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Multiobjective Optimization

Multiobjective Optimization. Chapter 7 Luke, Essentials of Metaheuristics, 2011 Byung-Hyun Ha. R1. Outline. Introduction Naive methods Pareto dominance Non-Dominated Sorting Genetic Algorithm Strength Pareto Evolutionary Algorithm Summary. Introduction. Multiobjective optimization

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Multiobjective Optimization

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  1. Multiobjective Optimization Chapter 7 Luke, Essentials of Metaheuristics, 2011 Byung-Hyun Ha R1

  2. Outline • Introduction • Naive methods • Pareto dominance • Non-Dominated Sorting Genetic Algorithm • Strength Pareto Evolutionary Algorithm • Summary

  3. Introduction • Multiobjective optimization • Finding the solution that optimizes multiple functions • Examples • Building with multiple objective, i.e., cheaper, taller, safer, efficient • Product with low cost and high quality • Symbolic regression with high fitness and small size of tree • Trade-offs between objectives • To consider multiobjectives, we need to decide • How to define fitness of individual, and/or • How individuals to be selected • Two different levels of diversity, required • That of individual, as usual • That in perspective of multiobjectives

  4. Naive Methods • Aggregation • Bundling all objectives into a single fitness • e.g., weighted sum of each quality of a building • c.f., linear parsimony pressure for bloat problem of variable-size encoding • Problems • Weight? • c.f., Analytic Hierarchy Process (AHP) • Linearity? • Effective search? • Distance from ideal solutions? weighted objective feasible

  5. Naive Methods • Picking individuals by tournament selection • Giving up linear combination • Assuming clear preferences among objectives • Multiobjective Lexicographic Tournament Selection • c.f., goal programming • Random objective each time • Multiobjective Ratio Tournament Selection • Using voting • Multiobjective Majority Tournament Selection • Multi-stage tournament by each objective • Multiple Tournament Selection • Other sophisticated ways..?

  6. Pareto Dominance • One way of defining ‘better’ • Solution MPareto-dominatessolution N, • if M is at least as good as N in all objectives, and superior to N in at least one objective. • Pareto front (best options) • Solutions not Pareto-dominated by others

  7. Pareto Dominance • Pareto front (cont’d) • Types of Pareto front • Spread • Number of objectives? • Size of population for accurately sampling Pareto front grows exponentially • e.g., less than 4 or 5 are good. theoreticaloptima

  8. Non-Dominated Sorting Genetic Algorithm • Evaluation of individuals (simply approach) • By tournament selection based on Pareto domination • Algorithm: Pareto Domination Binary Tournament Selection • Selecting one that Pareto-dominates the other • Choosing either on at random, if each does not dominated by the other • Disadvantages • One is still preferred even in case no dominance between two. • Pareto front rank • Rank 1: Pareto front of P • Rank 2: Pareto front of (P – Rank 1) • Rank 3: Pareto front of (P – Rank 1– Rank 2) • ... • Better way of evaluation • Using individual’s Pareto front rank as its fitness

  9. Non-Dominated Sorting Genetic Algorithm • Sparsity • Distance from closest individuals • Using Manhattan distance as measure • Sum of distance along rank • Employed for spread of individuals • c.f., crowding of coevolution • Algorithms • Multiobjective Sparsity Assignment • Non-Dominated Sorting LexicographicTournament Selection With Sparsity • NSGA-II • Non-Dominated Sorting Genetic Algorithm II • Sort of (+) and elitism • Looking for entire Pareto front which is spread throughout the space • Fitness by considering Pareto front rank • Crowding by considering sparsity

  10. Strength Pareto Evolutionary Algorithm • Pareto strength of i • Number of individuals in population that i Pareto-dominates • Problem? • How about weakness? • Wimpiness of i • Sum of total strength of everyone who dominates i • SPEA2 • Strength Pareto Evolutionary Algorithm 2 • Fitness by considering wimpiness • Crowding by considering Euclideandistance • Distance to k-nearest individual • e.g., k = ||P||

  11. Notes (Talbi, 2009) • Interactions in multicriteria decision making • A prior, a posterior, interactive • Design issues of multiobjective metaheuristics • Fitness assignment strategies • Scalar approaches • Aggregation, goal programming, ... • Criterion-based approaches • Dominance-based approaches • Using Pareto dominance, ... • Indicator-based approaches • Diversity preservation • Kernel methods • Fitness sharing, ... • Nearest-neighbor methods • Crowding, ... • Histograms preference results a prioriknowledge decisionmaker solver a posterior knowledge learning

  12. Summary • Multiobjective optimization • How to define fitness and/or to select individuals? • Naive approaches • Aggregation of multiobjectives • Selecting randomly considering each objective • Pareto dominance • Exploiting Pareto dominance for search • Tournament selection based on Pareto domination • Non-Dominated Sorting Genetic Algorithm • Pareto front rank, Sparsity • Strength Pareto Evolutionary Algorithm • Wimpiness

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