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

2. 1 Outline  Introduction  Naive methods  Pareto dominance  Non-Dominated Sorting Genetic Algorithm  Strength Pareto Evolutionary Algorithm  Summary

3. 2 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. 3 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? feasible weighted objective

5. 4 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. 5 Pareto Dominance  One way of defining ‘better’  Solution M Pareto-dominates solution 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. 6 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. theoretical optima

8. 7 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. 8 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 Lexicographic Tournament 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. 9 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 Euclidean distance Distance to k-nearest individual e.g., k =  ||P||

11. 10 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 decision maker solver preferenceresults a priori knowledge a posterior knowledge learning

12. 11 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