Evolutionary Algorithms EVO Satisfying Multiple Objectives L10

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Evolutionary Algorithms (EVO) Satisfying
Multiple Objectives (L10)

• John A Clark
• Professor of Critical Systems
• Non-Standard Computation Group Dept. of Computer
  Science
• University of York, UK

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References

• A Comprehensive Survey of Evolutionary-Based
  Multiobjective Optimization Techniques.
• by Carlos A. Coello Coello (on web, and link
  from web page)
• Multi-objective Evolutionary Algorithms
  Analysing the State of the Art
• David A. van Veldhuizen and Gary Lamont
• A Short Tutorial on Evolutionary Multiobjective
  Optimization
• by Carlos A. Coello Coello (on web, and link
  from web page)
• www.cs.cinvestav.mx/emooworkgroup/tutorial-slides
  -coello.pdf
• Evolutionary Algorithms for Multi-criterion
  Optimization A Survey.
• By Ashish Ghosh and Satchidananda Dehuri

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Definition of Multi-objective Problem

• Can be defined as the problem of finding
• A vector of decision variables which satisfies
  constraints and optimizes a vector function whose
  elements represent the objective functions. These
  functions form a mathematical description of
  performance criteria which are usually in
  conflict with each other. Hence the term optimize
  means finding such a solution which would give
  the values of all the objective functions
  acceptable to the designer.

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Definition

• More formally. Find the vector which will
  satisfy
• the m inequality constraints
• d
• and the p equality constraints
• d
• and optimizes the vector of functions
  (objectives)

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Examples of Multiple Objectives

• Pervasive computing Chips with Everything.
  Limited amount of processing power available
  (often very small chips)
• Amount of memory (e.g. stack usage)
• Time taken to reach a solution.
• Precision of the answer.
• Power consumption yes, if you have a smart dust
  mote with 1mm3 battery this is important.
• Also has effects on on-chip temperature
• Note multiple objectives may be in conflict.

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Examples of Multiple Objectives

• Car design
• Need to reduce drag since this impacts on petrol
  consumption.
• Needs to be able to accommodate people (!!!) and
  provide reasonable amount of other carrying
  space, e.g. for luggage.
• Needs to have an aesthetic appeal (but peoples
  tastes differ).
• Performance related factors speed, acceleration
  and weight of vehicle.
• Strengthbut also ability to absorb impacts.
• And so on
• Again multiple objectives in conflict.

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Examples of Multiple Objectives

• Dependable systems task distribution. Want
• Low communications overheads - so put them on
  the same processor.
• Fault tolerance bad idea to place on same
  processor.
• Timing of end to end transactions.

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Pareto Optimality (Minimisation)

• Pareto optimality is a major concept in
  multi-objective optimisation.
• A solution is Pareto optimal if for every
  either

or there is at least one i e I such thatThus,
if any vector causes a decrease in the
value of some objective it causes a simultaneous
increase in some other objective
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Pareto Optimality 2 Objectives
Objective 2
For any two Pareto optimal vectors, neither is
better than the other.
dominated
Pareto optimal
Gives rise to the notion of a Pareto
front. Generally difficult to compute a
mathematical expression for this front.
Objective 1
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Aggregating Functions
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Weighted Sums

• An obvious way to couch a multi-objective problem
  is as a minimisation of a weight sum of objective
  values

In practice the magnitude of the weights matters,
even when they are all scaled by some factor (due
to the interaction with specifics of the
optimisation technique).
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Strengths and Weaknesses

• Computationally fairly straightforward and
  efficient.
• Can also use the results as inputs into other
  techniques, i.e. seed other techniques with
  generated solutions.

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Strengths and Weaknesses

• Difficulty finding appropriate weights.
• Need to choose weights so that one objective does
  not dominate.
• Need to understand to some extent the ranges of
  the possible objectives.
• This can be very hard for many difficult
  problems.
• Needs to be chosen by designer.
• Often a trial and error approach is needed.
• Weights do not reflect importance
• Weights are just a means to an end
• We just need them to identify Pareto optimal
  points.
• Optimality by reducing to a single meta (sum)
  objective is a function of the weights.
• Using linear combinations cannot handle
  non-convex regions of the search space.
• Can give a single solution - what if you want
  more?

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Goal Programming

• Here the designer sets targets for the
  optimisation

Targets for each objective
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Strengths and Weaknesses

• Efficient.
• Designer needs to determine targets
• This may require significant knowledge in its own
  right.
• Can also use a weighting regime as previous
  method
• But subject to many of the same problems.
• May prove inefficient if feasible range is
  difficult to approach

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Non-Pareto Optimality Based Methods
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Vector Evaluated Genetic Algorithm (VEGA)

• k sub-populations are selected of size N/k
• The ith subpopulation is selected according to
  fitness of the ith objective.
• Once selected these populations are merged and
  shuffled and the usual cross-over and mutation
  operators applied.
• So selection is varied in its objective and good
  performing solutions in each objective go through
  to be further processed.

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Vector Evaluated Genetic Algorithm (VEGA)
Based around the Genesis program
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3
4
5
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7
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N
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VEGA

• Very simple.
• Work by Schaffer used proportional fitness
  selection (and fitnesses were proportional to the
  objectives).
• Some parts of the search space will not be
  sampled.
• Possible dangers arising of speciation, where
  species arise that are good at one particular
  objective but not necessarily others.

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Lexicographic Ordering

• Rank objectives in order of importance and adopt
  a stepwise optimization approach.

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Lexicographic Ordering

• Proceeds in criticality or importance order.
  Optimizing objectives introduces constraints for
  the subsequent optimisations.
• Optimise the most important.
• Now do the best you can on the second but but get
  worse on the first
• And so on
• Major issue is a tendency to favour certain
  objectives disproportionately.
• Ranking objectives and then dealing with each in
  turn has consequences you might prefer a more
  global view when carrying out the optimisation
  depends on the problem.
• As with any technique, it really depends on
  whether you are happy with the answers.

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Pareto Optimality Based Methods
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Pareto Ranking by Fronts (Goldbergs scheme)

• curr_rank1
• mn
• While n! do
• for(i1..m)
• if(xu) is non-dominated then rank(x,t)curr_rank
• for(i1..m)
• if rank(x,t)curr_rank then remove x from
  population. nn-1
• curr_rankcurr_rank-1
• mn

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Pareto Ranking by Fronts (Goldbergs scheme)

• Identify all the non-dominated individuals and
  give them rank 1.
• Remove these from the population.
• Now identify all the non-dominated individuals
  from the remaining population and give them rank
  2, and so on

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Pareto Ranking by Fronts (Goldbergs scheme)
Objective 2
4
3
2
1
Objective 1
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Further Domination Based Ranking (Fonseca and
Flemming)

• Identify all non-dominated individuals in the
  population and give them rank 1.
• For each other member determine the number of
  individuals pi(t) in the rest of the population
  that dominate it.
• Rank based fitness much as before
• Sort according to rank.
• Use linear gradient of probabilities for
  selection.

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Cute Examples of Curve and Surface Fitting!

• Some nice pictures!
• Work by Rony Goldenthal
• Michel Bercovier
• www.cs.huji.ac.il/ronygold/multi_objective.html

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Problem

• The work is about curves and surfaces
• Fitting finding surfaces (curves) as close as
  possible to a given set of points.
• Design/Fairing generate a surface achieving
  certain quality measures design objective
  (minimal length, appropriate curvature,).

NURBS Non-Uniform Rational B-Spline
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NURBS Surface

• Surface Area 155.77 Surface Curvature 0.099

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NURBS Surface

• Surface Area 155.16 Surface Curvature 0.04

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NURBS Surface

• Surface Area 154.72 Surface Curvature 0.12

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NURBS Surface II

• Order 4,4
• Input points 16,16
• Control Points 5,5
• Cost functions
• Approximation Error
• Surface Curvature

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NURBS Surface

• Approximation Error 5.807 Surface Curvature 3.39

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NURBS Surface

• Approximation Error 5.19 Surface Curvature 3.56

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NURBS Surface

• Approximation Error 2.46 Surface Curvature 5.23

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NURBS Surface

• Approximation Error 1.348 Surface Curvature 9.95

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NURBS Surface

• Approximation Error 1.256 Surface Curvature
  11.51

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NURBS Surface

• Approximation Error 0.937 Surface Curvature
  20.84

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Many Many More

• There are many more multi-objective techniques.
• Paper by Ghosh and Dehuri talks about many.
• There are many good survey papers around on the
  issue of multiple objectives.
• Have referenced several at the beginning of the
  talk.