GA Approaches toMulti-Objective Optimization

Scott Noble

Fred Iskander

18 March 2003

Multi-Objective Optimization Problems (MOPs)

• Multiple, often competing objectives

• In the case of a commensurable variable space, can often be reduced to a single objective function (or sequence thereof) and solved using standard methods

• Some problems cannot be reduced and must be solved using pure MO techniques

Three General Approaches

• Preemptive Optimization• sequential optimization of individual objectives

(in order of priority)

• Composite Objective Function• weighted sum of objectives

• Purely Multi-Objective• Population-Based• Pareto-Based

Preemptive Optimization Steps

1.Prioritize objectives according to predefined criteria (problem-specific)

2.Optimize highest-priority objective function

3.Introduce new constraint based on optimum value just obtained

4.Repeat steps 2 & 3 for every other objective function, in succession

Composite Objective Functions

1.Assign weights to each function according to predefined criteria (problem-specific)• MAX and MIN objectives receive opposite signs

2.Sum weighted functions to create new composite function

3.Solve as a regular, single-objective optimization problem

Transformation Approaches

• Advantages:• Easy to understand and formulate

• Simple to solve (using standard techniques)

• Disadvantages:• A priori prioritization/weighting can end up

being arbitrary (due to insufficient understanding of problem): oversimplification

• Not suited to certain types of MOPs

Pure MOPs:Population-Based Solutions

• Allow for the investigation of tradeoffs between competing objectives

• GAs are well suited to solving MOPs in their pure, native form

• Such techniques are very often based on the concept of Pareto optimality

Pareto Optimality

• MOP tradeoffs between competing objectives

• Pareto approach exploring the tradeoff surface, yielding a set of possible solutions• Also known as Edgeworth-Pareto optimality

Pareto Optimum: Definition

• A candidate is Pareto optimal iff:• It is at least as good as all other candidates for

all objectives, and

• It is better than all other candidates for at least one objective.

• We would say that this candidate dominates all other candidates.

Dominance: Definition

Given the vector of objective functions ))(,),(()( 1 xfxfxf k

we say that candidate dominates , (i.e. ) if:1x

2x

21 xx

)()(:},,1{

},,1{)()(

21

21

xfxfki

and

kixfxf

ii

ii

(assuming we are trying to minimize the objective functions). (Coello Coello 2002)

Pareto Non-Dominance

• With a Pareto set, we speak in terms of non-dominance.

• There can be one dominant candidate at most. No accommodation for “ties.”

• We can have one or more candidates if we define the set in terms of non-dominance.

)()(|: xfxfFxFxP

Pareto Optimal Set

The Pareto optimal set P contains all candidates that are non-dominated. That is:

where F is the set of feasible candidate solutions

(Coello Coello 2002)

Examples

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7

(Fonseca and Fleming 1993)

Examples

Candidate f1 f2 f3 f4

1 (dominated by: 2,4,5) 5 6 3 10

2 (dominated by: 5) 4 6 3 10

3 (non-dominated) 5 5 2 11

4 (non-dominated) 5 6 2 10

5 (non-dominated) 4 5 3 9

Example: Pareto Ranking

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7

(Fonseca and Fleming 1993)

(1)

(1)

(1)

(1)

(3)

(2)

(6)

Pareto Front

• The Pareto Front is simply values of the optimality vector evaluated at all candidates in the Pareto Optimal Set

f

Pareto Front

(Tamaki et al. 1996)

Non-Pareto Selection

• VEGA (Parallel Selection)• Vector Evaluated Genetic Algorithm• Next-generation sub-populations formed from separate

objective functions

• Tournament Selection• Pair wise comparison of individuals w.r.t. objective

functions (pre-prioritized or random)

• Random Objective Selection• Repetitive selection using a randomly selected

objective function (predetermined probabilities)

Pareto-Based Selection

• Pareto Ranking• Tournament Selection with Dominance

• pair wise comparison against a comparison set based on dominance

• Pareto Reservation (Elitism)• carry non-dominated candidates forward from previous

generation• use additional selection method to regulate population

size

• Pareto-Optimal Selection

Diversity

• Lack of genetic diversity is an inherent issue with GAs

• Fitness sharing encourages diversity by penalizing candidates from the same area of the solution or function space

Summary

• There are multiple approaches to MOPs.

• GAs are well suited to exploring a multi-objective solution space.

• They provide insight into the tradeoffs associated with MOPs, not necessarily a particular solution.

Further Reading

• Coello Coello, C.A. 2002. “Introduction to Evolutionary Multiobjective Optimization.” www.cs.cinvestuv.mx/~EVOCINV/download/class1-emoo-eng.pdf

• Fonseca, C.M. and P.J. Fleming. 1993. Genetic Algorithms for Multiobjective Optimization: Formulation, Discussion and Generalization. Genetic Algorithms: Proceedings of the Fifth International Conference. S. Forrest, ed. San Mateo, CA, July 1993.

• Tamaki, H., H. Kita and S. Kobayashi. 1996. Multi-Objective Optimization by Genetic Algorithms: A Review. Proceedings of the IEEE Conference on Evolutionary Computation, ICEC 1996, pp 517-522.

• Younes, A., H. Ghenniwa and S. Areibi. 2002. An Adaptive Genetic Algorithm for Multi-Objective Flexible Manufacturing Systems. GECCO, New York, July 2002.