GA Approaches to MultiObjective Optimization Scott Noble Fred
GA Approaches to Multi-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 formulate • Simple to solve (using standard techniques) • Disadvantages: • A 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 we say that candidate dominates , (i. e. ) if: (assuming we are trying to minimize the objective functions). (Coello 2002)
Pareto Non-Dominance • With a Pareto set, we speak in terms of nondominance. • 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.
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 2002)
Examples (Fonseca and Fleming 1993)
Examples Candidate f 1 f 2 f 3 f 4 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 (1) (6) (3) (1) (2) (1) (Fonseca and Fleming 1993)
Pareto Front • The Pareto Front is simply values of the optimality vector evaluated at all candidates in the Pareto Optimal Set
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 multiobjective solution space. • They provide insight into the tradeoffs associated with MOPs, not necessarily a particular solution.
Further Reading • Coello, C. A. 2002. “Introduction to Evolutionary Multiobjective • • • Optimization. ” www. cs. cinvestuv. mx/~EVOCINV/download/class 1 -emooeng. 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. 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.
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