Basics of problem Solving-Evaluation Function

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MAE 552 Heuristic Optimization Lecture
8 February 8, 2002
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http//www.statslab.cam.ac.uk/richard/tmp/mcp/jav
a/ANNEAL/annealing.html
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• Start with a ball at point A. Shake it up and
  it might jump out of A and into B.
• Give it another shake (adding energy) and it
  might go to C.
• This is the general idea behind SAs.

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The SA Algorithm
T0 m10, m20, m30, m40, mm0 T1 m11,
m21, m31, m41, mm1 T2 m12, m22,
m32, m42, mm2 T3 m13, m23, m33,
m43, mm3 T4 m14, m24, m34, m44,
mm4 T5 m15, m25, m35, m45,
mm5 .. Tn m1n, m2n, m3n, m4n,
mmn nnumber of levels in cooling
schedule mnumber of transitions in each Markov
chain
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Simulated Annealing Parts of the SA

• The following musty be specified in implementing
  SA
• 1. An unambiguous description for the objective
  function f (analogous to energy) and possible
  constraints.
• 2. A clear representation of the design vector
  (analogous to the configuration of a solid) over
  which an optimum is sought.
• 3. A cooling schedule this includes the
  starting value of the control parameter, To, and
  rules to determine when the current value of the
  control parameter should be reduced and by how
  much (the decrement rule) and a stopping
  criterion to determine when the optimization
  process should be terminated.

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Simulated Annealing Parts of the SA
4. A move set generator which generates
candidate points. 5. An acceptance criterion
which decides whether or not a new move is
accepted. 4 and 5 together are called a
transition mechanism which results in the
transformation of a current state into a
subsequent one.
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Simulated Annealing Cooling Schedule

• SA generates a series of points towards the
  optimum as it proceeds
• X0, X1, X2, X3.
• With corresponding function values
• f(X0), f(X1), f(X2), f(X3).
• Because of the stochastic nature of SA, the
  sequence of the fs is random and not monotonic.
• However it does drift towards the optimum because
  of the gradual reduction in the control
  parameter.

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Cooling Schedules

• A cooling schedule is used to achieve convergence
  to a global optimum in function optimization.
• Cooling schedule describes how control parameter
  T changes during optimization process.
• First let us look at the concept of acceptance
  ratio, X(Tk).
• X(Tk) ( of Accepted Moves / of Attempted
  Moves)
• If T is large almost all moves are accepted
• X(Tk)-gt1
• As T decreases
• X(Tk)-gt1
• For maximum efficiency, it is important to set
  the proper value of To.

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Simulated Annealing Cooling Schedule

• 3 Parts in a cooling schedule
• 1. Choose the starting value of the control
  parameter, T0.
• It should be large enough to melt the objective
  function, to leap over all peaks.
• This is accomplished by ensuring that the initial
  X(T0) is close to 1.0 (most random moves are
  accepted).
• Start the SA Algorithm
• At some T0 and execute for some number of
  transitions and check X(T0).
• If not close to 1.0 multiply Tk by a factor
  greater than 1.0 and execute again.
• Repeat until X(T0) close to 1.0.

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Simulated Annealing Cooling Schedule

• 2. The decrement rule.
• Two parts to this - the time when the control
  parameter reduction should occur and the rate by
  which it should be reduced.
• If using fixed length Markov Chains of fixed
  length, that is once the total number of
  attempted moves at each value of the control
  parameter (i.e. inner loop) reaches a
  predetermined value, it is time to reduce the
  control parameter.
• A frequently used decrement function is
• Tk1rTk k0,1,2,........
• r control parameter reduction coefficient.
• Generally this is a constant between.8 and .99.

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Simulated Annealing Cooling Schedule

• rt can also be set based on the problem size and
  characteristics.
• SPEARS set r 1/(Num_dvsk)
• k current step in the cooling schedule
• All settings of Simulated Annealing will entail a
  tradeoff between searching thoroughly at a
  particular level of T and the number of steps in
  the cooling schedule.

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The SA Algorithm
Increase the number of transitions in each Markov
Chain
Number of steps in the Cooling Schedule
T0 m10, m20, m30, m40, mm0 T1 m11,
m21, m31, m41, mm1 T2 m12, m22,
m32, m42, mm2 T3 m13, m23, m33,
m43, mm3 T4 m14, m24, m34, m44,
mm4 T5 m15, m25, m35, m45,
mm5 nnumber of levels in cooling
schedule mnumber of transitions in each Markov
chain
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Simulated Annealing Cooling Schedule

• No matter how sophisticated the decrement rule -
  important to reach a balance between rapid
  decrement of the control parameter and short
  length of Markov Chains.
• 3. Stopping criterion.
• Rule of thumb if the improvement in objective
  function after a period of execution remains
  fairly constant then stop the algorithm.
• If the last configuration of several consecutive
  inner loops have been very close to each other
  then it is time to stop

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Simulated Annealing Transition Mechanism

• A transition mechanism transforms a current state
  into a subsequent one. It consists of two parts
  
• (a) move set generator and
• (b) an acceptance criterion

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Simulated Annealing Move Set Generator

• a move set generator
• Generates a random point X from the neighborhood
  of xc.
• Its move (step) generation depends on the data
  type and the corresponding value of the control
  parameter Tk.
• For high values of Tk, almost all attempted moves
  are accepted and it is inefficient to use a small
  neighborhood because it will cause slow progress
  of the algorithm.
• On the contrary, for small values of Tk, more
  attempted moves are rejected if a neighborhood is
  used.
• The size of the move should decrease as the
  control parameter is reduced. This improves
  computational efficiency.

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Simulated Annealing Move Set Generator

• Large Value of T, large neighborhood.

x2
vc
x1
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Simulated Annealing Move Set Generator

• Small Value of T, small neighborhood.

x2
g1
x1
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Simulated Annealing Move Set Generator

• Gaussian Neighborhoods

Choose a candidate from the neighborhood based
on a gaussian distribution.
x2
g1
x1
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Simulated Annealing Move Set Generator

• Depending on the type of representation
  controlling the size of the neighborhood is going
  to entail different things.
• For the SAT problem the representation is a
  string on binary numbers TRUE, FALSE
• A one-flip neighborhood is defined as all of the
  points that could be arrived at by flipping one
  of the bits.
• X010111100011-gtX110111100011
• Two-flip neighborhood
• X010111100011-gtX100111100011
• Less than one-flip neighborhood
• X010111100011-gtX100111100011

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Simulated Annealing Move Set Generator

• For the NLP there are an infinite choice of move
  directions and magnitudes.
• One approach is to generate a random move each
  time along a single design variable keeping all
  others constant.
• Xcx1,x2,x3,x4-gt Xnx1new,x2,x3,x4
• Another approach is to change all design
  variables simultaneously.
• Xcx1,x2,x3,x4-gt Xnx1new, x2new, x3new,
  x4new

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Simulated Annealing - Constraint Handling

• Exterior Penalty Function
• Where rp generally starts small and is gradually
  increased to ensure feasibility.
• Interior Penalty Function
• Here rp for the second term is the same as before
  but for the
• first terms it starts large and is gradually
  decreased.