Incremental Recursive Ranking Grouping – A Decomposition
Strategy for Additively and Nonadditively Separable Problems
Marcin M. Komarnicki
Department of Systems and Computer Networks
Wroclaw University of Science and Technology
Wroclaw, Poland
marcin.komarnicki@pwr.edu.pl
Michal W. Przewozniczek
Department of Systems and Computer Networks
Wroclaw University of Science and Technology
Wroclaw, Poland
michal.przewozniczek@pwr.edu.pl
Halina Kwasnicka
Department of Artifcial Intelligence
Wroclaw University of Science and Technology
Wroclaw, Poland
halina.kwasnicka@pwr.edu.pl
Krzysztof Walkowiak
Department of Systems and Computer Networks
Wroclaw University of Science and Technology
Wroclaw, Poland
krzysztof.walkowiak@pwr.edu.pl
ABSTRACT
Many real-world optimization problems may be classifed as Large-
Scale Global Optimization (LSGO) problems. When these high-
dimensional problems are continuous, it was shown efective to
embed a decomposition strategy into a Cooperative Co-Evolution
(CC) framework. The efectiveness of the method that decomposes
a problem into subproblems and optimizes them separately may
depend on the decomposition accuracy and cost. Recent decomposi-
tion strategy advances focus mainly on Diferential Grouping (DG).
However, when a considered problem is nonadditively separable,
DG-based strategies may report some variables as interacting, al-
though the interaction between them does not exist. Monotonicity
checking strategies do not sufer from this disadvantage. However,
they sufer from another decomposition inaccuracy ś monotonicity
checking strategies may miss discovering many existing interac-
tions. Therefore, Incremental Recursive Ranking Grouping (IRRG)
is a new proposition that accurately decomposes both additively
and nonadditively separable problems. The decomposition cost of
IRRG is higher when compared with Recursive DG 3 (RDG3). Since
the higher cost was a negligible part of the overall computational
budget, optimization results of the considered CC frameworks were
afected mainly by the decomposition accuracy.
CCS CONCEPTS
· Computing methodologies → Artifcial intelligence; Con-
tinuous space search.
KEYWORDS
Large-Scale Global Optimization, Problem Decomposition, Mono-
tonicity Checking, Nonadditive Separability
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For all other uses, contact the owner/author(s).
GECCO ’23 Companion, July 15–19, 2023, Lisbon, Portugal
© 2023 Copyright held by the owner/author(s).
ACM ISBN 979-8-4007-0120-7/23/07.
https://doi.org/10.1145/3583133.3595846
ACM Reference Format:
Marcin M. Komarnicki, Michal W. Przewozniczek, Halina Kwasnicka,
and Krzysztof Walkowiak. 2023. Incremental Recursive Ranking Grouping
ś A Decomposition Strategy for Additively and Nonadditively Separable
Problems. In Genetic and Evolutionary Computation Conference Companion
(GECCO ’23 Companion), July 15–19, 2023, Lisbon, Portugal. ACM, New York,
NY, USA, 2 pages. https://doi.org/10.1145/3583133.3595846
1 INCREMENTAL RECURSIVE RANKING
GROUPING
Incremental Recursive Ranking Grouping (IRRG) [2] is a recursive
decomposition strategy [5] that is derived from monotonicity check-
ing [1]. Its typical time complexity is O( log()) . According to [1]
and when a -dimensional function : Ω → R is considered, two
disjoint groups of variables
1
and
2
are interacting if ∃
1
,
2
> 0,
u
1
∈
1
, u
2
∈
2
, x
*
∈ Ω such that
( x
*
)≤ ( x
*
+
1
u
1
)∧ ( x
*
+
2
u
2
) > ( x
*
+
1
u
1
+
2
u
2
) (1)
where unit vector u
j
= [
1
, ...,
] is a member of
only when
∀
∈1,...,
= 0 ⇔
∉
. Monotonicity checking strategies may
be classifed as empirical linkage learning techniques [4] because
the above interaction check ensures that only existing interactions
between decision variables can be discovered [2].
In comparison to other decomposition strategies, IRRG intro-
duces mainly three new mechanisms. (I) Creation of two rankings
r
1
= [
1,1
, ...,
1,
] and r
2
= [
2,1
, ...,
2,
] , where
is a user-
defned parameter that indicates the number of samples, to check
if
1
and
2
interact. To generate
samples, at frst we need
values for each variable
∈
1
that are evenly taken from the
feasible set. Let us denote these values as ˜
1,
, ..., ˜
,
. Then,
vectors ˜ x
i
= [ ˜
,1
, ..., ˜
,
] can be created. Value ˜
,
is defned as
˜
,
=
˜
( ) ,
,
∈
1
*
,
∉
1
(2)
where
is a random permutation of {1, ...,
} for the th variable
and
*
is the th element of a given vector x
*
. The ranking are gen-
erated based on these vectors afterward. The th value of r
1
(
1,
)
is computed for ( ˜ x
i
) . Value
2,
is based on ( ˜ x
i
+
2
u
2
) . Two dis-
joint sets of variables
1
and
2
interact if ∃
*
∈{ 1,...,
}
1,
* ≠
2,
* .
(II) Applying an initial optimization to fnd a high-quality solution
27