Diapositiva 1

1
Clase 3 Heurísticas Ascenso de Colina y
Recocido Simulado
Gabriela Ochoa http//www.ldc.usb.ve/gabro/
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Fitness Landscape (Paisaje Adaptativo)

• Can envisage population with n traits as
  existing in a n1-dimensional space (landscape)
  with height corresponding to fitness
• Each different individual (phenotype) represents
  a single point on the landscape
• Population is therefore a cloud of points,
  moving on the landscape over time as it evolves
  - adaptation

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Fitness Landscape (2 traits)
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Métodos de Ascenso de Colina - 1

• Usan una técnica de mejoramiento iterativo
• Comienzan a partir de un punto (punto actual) en
  el espacio de búsqueda
• En cada iteración, un nuevo punto es seleccionado
  de la vecindad del punto actual
• Si el nuevo punto es mejor, se transforma en em
  punto actual, sino otro punto vecino es
  seleccionado y evaluado
• El método termina cuando no hay mejorías, o
  cuando se alcanza un numero predefinido de
  iteraciones

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Hillclimbing Methods - 2

• May converge to local optima
• usually have to start search from various
  starting points
• Initial starting points may be chosen,
• randomly
• according to some regular pattern
• based on other information (e.g. results of a
  prior search)

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Hillclimbing Methods - 3

• Variations of hillclimbing algorithms differ in
  the way a new string is selected for comparisons
  with the current string
• One version of simple (iterated) hillclimbing
  method is the steepest ascent hillclimbing

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Hillclimbing Methods - 4

• Example problem
• The search space is a set of binary strings v of
  length 30
• The objective function f (to be maximized)
• f(v)11one(v)-150
• where one(v) returns the number of ones in v.
• e.g. v1(110111101111011101101111010101)
• f(v1) 1122 - 150 92

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Hillclimbing Methods - 5

• procedure iterated hillclimber
• begin
• t ? 0
• repeat
• local ? FALSE
• select a curent string vc at random
• evaluate vc
• repeat
• form 30 new strings in the neigborhood of
  vc by
• flipping single bits of vc
• select vn from the set of new strings with
  the
• largest value of the objective function f
• if f(vc) lt f(vn) then vc ? vn
• else local ? TRUE
• until local
• t ? t1
• until tMAX
• end

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Hillclimbing Methods - 6

• success/failure of each iteration depends on
  starting point
• success defined as returning a local or a global
  optimum
• in problems with many local optima a global
  optimum may not be found

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Hillclimbing Methods - 7

• Weaknesses
• Usually terminate at solutions that are local
  optima
• No information as to how much the discovered
  local optimum deviates from the global (or even
  other local optima)
• Obtained optimum depends on starting point
• Usually no upper bound on computation time

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Hillclimbing Methods - 8

• Advantages
• Very easy to apply (only a representation, the
  evaluation function and a measure that defines
  the neigborhood around a point is needed)

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Search Techniques Revisited - 1

• Effective search techniques provide a mechanism
  to balance exploration and exploitation
• exploiting the best solutions found so far
• exploring the search space

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Search Techniques Revisited - 2

• Hillclimbing methods exploit the best available
  solution for possible improvement but neglect
  exploring a large portion of the search space
• Random search (points in the search space are
  sampled with equal probability) explores the
  search space thoroughly but misses exploiting
  promising regions.

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Search Techniques Revisited - 3

• Aim is to design search algorithms that can
• escape local optima
• balance exploration and exploitation
• make the search independent from the initial
  configuration

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Simulated Annealing - 1

• derived from statistical mechanics
• based on analogy between annealing of solids and
  the solving of combinatorial optimization problems

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Simulated Annealing - 2

• annealing the physical process of heating up a
  solid and then cooling it down until it
  crystallizes
• at higher temperatures atoms have higher energies
  and more freedom to arrange themselves
• as temperature is decreased atomic energies
  decrease

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Simulated Annealing - 3

• a crystal with regular structure is obtained at
  minimum energy state
• rapid quenching very rapid cooling which causes
  widespread irregularities in the crystal
  structure and system does not reach minimum
  energy state

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Simulated Annealing - 4

• the states of the solid represent feasible
  solutions of the optimization problem
• the energies of the states correspond to values
  of the objective function calculated at those
  states
• minimum energy state corresponds to optimal
  solution
• rapid quenching corresponds to converging to
  local optima

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Simulated Annealing - 5

• Four principle choices to make
• representation of solutions
• definition of cost function
• definition of generation function for neighbors
  (can be random)
• designing a cooling schedule four parameters
  must be specified
• initial temperature, temperature update rule,
  number of iterations to be performed at each
  temperature, stopping criteria

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Simulated Annealing - 6

• procedure simulated annealing
• begin
• create an initial solution
• repeat
• evaluate solution
• if accepted then update current solution
• if time to change temperature then
• decrease temperature
• if stopping-criteria NOT satisfied then
• generate new solution
• until stopping-criteria satisfied
• end

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Problema Optimización Numérica

• Función
• Vencindad
• x (x1, , xn), li xi ui
• x xi N(0,si), si (ui - li)/6

• Hillclimbing iterado, generar vecindad de tamaño
  fijo
• Simulated Annealing, dos tipos de puntos vecinos.

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Simulated Annealing in Matlab
for i1TRIES, nAccep 0 for
j1NOVER, solNew NewSolution(solCurr,bo
unds,i,TRIES) Generate new solution
solNew(1,) solNew(1,totLen)
EvalSol(solNew,evalOps) Evaluate new Solution
deltaE solCurr(1,totLen) -
solNew(1,totLen) if (deltaE gt 0)
(rand lt exp(deltaE/T)) solCurr
solNew Accept Solution
nAccep nAccep 1
end if (solCurr(1,totLen) lt
solBest(1,totLen)) Minimization Problem
solBest solCurr bestTrace
bestTrace solBest(1,totLen) end
currTrace currTrace solCurr(1,totLen)
end T T TFACTR
Annealing Schedule cbTrace cbTrace
solBest(1,totLen) Current best trace end
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Parámetros y variables del SA

• solCurr zeros(1,totLen) Allocate the
  current solution
• solNew zeros(1,totLen) Allocate the new
  solution
• solBest zeros(1,totLen) Allocate the best
  solution
• TFACTR 0.85 Ann Sched
  Reduce T by this factor on each step
• TRIES 100 Number of
  Temperature Steps to try
• TMAX 10 Maximun
  Initial Temperature
• T TMAX Current
  Temperature
• NOVER 30 Max. No. of
  solutions tried at any temp
• NLIMIT 10 Max. No. of
  successful solutions before continuing
• nAccep 0 Counter for
  the number of accepted solutions
• deltaE 0 Maintains
  difference in performance between current and new
  solution
• bestFitness 0 Maintais
  best-so-far fitness

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EAs and Other Search Heuristics

• avoid converging to local optima
• use exploration of the search space and
  exploitation of promising areas
• not dependent on initial starting point(s)
• start search from many points in the search space
  space

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EAs and Other Search Heuristics

• conduct search in parallel over the search space
• implicit parallelism
• through recombination operators, reach better
  solutions by combining already found good
  solutions (building block hypothesis and the
  schema theorem)
• may be used together with other approaches
  (hybrids)