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)
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)
Assume all f,g,h are differentiable
Multiple Objective Functions
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
rugged fitness landscape sensitivity issue
http://www.calresco.org/lucas/pmo.htm
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
a priori articulation of preferencesa posteriori articulation of preferencesprogressive articulation of preferences
genetic algorithms
Organization
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
x1 is superior to x2 iff
x1 dominates x2
x1 > x2
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
Weakly Pareto Optimalno other point with better object values
Properly Pareto Optimal
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
Solving multiple objective optimization provides:
Necessary condition for Pareto optimalityand / or
Sufficient condition for Pareto optimality
Common function transformation methodsto remove dimensions or balance magnitude differences
Methods with a priori articulation of preferences
Allow user to specify preferences for, or relative importance of, objective functions
Weighted Sum Method
Sufficient for Pareto optimality
no guarantee of final result acceptableimpossible to find points in non-convex sections
not even distribution
Weighted global criterion method
Lexicographic Method
objective functions arranged in order of importance
solve following optimization problems one at a time
Goal Programming Method
Goal Attainment Method
computationally faster than typical goal programming methods
Physcial Programming
Class function for each metricmonotonically increasing, monotonically decresing, or unimodal function
specify numeric ranges for degrees of preferencedesirable, tolerable, undesirable, etc.
Methods for a posteriori articualtion of preference
generate first, choose later approaches
generate representative Pareto optimal setuser selects from palette of solutions
Physical Programming
systematically vary parameters
traverses criterion space
Normal boundary intersection method
Normal constraint method
determine utopia point
normalize objective functions
individual minimization of objective functionsform vertices of utopia hyperplane
Methods no articulation of preferences
Global criterion methods
with wi = 1.0
similar to a priori techniques with no weights
Min max method
provides weakly Pareto optimal point
treat as single objective function
Objective sum method
To avoid additional constraints and discontinuities
Nast arbitration and objective product method
Maximize
where si >= Fi(x)
Rao’s method
normalize so Finorm is between zero and oneand Finorm=1 is worst possible
Genetic Algorithms
no derivative information needed
global optimization
e.g., generate sub-populations by optimizing one objective function
directions in shaded area reduce both objective functions