g(x)/g1(x)?
• -x must be found so that the gradient is zero.
  Thus g(x) in Newton's formula becomes ?f(x) and
  the updated equation can be written as
• x lt- x Hf-1(x)?f(x)?
• where Hf (x) is the Hessian matrix of second
  derivatives

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Constrained optimization

• -An optimization problem is constrained if
  solutions must satisfy some hard constraints on
  the values of each variable.
• -The difficulty of constrained optimization
  problems depends upon the nature of the
  constraints and the objective function

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Sources

• -Textbook
• -Wikipedia