Optimization with Genetic Algorithms

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Optimization withGenetic Algorithms

• Walter Reade
• October 31, 2002
• TAG Meeting

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Outline

• Background on Optimization
• Introduction to Genetic Algorithms
• Using GAs to Solve Difficult Problems
• A MatLab Implementation
• Summary / Questions

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How Do We Find the Minimum?
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Gradient Methods(Steepest Descent)

• Move in the direction of steepest gradient.
• Simple to implement, guaranteed convergence.
• Must know something about the derivative.
• Can easily get stuck in a local minimum.

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Stochastic Methods

• Heuristic
• Using Rules of Thumb
• Metaheuristic
• A framework of heuristics used to update a set of
  solutions during a search.
• Simulated Annealing
• Tabu Search
• Ant Colony Systems

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Genetic Algorithms

• Use a population of possible solutions to the
  search space.
• Each solution is encoded in a string called a
  chromosome (or genome).
• Chromosomes are evaluated for fitness each
  generation (iteration) chromosomes that are more
  fit have a high probability of surviving.

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Genetic Algorithms (cont.)

• Once the surviving population is chosen,
  different parent chromosomes are combined to
  form child chromosomes.
• Chromosomes may undergo mutation.
• A new generation is formed, the process is
  repeated.
• By selection, cross-over, and mutation, GAs
  search the solution space while creating stronger
  solutions over each generation.

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Fitness and Selection

• Roulette Wheel
• Competition
• Etc.

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Cross-Over

• Replaces two parent solutions with two children
  solutions.
• Mechanism for covering large area of search space.

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Mutation

• Operates on a single chromosome.
• Mechanism to improve local search space.

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Advantages to using GAs

• Flexible and adaptive to a wide variety of
  problems.
• Robust, global search capability.
• Does not require the solution space to be smooth,
  continuous, or differentiable.
• Can be used in permutation problems.
• No practical drawbacks.
• Slow local convergence
• Perceived learning overhead

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Applications

• Function optimization
• Job shop scheduling
• Process planning
• Assembly line optim.
• Process control
• Airplane landing
• Nested design
• Keyboard layout

• Creativity
• VLSI
• Traveling Sales Man
• Chemical kinetics
• Etc.
• Etc.
• Etc.

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Solving Difficult Problems
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Difficult Problems

• Appeared in Jan/Feb 2002 SIAM News in the
  100-Dollar, 100-Digit Challenge.
• exp(sin(50x)) sin(60exp(y)) sin(70sin(x))
  sin(sin(80y)) - sin(10(xy)) 0.25(x2
  y2)
• The genetic algorithm was able to solve this to
  10 digits of precision in 2000 generations, which
  took 25 seconds on a P-III 1.0 GHz. (35 success
  rate)

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Permutation (Order-based) Problems

• Time-share Example
• One condo building at a ski resort
• Four identical condo units
• 16 week ski season 64 total owners
• 2nd choice 2 free lift tickets per person, 3rd
  choice 5 free tickets, otherwise and 7 free
  tickets.
• 5 out of 16 weeks are twice as popular
• Maximum occupancy 22
• Possible solutions 1x1067

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Results of GA

• A previous published result (using SA) found a
  minimum of 224 after 261 iterations, and no
  improvement after 1,000,000 iterations.
• The GA found a cost of 200 after 2,150
  iterations, and a minimum of 172 after 250,000
  iterations.
• (Author of previous work was astonished at the
  new result.)

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Using GAs in MatLab

• http//www.ie.ncsu.edu/mirage/GAToolBox/gaot/

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MatLab Code

• Bounds on the variables
• bounds -5 5 -5 5
• Evaluation Function
• evalFn 'Four_Eval'
• evalOps
• Generate an intialize population of size 80
• startPopinitializega(80,bounds,evalFn,1e-10
  1)
• GA Options epsilon float/binary display
• gaOpts1e-10 1 0
• Termination Operators -- 500 Generations
• termFns 'maxGenTerm'
• termOps 500

• Selection Function
• selectFn 'normGeomSelect'
• selectOps 0.08
• Crossover Operators
• xFns 'arithXover heuristicXover simpleXover'
• xOpts 1 0 1 3 1 0
• Mutation Operators
• mFns 'boundaryMutation multiNonUnifMutation
  nonUnifMutation unifMutation'
• mOpts 2 0 03 200 32 200 32 0 0
• Apply the genetic algorithm
• soln endPop bestPop tracega(bounds,evalFn,evalO
  ps,startPop,gaOpts,termFns,termOps,selectFn,select
  Ops,xFns,xOpts,mFns,mOpts)

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Evaluation Function

• function x, soln Four_Eval(x, options)
• soln -(exp(sin(50x(1))) sin(60exp(x(2)))
  sin(70sin(x(1))) ...
• sin(sin(80x(2))) - sin(10(x(1)x(2)))
  0.25(x(1)2 x(2)2))

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Time Share Evaluation Function

• function assignment, soln local_min(assignment
  , options)
• global family_info
• cost 0
• occupancy zeros(16,1)
• for i 164
• if ceil(assignment(i)/4) family_info(i,2)
  first choice
• cost cost 0
• elseif ceil(assignment(i)/4)
  family_info(i,3) second choice
• cost cost 2family_info(i,1)
• elseif ceil(assignment(i)/4)
  family_info(i,4) third choice
• cost cost 4family_info(i,1)
• else didn't get any choice
• cost cost 50 7family_info(i,1)
• end

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Summary

• Genetic Algorithms are
• Powerful
• Flexible
• Easy to use and understand
• Consider using a GA for your next optimization
  problem!

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