Multiobjective Optimization

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Multiobjective Optimization

• Chapter 7
• Luke, Essentials of Metaheuristics, 2011
• Byung-Hyun Ha

R1
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Outline

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

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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

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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
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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..?

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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

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Pareto Dominance

• Pareto front (contd)
• 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
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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 individuals Pareto front rank as its
  fitness

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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

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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

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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
decisionmaker
solver
a prioriknowledge
a posterior knowledge learning
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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