2200 2400 2600 2800 3000 3200 3400 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 values objectives Are All Objectives Necessary? On Dimensionality Reduction in Evolutionary Multiobjective Optimization Dimo Brockhoff and Eckart Zitzler Computer Engineering Computer Engineering and Networks Laboratory and Networks Laboratory Results 0 10 20 30 40 50 5 10 15 20 2 4 6 8 10 12 14 16 number of objectives needed exact greedy delta k number of objectives needed 0 10 20 30 40 50 5 10 15 20 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 runtimes [ms] exact greedy delta k runtimes [ms] entire search space of 0-1-knapsack problem with 7 items Exact algorithm vs. heuristic Different problems act differently Pareto front approximations for 0-1-knapsack and DTLZ7 Motivation Multiobjective Problem EA Approximation of Pareto front which solution is best? reduce number of objectives Assist decision maker error: δ = 0 Conclusions Approach 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 error general statements on redundancy impossible ⇒ δ−MOSS k-EMOSS 0 5 10 15 20 25 30 35 40 20 10 0 number of objectives needed delta [% of population’s spread] knapsack, 15 objectives knapsack, 25 objectives DTLZ7, 15 objectives DTLZ7, 25 objectives 0 0.2 0.4 0.6 0.8 1 90 60 30 error in computed set k [% of entire objectives] knapsack, 15 objectives knapsack, 25 objectives DTLZ7, 15 objectives DTLZ7, 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? 2500 2600 2700 2800 2900 3000 3100 3200 19 14 4 2 1 values objectives 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 slightly changed relation with error δ -EMOSS: Compute an objective set with objectives, changing the relation least Objective Conflicts Preservation of dominance structure • Pairwise objective conflicts non-redundancy • Omission of objectives possibly additional edges dominance structure preserved slightly changed Dimensionality Reduction Changes in dominance structure omit omit and 1 3 2 4 obj. values obj. values
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2200
2400
2600
2800
3000
3200
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2019181716151413121110987654321
valu
es
objectives
Are All Objectives Necessary? On Dimensionality Reduction in Evolutionary Multiobjective Optimization
Dimo Brockhoff and Eckart Zitzler Computer EngineeringComputer Engineeringand Networks Laboratoryand Networks Laboratory
ResultsResults
0 10
20 30
40 50 5
10
15
20
2
4
6
8
10
12
14
16
number of objectives needed
exactgreedy
delta
k
number of objectives needed
0 10
20 30
40 50 5
10
15
20
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
0
5
10
15
20
25
30
35
4020100
num
ber
of o
bjec
tives
nee
ded
delta [% of population’s spread]
knapsack, 15 objectivesknapsack, 25 objectives
DTLZ7, 15 objectivesDTLZ7, 25 objectives
0
0.2
0.4
0.6
0.8
1
906030
erro
r in
com
pute
d se
t
k [% of entire objectives]
knapsack, 15 objectivesknapsack, 25 objectives
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?
2500
2600
2700
2800
2900
3000
3100
3200
1914421
valu
es
objectives
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
1
3
2
4
obj.
values
obj.
values
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