ga approaches to multi-objective optimization scott noble fred iskander 18 march 2003
TRANSCRIPT
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.