Local Optimal Sets and Bounded Archiving on
Multi-objective NK-Landscapes with Correlated Objectives
Manuel López-Ibáñez1, Arnaud Liefooghe2,3, and Sébastien Verel4
Pareto local optimal set
MO and Pareto set
x2
f2
f2
nondominated
vector
dominated
solution
nondominated
solution
PLO-set
PLO
solution
A := {x0}
2:
repeat
select x ∈ A at random s.t. x is
not visited
for all x0 ∈ N(x) do
Update Archive(A, x0)
set x to visited
until ∀x ∈ A, x is visited
3:
Decision space
4:
Objective space f1
5:
neighborhood
relation
better
6:
worse
x2
f2
Pareto front
7:
f1
efficient
solution
Maximization problem on binary strings
1:
dominated
vector
x1
ρ MNK-landscapes
Pareto local search (PLS)
fi(x) = 1n ∑nj=1 cij (x j , x j1 , . . . , x jk )
Parameters to define pb. instances:
ρ : correlation between objectives
m : number of objectives
n : bit string length
x0 PLO iff ∀x ∈ N(x0), x0 is not dominated by x
+ PLS stops on a PLO-set
S is a PLO-set iff ∀x ∈ S, x is PLO,
Efficient set
k : degree of non-linearity
and all solutions are mutually non-dominated
Decision space
x1
Objective space f1
S is also a maximal PLO-set iff any neighbor
of S is weakly dominated by a solution in S
x? is PO iff ∀x ∈ X , x? is not dominated by x
How problem characteristics
affect PLO-sets?
Bounded archiving: |A| ≤ µ
UNB: unbounded (µ = ∞) ⇒ maximal
HVA: hypervolume-based archiver
MGA: multi-level grid archiver
+ Characterizing local optimality in
multi-objective set-based local
search
Running time
PLO-set size (PLS with unbounded archive)
Running time = PLS length (number of iteration of PLS)
m
100
50
●
●
●
10
2
3
5
●
10
●
200
5
2
4
UNB
HVA
MGA
●
150
●
2
●
●
1
140
PLS length
1000
500
20
PLO−set size
PLO−set size
m
2
3
5
●
+ The larger the length, the larger the basin diameter, the lower the number of LO
8
1
2
4
k
8
100
50
120
80
60
40
20
k
n = 16, ρ = −0.2
UNB
HVA
MGA
100
PLS length
10000
5000
n = 16, ρ = 0.7
10 20
40
80
10 20
40
µ
1000
500
100
50
2
3
5
●
●
10
5
●
●
●
●
−0.7
−0.2 0
0.2
1000
500
●
●
100
50
n = 16, m = 3, k = 8, ρ = −0.2
1
2
4
8
1000
UNB
HVA
MGA
6000
●
10
5
0.7
−0.2
0
0.2
ρ
0.7
ρ
n = 16, k = 4
n = 16, m = 3, k = 8, ρ = 0
8000
PLS length
●
k
4000
2000
0
n = 16, m = 5
UNB
HVA
MGA
800
PLS length
m
µ
●
10000
5000
PLO−set size
PLO−set size
10000
5000
600
400
200
0
10 20
40
80
10 20
40
µ
N
K
M
0.5
UNB
HVA
MGA
0.4
0.2
0.3
●
0.1
0.0
0.0
40
80
40
n = 16, m = 5, k = 8, ρ = 0
●
0.10
●
0.3
hvr
hvr
0.15
●
●
1
2
0.00
0.2
0.0
4
8
k
n = 16, m = 3, ρ = 0.0, µ = 10
20
●
●
●
1
2
4
8
k
n = 16, m = 5, ρ = 0.0, µ = 10
• Nearly logarithmic improvement in quality w.r.t. µ
• Quality decreases with k
• Marginal difference between PLSHVA and PLSMGA
600
UNB
HVA
MGA
400
200
0
1
2
4
8
1
2
4
8
k
n = 16, m = 5, ρ = 0.0, µ = 10
-
reduces quality
Conclusions
●
●
40
800
●
UNB
HVA
MGA
0.1
0.05
60
●
●
– In contradiction with the single-objective case ! . . .
– . . . but, two effects: (1) # PLO-solutions and (2) neighborhood size
0.4
UNB
HVA
MGA
●
1000
• Marginal difference between PLSHVA and PLSMGA
• Bounding the archive size ⇒ + reduces running time and
• PLS-length increases with k:
80
µ
●
80
UNB
HVA
MGA
n = 16, m = 3, ρ = 0.0, µ = 10
µ
0.20
●
●
●
●
k
10 20
n = 16, m = 5, k = 8, ρ = −0.2
●
0.2
0.1
10 20
UNB
HVA
MGA
0.4
hvr
0.3
hvr
PLS length
Quality (hypervolume- or epsilon-indicator)
1200
PLS length
100
0.5
n = 16, m = 5, k = 8, ρ = 0
140
120
80
µ
n = 16, m = 5, k = 8, ρ = −0.2
ρ
PLO-set size
0.6
80
• Cardinality: main factors are m and ρ
• Most maximal PLO-sets have roughly the same size
• Shorter PLS-length ⇒ lower quality
• Are PLO-sets obtained by PLSHVA and PLSMGA similar?
• Relationship between PLO-solutions and number and size of PLO-sets?
1IRIDIA, Université Libre de Bruxelles, Belgium — 2Université Lille 1, LIFL, UMR CNRS 8022, France
3Inria Lille–Nord Europe, France — 4Université du Littoral Côte d’Opale, LISIC, France