SLIDE 1 Multiple objective function optimization
R.T. Marker, J.S. Arora, “Survey of multi-objective optimization methods for engineering”
Structural and Multidisciplinary Optimization Volume 26, Number 6, April 2004 , pp. 369-395(27)
SLIDE 2 Assume all f,g,h are differentiable
Multiple Objective Functions
SLIDE 3 Feasible design space - satisfies all constraints Preliminaries Feasible criterion space - objective function values
- f feasible design space region
Preferences - user’s opinion about points in criterion space Scalarization methods v. vector methods
SLIDE 4 rugged fitness landscape sensitivity issue
http://www.calresco.org/lucas/pmo.htm
SLIDE 5 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 resources money ideas time
http://www.calresco.org/lucas/pmo.htm
SLIDE 6 a priori articulation of preferences a posteriori articulation of preferences progressive articulation of preferences genetic algorithms
Organization
SLIDE 7 compromise solution utopia (ideal) point point that optimizes all objective functions
- ften doesn’t exist
- ne or more objective functions not optimal
close as possible to utopia point F0
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x1 is superior to x2 iff x1 dominates x2 x1 > x2
SLIDE 9 Pareto optimal solution if there does not exist another feasible design
- bjective vector such that all objective functions
are better than or equal to and at least one
- bjective 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 Optimal no 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 set Identify some subset of optimal set seek a single final point
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Solving multiple objective optimization provides: Necessary condition for Pareto optimality and / or Sufficient condition for Pareto optimality
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Common function transformation methods to 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 acceptable impossible to find points in non-convex sections not even distribution
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Weighted global criterion method
SLIDE 17 Lexicographic Method
- bjective functions arranged in order of importance
solve following optimization problems one at a time
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Goal Programming Method
SLIDE 19 Goal Attainment Method
computationally faster than typical goal programming methods
SLIDE 20 Physcial Programming
Class function for each metric monotonically increasing, monotonically decresing, or unimodal function
specify numeric ranges for degrees of preference desirable, tolerable, undesirable, etc.
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Methods for a posteriori articualtion of preference generate first, choose later approaches generate representative Pareto optimal set user selects from palette of solutions
SLIDE 22 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 functions form vertices of utopia hyperplane
SLIDE 25 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)
SLIDE 29 Rao’s method
normalize so Finorm is between zero and one and Finorm=1 is worst possible
SLIDE 30 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
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