Global Optimization: Visualizing Heuristic Strategies

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Global Optimization Visualizing Heuristic
Strategies

• Rob Dimeo
• IDL/DAVE Lunchtime Seminar
• December 14, 2004

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Conventional Optimization Algorithms

• Minimization methods require you choose between
• one that requires only function evaluations
• or one that requires function evaluations and
  derivatives

For functions that have multiple minima these
algorithms can get caught in one of the local
minima
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Heuristic Global Optimization Algorithms

• Many algorithms borrow from a natural paradigm
• Simulated Annealing
• Genetic Algorithm
• Particle Swarm Optimization
• Ant Colony Optimization

• Some are artificial constructs or ad-hoc
• Stochastic tree optimization
• Stochastic downhill simplex

A common problem with global optimization
algorithms Exploration vs. exploitation
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The Simple Genetic Algorithm
Creation of initial population

• Based on Darwinian survival-of-the-fittest
• Search space encoded as chromosomes made of bits
• Solutions are bred using rules for reproduction,
  crossover, and mutation
• Population of solutions evolve for some number of
  generations (undergoing reproduction, crossover,
  and mutation) and the best fit solution is
  determined in the last generation
• Example Solution of the 1-d Ising model

Fitness (function) evaluation
Termination criteria met?
yes
Done
no
Selection
Crossover
Mutation
Determination of new population
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Stochastic Downhill Simplex

• Standard downhill simplex (reflections,
  expansions, and contractions) augmented with a
  random restart
• At conclusion of a series of downhill simplex
  moves, the simplex is exploded and another
  series of downhill simplex moves proceeds
• Example function minimization

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Stochastic Tree Search

• Start at the original node
• Create branches to 2 new nodes
• Evaluate the function at each new node (save the
  minimum). Assign branch probabilities based on
  function evaluation
• In the next iteration begin at the original node
• Choose branch based on branch probability until
  you reach a terminal node
• At the terminal node create 2 new nodes
• Repeat until some depth has been reached
• Enhancement Specify a branch decay that reduces
  the probability based on the number of times that
  node has been visited

• One point in search space defines a node
• From starting node branches are constructed to
  two new nodes
• Function (c2) evaluated and stored at each new
  node
• Thickness of branch is related to the measure of
  probability for taking that branch in subsequent
  iterations

Example Function minimization