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Evolutionary multi-objective algorithm design issues. Karthik Sindhya , PhD. Postdoctoral Researcher Industrial Optimization Group Department of Mathematical Information Technology [email protected] http://users.jyu.fi/~kasindhy/. Objectives The objectives of this lecture are to:

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evolutionary multi objective algorithm design issues

Evolutionary multi-objective algorithm design issues

KarthikSindhya, PhD

Postdoctoral Researcher

Industrial Optimization Group

Department of Mathematical Information Technology

[email protected]

http://users.jyu.fi/~kasindhy/

slide2

Objectives

The objectives of this lecture are to:

  • Address the design issues of evolutionary multi-objective optimization algorithms
    • Fitness assignment
    • Diversity preservation
    • Elitism
  • Explore ways to handle Constraints
slide3

References

  • K. Deb. Multi-Objective Optimization using Evolutionary Algorithms. Wiley, Chichester, 2001.
  • E. Zitzler, M. Laumanns, S. Bleuler. A Tutorial on Evolutionary MultiobjectiveOptimization, in Metaheuristicsfor MultiobjectiveOptimisation, 3-38, Springer-Verlag, 2003.
slide4

Algorithm design issues

  • The approximation of the Pareto front is itself multi-objective.
    • Convergence: Compute solutions as close as possible to Pareto front quickly.
    • Diversity: Maximize the diversity of the Pareto solutions.
  • It is impossible to describe
    • What a good approximation can be for a Pareto optimal front.
    • Proximity to the Pareto optimal front.
slide5

Fitnessassignment

  • Unlike single objective, multiple objectives exists.
    • Fitness assignment and selection go hand in hand.
  • Fitness assignment can be classified in to following categories:
    • Scalarization based
      • E.g. Weighted sum, MOEA/D
    • Objective based
      • VEGA
    • Dominance based
      • NSGA-II
slide6

Fitnessassignment

  • Scalarization based (Aggregation based):
    • Aggregate the objective functions to form a single objective.
    • Vary the parameters in the single objective function to generate multiple Pareto optimal solutions.

Parameters - weights

w1f1(x)+ w2f2(x),…, wkfk

Or, max(wi(fi- zi ))

  • f1(x), f2(x),…, fk(x)

F

slide7

Fitnessassignment

  • Advantages – Weighted sum
    • Easy to understand and implement.
    • Fitness assignment is computationally efficient.
    • If time available is short can be used to quickly provide a Pareto optimal solution.
  • Disadvantages - Weighted sum
    • Non-convex Pareto optimal fronts cannot be handled.
slide8

Fitnessassignment

  • Objective based
    • Switch between objectives in the selection phase.
      • Every time an individual is chosen for reproduction, a different objective decides.
    • E.g. Vector evaluated genetic algorithm (VEGA) proposed by David Schaffer.
      • First implementation of an evolutionary multi-objective optimization algorithm.
      • Subpopulations are created and each subpopulation is evaluated with a different objective.

Mating pool

Population

New population

f1

Selection

Reproduction

f2

f3

slide9

Fitnessassignment

  • Advantages
    • Simple idea and easy to implement.
    • Simple single objective genetic algorithm can be easily extended to handle multi-objective optimization problems.
    • Has tendency to produce solutions near the individual best for every objective.
      • An advantage when this property is desirable.
  • Disadvantages
    • Each solution is evaluated only with respect to one objective.
      • In multi-objective optimization algorithm all solutions are important.
    • Individuals may be stuck at local optima of individual objectives.
slide10

Fitnessassignment

  • Dominance based
    • Pareto dominance based fitness ranking proposed by Goldberg in 1989.
  • Different ways
    • Dominance rank: Number of individuals by which an individual is dominated.
      • E.g. MOGA, SPEA2
    • Dominance depth: The fitness is based on the front an individual belongs.
      • NSGA-II
    • Dominance count: Number of individuals dominated by an individual.
      • SPEA2
slide11

Fitnessassignment

0

4

1

1

1

1

0

2

4

0

2

0

Dominance rank

Dominance count

3

2

1

Dominance depth

slide12

Diversitypreservation

  • Chance of an individual being selected
    • Increases: Low number of solutions in its neighborhood.
    • Decreases: High number of solutions in its neighborhood.
  • There are at least three types:
    • Kernel methods
    • Nearest neighbor
    • Histogram
slide13

Diversitypreservation

  • Kernel methods:
    • Sum of f values, where f is a function of distance.
    • E.g. NSGA
  • Nearest neighbor
    • The perimeter of the cuboid formed by the nearest neighbors as the vertices.
    • E.g. NSGA-II

f

f

f

i-1

i

i+1

slide14

Diversitypreservation

  • Histogram
    • Number of elements in a hyperbox.
    • E.g. PAES
slide15

Elitism

  • Elitism is needed to preserve the promising solutions

No archive strategy

Old population

Offspring

Old population

Offspring

New Archive

New population

New population

Archive

slide16

Constraint handling

  • Penalty function approach
    • For every solution, calculate the overall constraint violation, OCV (sum of Constraint violations).
    • Fm(xi) = fm(xi) + OCV
      • Solution - (xi), fm(xi)- mth objective value for xi, , Fm(xi) – Overall mthobjective value for xi .
      • OCV is added to each of the objective function values.
  • Use constraints as additional objectives
    • Usually used when feasible search space is very narrow.
slide17

Constraint handling

  • Deb’s constraint domination strategy
    • A solution xiconstraint dominates a solution xj, if any is true:
      • xi is feasible and xj is not.
      • xi and xj are both infeasible, but xi has a smaller constraint violation.
      • xi and xj are feasible and xi dominates xj.
    • Advantages:
      • Penalty less approach.
      • Easy to implement and clearly distinguishes good from bad solutions.
      • Can handle, if population has only infeasible solutions.
    • Disadvantages:
      • Problem to maintain diversity of solutions.
      • Slightly infeasible and near optimal solutions are preferred over feasible solutions far from optima.
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