Part II.2 A-Posteriori Methods and Evolutionary Multiobjective Optimization

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Part II.2 A-Posteriori Methods and Evolutionary
Multiobjective Optimization

• Scalar solution methods
• A-posteriori methods
• Evaluation of algorithms

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A posteriori-methods

• A priori-method
• First define preferences, including decisions in
  case of conflicts (e.g. by specifying a utility
  function)
• Let the algorithm search for a single best
  solution
• User selects the obtained solution
• gt Single objective optimization can be used
• A posteriori-method
• Specify general, possibly conflicting, goals
• Let algorithm find Pareto front
• User selects best solution among solution on
  Pareto front
• gt Algorithms for obtaining Pareto fronts needed!

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A-Posteriori methods (Pareto Optimization)

• A finite set of non-dominated solutions
  Approximation set
• Strive for good coverage and convergence to the
  pareto front !

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Overview Approaches

• Continuation methods
• Starting from a Karush Kuhn Tucker point
  gradually extend Pareto front by including
  neighboring Karush Kuhn Tucker points
• Problem Connected Pareto front is required and
  differentiability
• Epsilon-Constraint method
• Obtain all points on the Pareto front by solving
  constraint optimization problems.All but one
  objective is set to a constraint value
• The constraint values are changed gradually until
  the whole Pareto front is sampled
  density/position of points can be easily
  controlled
• Problem Effort growth exponentially.
• Population-based Metaheuristics (evolutionary,
  particle swarm, )
• Use selection/variation scheme to gradually move
  a population of search points to the Pareto front
• Very flexible method, easy to apply in different
  search spaces
• Problem Cannot guarantee optimality of result

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Overview Approaches

• Indicator-based method
• Approximation set A to pareto front of qA
  points (each one of dimension n) is viewed as nm
  dimensional vector
• A quality measure is defined for a set of points
  e.g. the dominated hyper-volume and functions as
  surrogate objective function
• Problem Reference point is required Dimension
  of problem may be to large if q is to large

An example of indicator for the quality of an
approximation set is the S-metric, Measuring the
area between the points in A and the reference
point to be maximized.
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Evolutionary Multiobjective Optimization (EMO)
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Overview of the EMO field
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State of the literature in EMOA
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Biologically inspired terminology

• Biological term
• Individual
• Fitness
• Population
• Generation
• Mutation Operator
• Recombination
• Parents, Offspring

• Mathematical term
• Element of search space S
• Objective function value (penalty)
• Multi-set of elements of S
• Iteration of main loop
• Operator generating a new solution by adding a
  small perturbation to a given solution
• Operator generating a new solution by combining
  information of at least two given solutions
• Given a set of variations generated from an
  original set, the original set is called parents,
  and the set of variations offspring

The concepts are used slightly differently,
depending on authors. This will be the way we use
them.
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General schema of evolutionary search
Application of variation operators
parents
offspring
Population of individuals
Evaluation of fitness
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Example Steady state evolutionary algorithm
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Variation Operators
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Initialization
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Stochastic variation operators
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Representation Independence
Variation Operators Representation Operators
Population Model and Selection Operator
The concept of PISA (ETH Zuerich)
Algorithms such as NSGA, SPEA2, PAES are widely
independent of represenations (search spaces and
variation operators)!
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Mutation
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Recombination Crossover Operators
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Advanced operators
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Selection operator in EMOA
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First generation EMOA
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NSGA-II Algorithm
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Non-dominated sorting genetic algorithm (NSGA-II)
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(ml)- Evolutionary algorithm
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A-Posteriori methods (Pareto Optimization)

• A finite set of solutions
• Strive for good coverage and convergence to the
  pareto front !

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A Non-dominated sorting
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A Non-dominated sorting
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B Crowding distance sorting
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B Crowding distance sorting
Objective space (NSGA-II)
Variable space (NSGA)
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B Crowding distance sorting
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NSGA-II Complete procedure
Download NSGA-II
http//www.iitk.ac.in/kangal/soft.htm
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NSGA-II ZDT1
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NSGA-II ZDT2
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Results ZDTL4
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Non-dominated sorting genetic algorithm
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SMS-EMOA
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Remarks

• Introduced by Beume, Emmerich, Naujoks, 2005
• Outperforms standard approaches on common
  Benchmarks for continuous multiobjective
  optimization ZDT and DTLZ
• Especially well suited for small approximation
  set
• Paradigm shift Indicator-based Pareto
  optimization
• Can be hybridized with S-Gradient method (Deutz,
  Beume, Emmerich 2007)

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SMS-EMOA Basic Algorithm
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Replacement
The following invariant holds
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SMS-EMOA
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Hypervolume in 2-D
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Computation of hypervolume in 3-D

• O(m q3) algorithm was proposed by Emmerich

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SMS-EMOA (20.000 Auswertungen)
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Pareto front in higher dimensons
Points demark finite Set approximation of Pareto
fronts
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Pareto front in higher dimensons
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SMS-EMOA Conclusions

• SMS EMOA tries to maximize the dominated
  hypervolume
• Increment in hypervolume are used as a selection
  criterion in an Evolutionary Algorithm
• The evolutionary algorithm gradually improves the
  dominated hypervolume of the population by
  variation-selection scheme
• The SMS EMOA is the best performing MCO
  techniques on standard benchmarks from literature
  (cf. EJOR 2006 Preprint, EMO Conference 2005)
• A bottleneck is the high computational cost of
  computing hypervolume increments, if no, of
  objectives is high

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Summary

• Large number of EMOA algorithms available today
• Most popular variants are NSGA-II, SPEA-II
• In EMOA field also other population-based
  algorithms (particle swarm optimization,
  simulated annealing) are discussed
• New generation of EMOA is developed (IBEA,
  SMS-EMOA) that directly addresses optimization of
  performance measure
• Statistical performance measuring on test
  problems crucial technique to engineer and select
  EMOA technique
• Bi-annual conference EMOO 2001, 2003, 2005
  (Lecture notes in Computer Science)
• Bi-annual conference MCDM Operations Research
  Oriented