Presentation on theme: "GA Approaches to Multi-Objective Optimization"— Presentation transcript:
1 GA Approaches to Multi-Objective Optimization Scott NobleFred Iskander18 March 2003
2 Multi-Objective Optimization Problems (MOPs) Multiple, often competing objectivesIn the case of a commensurable variable space, can often be reduced to a single objective function (or sequence thereof) and solved using standard methodsSome problems cannot be reduced and must be solved using pure MO techniques
3 Three General Approaches Preemptive Optimizationsequential optimization of individual objectives (in order of priority)Composite Objective Functionweighted sum of objectivesPurely Multi-ObjectivePopulation-BasedPareto-Based
4 Preemptive Optimization Steps 1. Prioritize objectives according to predefined criteria (problem-specific)2. Optimize highest-priority objective function3. Introduce new constraint based on optimum value just obtained4. Repeat steps 2 & 3 for every other objective function, in succession
5 Composite Objective Functions 1. Assign weights to each function according to predefined criteria (problem-specific)MAX and MIN objectives receive opposite signs2. Sum weighted functions to create new composite function3. Solve as a regular, single-objective optimization problem
6 Transformation Approaches Advantages:Easy to understand and formulateSimple to solve (using standard techniques)Disadvantages:A priori prioritization/weighting can end up being arbitrary (due to insufficient understanding of problem): oversimplificationNot suited to certain types of MOPs
7 Pure MOPs: Population-Based Solutions Allow for the investigation of tradeoffs between competing objectivesGAs are well suited to solving MOPs in their pure, native formSuch techniques are very often based on the concept of Pareto optimality
8 Pareto Optimality MOP tradeoffs between competing objectives Pareto approach exploring the tradeoff surface, yielding a set of possible solutionsAlso known as Edgeworth-Pareto optimality
9 Pareto Optimum: Definition A candidate is Pareto optimal iff:It is at least as good as all other candidates for all objectives, andIt is better than all other candidates for at least one objective.We would say that this candidate dominates all other candidates.
10 Dominance: Definition Given the vector of objective functionswe say that candidate dominates , (i.e ) if:(assuming we are trying to minimize the objective functions).(Coello Coello 2002)
11 Pareto Non-DominanceWith 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.
12 Pareto Optimal SetThe 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)
18 Non-Pareto Selection VEGA (Parallel Selection) Tournament Selection Vector Evaluated Genetic AlgorithmNext-generation sub-populations formed from separate objective functionsTournament SelectionPair wise comparison of individuals w.r.t. objective functions (pre-prioritized or random)Random Objective SelectionRepetitive selection using a randomly selected objective function (predetermined probabilities)
19 Pareto-Based Selection Pareto RankingTournament Selection with Dominancepair wise comparison against a comparison set based on dominancePareto Reservation (Elitism)carry non-dominated candidates forward from previous generationuse additional selection method to regulate population sizePareto-Optimal Selection
20 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
21 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.
22 Further ReadingCoello Coello, C.A “Introduction to Evolutionary Multiobjective Optimization.”Fonseca, C.M. and P.J. Fleming 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 Multi-Objective Optimization by Genetic Algorithms: A Review. Proceedings of the IEEE Conference on Evolutionary Computation, ICEC 1996, ppYounes, A., H. Ghenniwa and S. Areibi An Adaptive Genetic Algorithm for Multi-Objective Flexible Manufacturing Systems. GECCO, New York, July 2002.