MonteCarlo Optimization Simulated Annealing

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MonteCarlo Optimization(Simulated Annealing)

• Mathematical Biology Lecture 6
• James A. Glazier

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Optimization

• Other Major Application of Monte Carlo Methods is
  to Find the Optimal (or a nearly Optimal)
  Solution of an Algorithmically Hard Problem.
• Given want to find that minimizes
  f.
• Definition
• then is a Local Minimum of f.
• Definition
• then is the Global Minimum of f.
• Definition
• then f has Multiple Degenerate Global Minima,

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Energy Surfaces

• The Number and Shape of Local Minima of f
  Determine the Texture of the Energy Surface,
  also called the Energy Landscape, Penalty
  Landscape or Optimization Landscape.
• Definition The Basin of Attraction of a Local
  Minimum, is
• The Depth of the Basin of Attraction is
• The Radius or Size of the Basin of Attraction is
• If no local minima except global minimum, then
  optimization is easy and energy surface is Smooth.

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Energy Surfaces

• If multiple local minima with large basins of
  attraction, need to pick an
• In each basin, find the corresponding and
  pick the best. Corresponds to enumerating all
  states if is a finite set, e.g. TSP.
• If many local minima or minima have small basins
  of attraction, then the energy surface is Rough
  and Optimization is Difficult.
• In these cases cannot find global minimum.
  However, often, only need a pretty good
  solution.

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Monte Carlo Optimization

• Deterministic Methods, e.g. Newton-Rabson
• ( ) Only Move Towards Better Solutions and Trap
  in Basins of Attraction. Need to Move the Wrong
  Way Sometimes to Escape Basins of Attraction
  (also Called Traps).
• Algorithm
• Choose a
• Start at Propose a Move to
• If
• If
• Where,

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Monte Carlo OptimizationIssues

• Given an infinite time, the pseudo-random walk
• Will explore all of Phase Space.
• However, you never know when you have reached the
  global minimum! So dont know when to Stop.
• Can also take a very long time to escape from
  Deep Local Basins of Attraction.
• Optimal Choice of g(x) and d will depend on the
  particular
• If g(x)?1 for xltx0, then will algorithm will not
  see minima with depths less than x0 .
• A standard Choice is the Boltzmann Distribution,
• g(x)e-x/T,
• Where T is the Fluctuation Temperature.
• The Boltzmann Distribution has right equilibrium
  thermodynamics, but is NOT an essential choice in
  this application).

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Temperature and d

• Bigger T results in more frequent unfavorable
  moves.
• In general, the time spent in a Basin of
  Attraction is exp(Depth of Basin/ T).
• An algorithm with these Kinetics is Called an
  Activated Process.
• Bigger T are Good for Moving rapidly Between
  Large and Deep Basins of Attraction but Ignore
  Subtle (less than T) Changes in
• Similarly, large d move faster, but can miss deep
  minima with small diameter basins of attraction.
• A strategy for picking T is called an Annealing
  Schedule.

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Annealing Schedules

• Ideally, want time in all local minimum basins to
  be small and time in global minimum basin to be
  nearly infinite.
• A fixed value of T Works if depth of the basin of
  attraction of global minimumgtgtdepth of the basin
  of attraction of all local minima and radius of
  the basin of attraction of global minimumradius
  of the largest basin of attraction among all
  local minima.
• If so, pick T between these two depths.
• If multiple local minima almost degenerate with
  global minimum, then cant distinguish, but
  answer is almost optimal.
• If have a deep global minimum with very small
  basin of attraction (golf-course energy). Then
  no method helps!

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Annealing Schedules

• If Energy Landscape is Hierarchical or Fractal,
  then start with large T and gradually reduce t.
• Selects first among large, deep basins, then
  successively smaller and shallower ones until it
  freezes in one.
• Called Simulated Annealing.
• No optimal choice of Ts.
• Generally Good Strategy
• Start with T Df/2, if you know typical values of
  Df for a fixed stepsize d, or T typical f, if
  you do not.
• Run until typical Df ltltT. Then set TT/2. Repeat.
• Repeat for many initial conditions.
• Take best solution.

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ExampleThe Traveling Salesman Problem

• Simulated Annealing Method Works for
  Algorithmically Hard (NP Complete) problems like
  the Traveling Salesman problem.
• Put down N points in some space
• Define an Itinerary
• The Penalty Function or Energy or Hamiltonian is
  the Total Path Length
• for a Given Itinerary

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ExampleThe TSP (Contd.)

• Pick any Initial Itinerary.
• At each Monte Carlo Step, pick
• If the initial Itinerary is
• Then the Trial Itinerary is the Permutation
• Then
• Apply the Metropolis Algorithm.
• A Good Initial Choice of T is
• This Algorithm Works Well, Giving a Permutation
  with H within a Percent or Better if the Global
  Optimum in a Reasonable Amount of Time.