Multiple objective function optimizationweb.cse.ohio-state.edu/.../MultiObjective/slides.pdf ·...
Transcript of Multiple objective function optimizationweb.cse.ohio-state.edu/.../MultiObjective/slides.pdf ·...
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Multiple objective function optimization
R.T. Marker, J.S. Arora, “Survey of multi-objective optimization methods for engineering”
Structural and Multidisciplinary OptimizationVolume 26, Number 6, April 2004 , pp. 369-395(27)
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Assume all f,g,h are differentiable
Multiple Objective Functions
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Feasible design space - satisfies all constraints
Preliminaries
Feasible criterion space - objective function values of feasible design space region
Preferences - user’s opinion about points in criterion space
Scalarization methods v. vector methods
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rugged fitness landscape sensitivity issue
http://www.calresco.org/lucas/pmo.htm
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Strange Attractors
non-linear cross-coupling
M( t+1 ) = a * M(t) + b * I ( t ) + c * T ( t )I ( t+1 ) = d * I ( t ) + e * T ( t ) + f * M( t )T ( t+1 ) = g * T (t) + h * M( t ) + j * I ( t )
economic resourcesmoneyideastime
http://www.calresco.org/lucas/pmo.htm
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a priori articulation of preferencesa posteriori articulation of preferencesprogressive articulation of preferences
genetic algorithms
Organization
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compromise solution
utopia (ideal) point
point that optimizes all objective functionsoften doesn’t exist
one or more objective functions not optimalclose as possible to utopia point
F0
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x1 is superior to x2 iff
x1 dominates x2
x1 > x2
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Pareto optimal solution
if there does not exist another feasible design objective vector such that all objective functions
are better than or equal to and at least one objective function is better
i.e., there is no x’ such that x’ > x
i.e., it is not dominated by any other point
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Weakly Pareto Optimalno other point with better object values
Properly Pareto Optimal
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Pareto optimal set
Set of all Pareto optimal points
possibly infinite set
Various Approaches
Identify Pareto optimal setIdentify some subset of optimal set
seek a single final point
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Solving multiple objective optimization provides:
Necessary condition for Pareto optimalityand / or
Sufficient condition for Pareto optimality
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Common function transformation methodsto remove dimensions or balance magnitude differences
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Methods with a priori articulation of preferences
Allow user to specify preferences for, or relative importance of, objective functions
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Weighted Sum Method
Sufficient for Pareto optimality
no guarantee of final result acceptableimpossible to find points in non-convex sections
not even distribution
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Weighted global criterion method
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Lexicographic Method
objective functions arranged in order of importance
solve following optimization problems one at a time
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Goal Programming Method
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Goal Attainment Method
computationally faster than typical goal programming methods
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Physcial Programming
Class function for each metricmonotonically increasing, monotonically decresing, or unimodal function
specify numeric ranges for degrees of preferencedesirable, tolerable, undesirable, etc.
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Methods for a posteriori articualtion of preference
generate first, choose later approaches
generate representative Pareto optimal setuser selects from palette of solutions
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Physical Programming
systematically vary parameters
traverses criterion space
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Normal boundary intersection method
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Normal constraint method
determine utopia point
normalize objective functions
individual minimization of objective functionsform vertices of utopia hyperplane
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Methods no articulation of preferences
Global criterion methods
with wi = 1.0
similar to a priori techniques with no weights
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Min max method
provides weakly Pareto optimal point
treat as single objective function
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Objective sum method
To avoid additional constraints and discontinuities
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Nast arbitration and objective product method
Maximize
where si >= Fi(x)
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Rao’s method
normalize so Finorm is between zero and oneand Finorm=1 is worst possible
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Genetic Algorithms
no derivative information needed
global optimization
e.g., generate sub-populations by optimizing one objective function
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directions in shaded area reduce both objective functions