Multi Objective Optimization MOOP with iSIGHT 9'0

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Multi Objective Optimization (MOOP) with iSIGHT
9.0

• David J. Powell, PhD
• dpowell2_at_elon.edu

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Goals

• Be able to directly use with no programming and
  in some cases with iSIGHT programming Multiple
  Objective Optimization in iSIGHT (both classical
  and evolutionary)
• Classical Numerical Approaches
• Understand iSIGHT flexibility to help you
  generate multiple points or a single point on
  pareto optimal front.
• Techniques are valid for any iSIGHT optimization
  technique.
• Understand iSIGHT tools for data analysis of
  pareto front.
• Understand new Genetic Algorithm approaches for
  MOOP

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Examples of MOOP

• In bridge construction, a good design is
  characterized by low total mass and high
  stiffness.
• Aircraft design requires simultaneously
  optimization of fuel efficiency, payload and
  weight.
• A good sunroof design in a car could aim to
  minimize the noise the driver hears and maximize
  the ventilation.
• The traditional portfolio optimization problem
  attempts to simultaneously minimize the risk and
  maximize the return.
• (info from http//www-fp.mcs.anl.gov/otc/Guide/Opt
  Web/multiobj/ )

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Primary References

• Anderson, J. (2001), Multiobjective Optimization
  in Engineering Design, Linkoping University,
  Technical Report 675.
• Deb, K. (2001), Multi-Objective Optimization
  using Evolutionary Algorithms, John Wiley Sons.
• Osyczka, A. (1985). Multicriteria Optimization
  for Engineering Design, In Design Optimization,
  pp 193-225.
• Sen, P. (1998). Multiple Criteria Decision
  Support in Engineering, Springer-Verlag

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MOOP General Form
Minimize fm(x), m
1,2,,M Subject to gj(x) lt 0,
j 1,2,.,J hk(x) 0,
k 1,2,.,K
xi(L) lt xi lt xi(U) , i 1,2, , n
where Xi Rn
continuous variables
Xi In
integer variables Xi (Xi1,
Xi2, ) discrete variables
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Ideal or Utopian Solution Vector

• For each of the M objectives, there exists one
  different optimal solution.
• An objective vector constructed with these
  individual optimal objective values constitutes
  the ideal objective vector or utopian vector.
• In general, this is never obtainable
• What is its use
• Individual optimal objective values used for
  normalization
• Used by some classical techniques as solutions
  closer to ideal are better.

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Utopian Objective Vector
Nadir upperbound of eachobjective
Utopia lowest value of each objective
Figure from Deb p. 27
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Domination

• A solution x(1) is said to dominate the other
  solution x(2), if following 2 conditions are
  true
• The solution x(1) is no worse than x(2) in all
  objectives for j 1, 2, , M
• The solution x(1) is strictly better than x(2) in
  at least one objective

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Domination Example
1 dominates 25 dominates 1
Figure from Deb page 29
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Pareto Optimal

• Globally Pareto-optimal set. The non-dominated
  set of the entire feasible search space S is the
  globally Pareto-optimal set.

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Pareto Optimal Front
Figure from Anderson
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Pareto Optimal Fronts
Figure from Deb p. 32
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Classification of MOO Techniques

• No articulation of preference information
• Global criterion (SC)
• MinMax (SN)
• Benson (SN)
• Prior
• Weighted Sum (1) (C)
• Goal Programming (S)
• Lexicographic (S)

• Posterior
• Weighted Sum (2) (C)
• eConstraint (N)
• Genetic Algorithm (N)
• Weighted MinMax (2) (SN)
• Weighted Goal Programming (2)
• Progressive
• Satisficing Tradeoff Analysis (not covered. Mimic
  with other methods)

For clarity I will present in category order but
deviate on individual techniques
Simple programming required S Convex
objective space C Nonconvex
objective space N
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IBeam Example
Figure from Osyczka p 196
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IBeam MOOP
Minimize Cross Section Area Minimize Static
Deflection Subject to g1(x) lt 16 (strength
constraint) 10 lt x1 lt 80 10
lt x2 lt 50 0.9 lt x3 lt 5
0.9 lt x4 lt 5 Starting design x0 75, 45,
2, 2 Cross Section Area 322 Static Deflection
0.01669 g1 5.605
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IBeam Calculation
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Starting Point Problem Formulation Utopia
CrossSectionArea
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Calculate Utopia Cross Section Area
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Starting Point for Utopia Static Deflection
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Calculate Utopian Static Deflection
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Start of Standard Tradeoff Curve
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How about others?

• NLPJOB by Schittkowski
• Weighted sum
• Lexicographic
• eConstraint on Tradeoff Method
• Global Criterion p 1, p 2
• MinMax