are all objectives necessary? - sop.tik.ee.ethz.ch

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Are All Objectives Necessary? On Dimensionality Reduction in Evolutionary Multiobjective Optimization

Dimo Brockhoff and Eckart Zitzler Computer EngineeringComputer Engineeringand Networks Laboratoryand Networks Laboratory

ResultsResults

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number of objectives needed

exactgreedy

delta

k

number of objectives needed

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101102103104105106107108109

runtimes [ms]exact

greedy

delta

k

runtimes [ms]

entire search space of 0-1-knapsack problem with 7 itemsExact algorithm vs. heuristic

Different problems act differentlyPareto front approximations for 0-1-knapsack and DTLZ7

MotivationMotivation

Multiobjective Problem

EA

Approximation of Pareto front

whichsolution is

best?

reduce numberof objectives

Assistdecisionmaker

error:δ = 0

ConclusionsConclusions

ApproachApproach

heuristic slightly worse results, but clearly faster⇒

Benefits of the Approach:• Definition of conflict between objective sets can detect redundancy• Objective reduction is adjustable by defining error threshold or

largest allowed objective set size• Approach guarantees maximal error in dominance structure change

Take Home Message: Given a set of solutions, objective reduction is possible by preserving or only slightly changing the dominance structure. The omission of redundant information can assist the decision maker.

Key Contributions:• Generalization of conflict between objective sets• Framework for objective reduction to assist the decision maker

⇒ the smaller the objective set, the larger the errorgeneral statements on redundancy impossible⇒

δ−MOSS k-EMOSS

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delta [% of population’s spread]

knapsack, 15 objectivesknapsack, 25 objectives

DTLZ7, 15 objectivesDTLZ7, 25 objectives

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DTLZ7, 15 objectivesDTLZ7, 25 objectives

Problem: Decision making with many objectives is challenging

Questions:• Can objectives be omitted while the dominance structure is

preserved/only slightly changed?• How to compute a minimum objective set?

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Drawbacks of known dimensionality reduction approaches:• Not suitable for black-box optimization [Agrell 1997]

• No guarantee to preserve dominance structure [Deb and Saxena 2005]

Algorithms

exact• , and resp., for δ-MOSS and k-EMOSS

greedy• for δ-MOSS• for k-EMOSS

The Miminum Objective Subset Problems

Given: Solution set with objective values

δ−MOSS: Compute a minimum objective set, yielding a slightlychanged relation with error δ

-EMOSS: Compute an objective set with objectives, changing therelation least

Objective ConflictsPreservation of dominance structure

• Pairwise objective conflicts non-redundancy• Omission of objectives possibly additional edges

dominance structurepreserved slightly changed

Dimensionality Reduction

Changes in dominance structure

omit omit and

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