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 Functions
Assume all f,g,h are differentiable
Preliminaries
Feasible design space - satisfies all constraints
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
economic resources
money
ideas
time
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 )
non-linear cross-coupling
Strange Attractors
http://www.calresco.org/lucas/pmo.htm
Organization
a priori articulation of preferences
a posteriori articulation of preferences
progressive articulation of preferences
genetic algorithms
utopia (ideal) point
F0
point that optimizes all objective functions
often doesn’t exist
compromise solution
one or more objective functions not optimal
close as possible to utopia point
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 Optimal
no 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 set
Identify some subset of optimal set
seek a single final point
Solving multiple objective optimization provides:
Necessary condition for Pareto optimality
and / or
Sufficient condition for Pareto optimality
Common function transformation methods
to 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 acceptable
impossible 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 metric
monotonically increasing, monotonically decresing, or unimodal function
specify numeric ranges for degrees of preference
desirable, tolerable, undesirable, etc.
Methods for a posteriori articualtion of
preference
generate first, choose later approaches
generate representative Pareto optimal set
user 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 functions
form vertices of utopia hyperplane
Methods no articulation of preferences
similar to a priori techniques with no weights
Global criterion methods
with wi = 1.0
Min max method
treat as single objective function
provides weakly Pareto optimal point
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 one
and 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