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