MultiObjective GA Using Reduced Model

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Multi-Objective GA Using Reduced Model

• Liang Shi

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Genetic Algorithm

• Real world problem can be described by mathematic
  functions (NP-hard). GA is a fast optimization
  method for finding the solution.
• Representation binary, real etc.
• Initial population
• Single evaluation function example
• f(x) xsin(10?x) 1.0, find f(x0)gtf(x), for
  all x?-1,2
• Genetic operators (crossover, mutation)

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Multi-Objective GA

• Real world problems usually are too complicated
  to use only a single function to describe.
• GA is needed to optimize all objectives.
• Find the solutions close to so called
  Pareto-front.
• Our method OEGADO Objective Exchange Genetic
  Algorithm for Design Optimization

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Background

• For engineering design problems, using GA to find
  possible solutions for real simulation.
• Evaluation function is very complicated, having
  many constraints. Number of variables may become
  very large.
• Feasible areas are small and discontinuous.
• Complicated surfaces mix with simpler surfaces
• Need hours or even days to do one function
  evaluation.
• Reducing function evaluation is mainly concerned.

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Goal of OEGADO

• Converge close to the true Pareto-optimal region
• Find well distributed Pareto-optimal region
• Perform as few objective evaluations as possible

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Main ideas

• Run several single objectives concurrently
  (suitable for parallel processing)
• Each GA optimizes one of the objectives, sharing
  the same representation and constraints.
• Independent populations
• Each GA uses least square approximation to form a
  reduced model of its objective for other GAs.
• Reduced models are exchanged between GAs.
• Each GA not only focus on its own objective, but
  also gets bias towards other objectives.

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Informed operator

• Informed initialization (In real application, we
  can use the manually selected population
  according to expert experience)
• Informed mutation
• Inform crossover
• Dealing with un-evaluable and infeasible points.
  (penalty function)

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Reduced model formation

• Maintain a large sample of points
• Divided into clusters, keep clusters size uniform
• Fit discontinuous and complicated surfaces
• Measure of merit and violation are separately

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Reduced Model(cont)

• Use quadratic least-squares approximation
  functions of the form
• n is the dimension of the search space and xi is
  the ith design variable

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An example of OEGADO for two objectives

• Both the GAs are run concurrently for the same
  number of iterations, each GA optimizes one of
  the two objectives while also forming a reduced
  model of it.
• At intervals equal to twice the population size,
  each GA exchanges its reduced model with the
  other GA.
• The conventional GA operators such as
  initialization, mutation and crossover are
  replaced by informed operators (IOs). The IOs
  generate multiple children and use the reduced
  model to compute the approximate fitness of these
  children. The best individual based on this
  approximate fitness is selected to be the
  newborn.
• The true fitness function is then called to
  evaluate the actual fitness of the newborn
  corresponding to the current objective.
• The individual is then added to the population
  using the replacement strategy.
• Steps 2 through 5 are repeated till the maximum
  number of evaluations is exhausted.

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Extend to n objectives

• 1. Each GA forms its own reduced model as
  explained earlier.
• 2. After a given interval of evaluations each GA
  offers its reduced model to one of the other two
  GAs and obtains one of their reduced models to
  use by its informed operators.
• 3. After the second interval each GA exchanges
  the reduced model with the other remaining GA.
• 4. This process continues and the GAs continue to
  exchange their reduced models in a round-robin
  fashion.
• Its still an important direction for future work

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Another excellent GA-NSGA2

• Non-dominated sorting based multi-objective
  evolutionary algorithm with a computational
  complexity of O(MN2) (where M is the number of
  objectives and N is the population size)
• Sort a population of size N according to the
  level of non-domination
• It proved to be better than many other
  multi-objective optimization GAs.

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Comparison between OEGADO and NSGA2

• Give each of them the same objective evaluation
  times
• On some benchmark or often referenced problems
• Parameters are set as recommended by both papers.

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Problem description

• TNK problem

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Problem description(cont)

• Welded Beam Design

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Problem description(cont)

• DTLZ8 (M Objectives, n dimensions)

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TNK
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Welded-Beam
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DTLZ8(3 objectives, 30 variables)
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Thank you!