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?

feasible weightedobjective

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 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

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 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

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

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

decisionmaker

solverpreference resultsa priori

knowledge

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