Computing factorized approximations of Pareto-fronts
using mNM-landscapes and Boltzmann distributions

arXiv:1512.03466v1 [] 10 Dec 2015

Roberto Santana, Alexander Mendiburu, Jose A. Lozano
Intelligent Systems Group. University of the Basque Country (UPV/EHU)
{roberto.santana,alexander.mendiburu,ja.lozano}@ehu.es
Abstract
NM-landscapes have been recently introduced as a class of tunable rugged models. They are a subset of the general interaction models where all the interactions
are of order less or equal M . The Boltzmann distribution has been extensively applied in single-objective evolutionary algorithms to implement selection and study
the theoretical properties of model-building algorithms. In this paper we propose
the combination of the multi-objective NM-landscape model and the Boltzmann
distribution to obtain Pareto-front approximations. We investigate the joint effect
of the parameters of the NM-landscapes and the probabilistic factorizations in the
shape of the Pareto front approximations.
keywords: multi-objective optimization, NM-landscape, factorizations, Boltzmann
distribution

1

Introduction

One important question in multi-objective evolutionary algorithms (MOEAs) is how the
structure of the interactions between the variables of the problem influences the different
objectives and impacts in the characteristics of the Pareto front (e.g. discontinuities,
clustered structure, etc.). The analysis of interactions is also important because there
is a class of MOEAs that explicitly capture and represent these interactions to make
a more efficient search [3, 10, 14]. In this paper, we approach this important question
by combining the use of a multi-objective fitness landscape model with the definition
of probability distributions on the search space and different factorized approximations
to these joint distributions. Our work follows a similar methodology to the one used in
[11, 12, 13, 16] to investigate the relationship between additively decomposable singleobjective functions and the performance of estimation of distribution algorithms (EDAs)
[6, 8].
Landscapes models are very useful to understand the behavior of optimizers under
different hypothesis about the complexity of the fitness function. Perhaps the best known
example of such models is the NK fitness landscape [5], a parametrized model of a fitness
landscape that allows to explore the way in which the neighborhood structure and the
strength of interactions between neighboring variables determine the ruggedness of the
landscape. One relevant aspect of the NK-fitness landscape is its simplicity and wide
usability across disciplines from diverse domains.
1

Another recently introduced landscape model is the NM-landscape [9]. It can be seen
as a generalization of the NK-landscape. This model has a number of attributes that
makes it particularly suitable to control the strength of the interactions between subsets
of variables of different size. In addition, it is not restricted to binary variables and allows
the definition of functions on any arity.
In [20], the NM-landscape was extended to multi-objective problems and used to
study the influence of the parameters in the characteristics of the MOP. We build on the
work presented in [20] to propose the use of the multi-objective NM-landscape (mNMlandscape) for investigating how the patterns of interactions in the landscape model
influence the shape of the Pareto front. We go one step further and propose the use
of factorized approximations computed from the landscapes to approximate the Pareto
fronts. We identify the conditions in which these approximations can be accurate.

2

NM-landscape

2.1

Definition

Let X = (X1 , . . . , XN ) denote a vector of discrete variables. We will use x = (x1 , . . . , xN )
to denote an assignment to the variables. S will denote a set of indices in {1, . . . , N},
and XS (respectively xS ) a subset of the variables of X (respectively x) determined by
the indices in S.
A fitness landscape F can be defined for N features using a general parametric interaction model of the form:
F (x) =

l
X
k=1

βUk

Y

xi

(1)

i∈Uk

where l is the number of terms, and each of the l coefficients βUk ∈ R. For k = 1, . . . , l,
Uk ⊆ {1, 2, . . . , N}, where Uk is a set of indices of the features in the kth term, and the
length |UQ
k | is the order of the interaction. By convention [9], it is assumed that when
Uk = ∅, j∈Uk xj ≡ 1. Also by convention, we assume that the model is defined for
binary variables represented as xi ∈ {−1, 1}.
The NM models [9] comprise the set of all general interactions models specified by
Equation 1, with the following constraints:
• All coefficients βUk are non-negative.
• Each feature value xi ranges from negative to positive values.
• The absolute value of the lower bound of the range is lower or equal than the upper
bound of the range of xi .
One key element of the model is how the parameters of the interactions are generated. In [9], each βUk is generated from e−abs(N (0,σ)) , where N (0, σ) is a random number
drawn from a Gaussian distribution with mean 0 and standard deviation σ. Increasing
σ determines smaller range and increasing clumping of fitness values. In this paper, we
use the same procedure to generate the βUk parameters.
We will focus on NM-models defined on the binary alphabet. In this case, the NMlandscape has a global maximum that is reached at x = (1, . . . , 1) [9].
2

3

Multi-objective NM-landscapes

The multi-objective NM-landscape model (mNM-landscape) is defined [20] as a vector
function mapping binary vectors of solutions into m real numbers f(.) = (f1 (.), f2 (.), . . . , fm (.)) :
BN → Rm , where N is the number of variables, m is the number of objectives, fi (.) is
the i-th objective function, and B = {−1, 1}. M = {M1 , . . . , Mm } is a set of integers
where Mi is the maximum order of the interaction in the i-th landscape. Each fi (x) is
defined similarly to Equation (1) as:
fi (x) =

li
X

βUki

Y

xj ,

(2)

j∈Uki

k=1

where li is the number of terms in objective i, and each of the li coefficients βUki ∈ R.
For k = 1, . . . , li , Uki ⊆ {1, 2, . . . , N}, where Uki is a set of indices of the features in the
kth term, and the length |Uki | is the order of the interaction.
Notice that the mNM fitness landscape model allows that each objective may have
a different maximum order of interactions. The mNM-landscape is inspired by previous
extensions of the NK fitness landscape model to multi-objective functions [1, 2, 7, 21].
One of our goals is to use the mNM-landscape to investigate the effect that the combination of objectives with different structures of interactions has in the characteristics
of the MOP. Without lack of generality, we will focus on bi-objective mNM-landscapes
(i.e., m = 2) and will establish some connections between the objectives. In this section
we explain how the constrained mNM-landscapes are designed.
As previously explained, the NM-model is defined for (x1 , . . . , xN ) ∈ {−1, 1}. However, we will use a representation in which (x1 , . . . , xN ) ∈ {0, 1}. The following transformation [20] maps the desired representation to the one used by the mNM-landscape.
Given the analysis presented in [9], it also guarantees that the Pareto set will comprise
at least two points, respectively reached at (0, . . . , 0) and (1, . . . , 1) for objectives f1 and
f2 .
f1 (y) : yi = −2xi + 1
f2 (z) : zi = 2xi − 1

(3)
(4)

where y = (y1 , . . . , yN ) ∈ {−1, 1} and z = (z1 , . . . , zN ) ∈ {−1, 1} are the new variables
obtained after the corresponding transformation have been applied to x = (x1 , . . . , xN ) ∈
{0, 1}.
When the complete space of solutions is evaluated, we add two normalization steps to
be able to compare landscapes with different orders of interactions. In the first normalization step, f (x) is divided by the number of the interaction terms (li ). In the second
step, we re-normalize the fitness values to the interval [0, 1], this is done by subtracting
the minimum fitness value among all the solutions, and dividing by the maximum fitness
value minus the minimum fitness value.
Another constraint we set in some of the experiments is that, if M1 < M2 then,
βUk1 = βUk2 for all |Uki | ≤ M1 . This means that all interactions contained in f1 are
also contained in f2 , but f2 will also contain higher order interactions. Starting from a
single mNM-landscape f of order M we will generate all pairs of models M1 , M2 , where
3

M1 ≤ M2 ≤ M. The coefficients βUk for f1 and f2 will be set as in f . The idea of
considering these pairs of objectives is to evaluate what is the influence in the shape of
the Pareto front, and other characteristics of the MOPs, of objectives that have different
order of interactions between their variables.

4

Boltzmann distribution

The relationship between the fitness function and the variables dependencies that arise
in the selected solutions can be modeled using the Boltzmann probability distribution
[11, 12]. The Boltzmann probability distribution pB (x) is defined as
pB (x) = P

e

g(x)
T

x′

e

g(x′ )
T

,

(5)

where g(x) is a given objective function and T is the system temperature that can be
used as a parameter to smooth the probabilities.
The key point about pB (x) is that it assigns a higher probability to solutions with
better fitness. The solutions with the highest probability correspond to the optima.
Starting from the complete enumeration of the search space, and using as the fitness
function the objectives of an mNM-landscape, we associate to each possible solution xi of
i
the search space m probability values (p1Bi (xi ), . . . , pm
Bi (x )) according to the corresponding Boltzmann probability distributions. There is one probability value for each objective
and in this paper we use the same temperature parameter T = 1 for all the distributions.
Using the Boltzmann distribution we can investigate how potential regularities of the
fitness function are translated into statistical properties of the distribution [11]. This
question has been investigated for single-objective functions in different contexts [15, 18,
17] but we have not found report on similar analysis for MOP. One relevant result in
single-objective problems is that if the objective function is additively decomposable in
a set of subfunctions defined on subsets of variables (definition sets), and the definition
sets satisfy certain constraints, then it is possible to factorize the associated Boltzmann
distribution into a product of marginal distributions [12]. Factorizations allow problem
decomposition and are at the core of EDAs.

5

Experiments

In our experiments we investigate the following issues:
• How the parameters of mNM model determine the shape of the Pareto front?
• How is the strength of the interactions between variables influenced by the parameters of the model?
• Under which conditions can factorized approximations of the Boltzmann probability
reproduce the shape of the Pareto front?
Algorithm 1 describes the steps of our simulations. We use a reference NM landscape (N = 10, M = 2) and create a bi-objective mNM model from it using different
combinations of parameters σ and |Uki |.
4

Algorithm 1: Simulation approach
1
2
3
4
5
6
7
8

5.1

Define the mMN model using its parameters.
For each objective:
Evaluate the 2N points of the search space.
Compute the Boltzmann distribution.
Compute the univariate marginals from the Boltzmann distribution.
For all solutions, compute univariate distribution as the product of univariate
marginals.
Determine the Pareto front using the objective values.
Determine the approximation of the Pareto front using the univariate factorizations of all the objectives.

Influence of the mNM-landscape parameters

We investigate how the parameters of mNM model determine the shape of the Pareto.
Figure 1 (column 1) shows the evaluation of the 210 solutions that are part of the
search space for N = 10 and different values of σ and |Uki |. From row 1 to row 4,
the figures respectively show the objective values of the mNM landscape for different
combination of its parameters: (σ = 1, |Uki | = 1), (σ = 1, |Uki | = 19), (σ = 2, |Uki | = 1),
(σ = 2, |Uki | = 19).
The influence of σ can be seen by comparing the figure in row 1 with the figure in
row 2, and doing a similar comparison with figures in row 3 and row 4. Increasing σ from
1 to 19 produces a clustering of the points in the objective space. One reason for this
behavior is that several genotypes will map to the same objective values. The clustering
effect in the space of objectives is a direct result of the clumpiness effect described for
the NM-model when σ is increased [9].
The effect of the maximum order of the interactions can be seen by comparing the
figure in row 1 with the figure in row 3, and the figures in rows 2 and 4. For σ = 1,
adding interactions transforms the shape of the Pareto front from a line to a boomeranglike shape. For σ = 19, the 8 points are transformed into a set of 8 stripes that seem to
be parallel to each other. In both cases, the changes due to the increase in the order of
the interactions are remarkable.
In the next experiments, and in order to emphasize the flexibility of the mNMlandscape, we allow the two objectives of the same mNM-landscape to have different
maximum order of interactions. Figure 2 shows the objective values and Pareto fronts
of the mNM model for σ = 36 for the situation in which f1 has a maximum order of
interactions |Uk1 | = o and f2 has a maximum order of interactions |Uk2 | = o + 1. It can be
observed that the shapes of the fronts are less regular than in the previous experiments
but some regularities are kept.

5.2

Boltzmann distribution

Figure 1 (column 2) shows the Boltzmann probabilities associated to each mNM-landscape
model described in column 1, i.e., (p1Bi (xi ), p2Bi (xi )).
The Boltzmann distribution modifies the shape of the objective space but it does not
modify the solutions that belong to the Pareto set. This is so because the dominance
5

−3

1

1.6

−3

x 10

1.6

x 10

0.9
0.8

1.4

1.4

1.2

1.2

0.7

2

i

q2B

i

0.5

pB

f2

0.6
1

1

0.4
0.3
0.2

0.8

0.8

0.6

0.6

0.1
0
0

0.2

0.4

f1

0.6

0.8

0.4
0.4

1

0.6

0.8

1
p1B

1.2

1.4

0.4
0.4

1.6

1.6

0.8

1
q1B

1.2

1
q1

1.2

1.4

1.6
−3

x 10

i

−3

1

0.6

−3

x 10

i

−3

x 10

1.6

x 10

0.9
0.8

1.4

1.4

1.2

1.2

0.7

i

q2

2

B

i

0.5

pB

f2

0.6
1

1

0.4
0.3
0.2

0.8

0.8

0.6

0.6

0.1
0
0

0.2

0.4

f

0.6

0.8

0.4
0.4

1

0.6

0.8

1
p1

1.2

1.4

0.4
0.4

1.6

2.2

1.4

1.6
−3

B

x 10

i

−3

−3

1

0.8

x 10

i

0.9

0.6

−3

B

1

x 10

x 10

2

0.8
1.8

0.7

i

q2

2

0.4

B

i

0.5

pB

f2

1.1

1.6

0.6

1.4

1

1.2

0.3
1

0.9

0.2
0.8

0.1
0
0

0.2

0.4

f

0.6

0.8

0.6
0.6

1

0.8

1

1.2

1.4
1
pB

1

1.6

1.8

2

0.8
0.8

2.2

1.8

−3

x 10

1.25

x 10

1.2

1.6

0.8

1.15

0.7

1.1

1.4

0.6
B

2

1.2

q2

0.5

i

i

1.05

pB

f2

1.1

−3

x 10

0.9

0.4

1
0.95

1

0.3
0.2

0.9
0.85

0.8

0.8

0.1
0
0

11
qB

i

−3

1

0.9

−3

x 10

i

0.2

0.4

f

0.6

0.8

1

0.6
0.6

0.8

1

1.2
p1
B

1

i

1.4

1.6

1.8
−3

x 10

0.75
0.7

0.8

0.9

1
q1

1.1

B

i

1.2

1.3
−3

x 10

Figure 1: Objective values, Boltzmann distribution, and univariate approximations for
different values of σ and different maximum orders of interactions. Column 1: Evaluation
of the 210 solutions that are part of the search space for N = 10 and different values of
σ and |Uki |. From row 1 to row 4, the figures respectively show the objective values
of the mNM model for (σ = 1, |Uki | = 1), (σ = 19, |Uki | = 1), (σ = 1, |Uki | = 2) and
(σ = 19, |Uki | = 2). Column 2: Boltzmann distributions computed from the objectives.
Column 3: Univariate approximations of the Boltzmann distributions.

6

1
0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

f2

2

f

1
0.9

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
0

0.2

0.4

f

0.6

0.8

0
0

1

0.2

1

0.4

f

0.6

0.8

1

1

Figure 2: Objective values and Pareto fronts of the mNM model for σ = 36 and different
maximum orders of interactions: Left) |Uk1 | = 1 and |Uk2 | = 2, Right) |Uk1 | = 2 and
|Uk2 | = 3.

relationships between the points are preserved by the Boltzmann distribution. However,
the Boltzmann distribution “bends” the original objective space. This effect can be
clearly appreciated in rows 1 and row 4. In the first case, the line is transformed into a
curve. In the second case the parallel lines stripes that appear in the original objective
space change direction.
The Boltzmann distribution can be used as an effective way to modify the shape of
the Pareto while keeping the dominance relationships. This can be convenient to modify
the spacing between Pareto-optimal solutions, for more informative visualization of the
objective space, and for investigating how changes in the strength of selection could be
manifested in the shape of the Pareto front approximations.

5.3

Factorized univariate approximations

Figure 1 (column 3) shows the approximations of the Boltzmann distributions for the
two objectives, each approximation computed using the corresponding product of the
univariate marginals, i.e., (qB1 i (xi ), qB2 i (xi )). For |Uk | = 1, the approximations are identical to the Boltzmann distribution. This is because the Boltzmann distribution can be
exactly factorized in the product of its univariate marginal distributions. Therefore, as a
straightforward extension of the factorization theorems available for the single-objective
additive functions, we hypothesize that if the structure of all objectives is decomposable and the decompositions satisfy the running intersection property [11, 12], then the
associated factorized distributions will preserve the shape of the Pareto front.
However, the univariate approximation does not always respect the dominance relationships and this fact provokes changes in the composition and shape of the Pareto front.
This can be appreciated in rows 3 and 4, where the univariate approximation clearly departs from the Boltzmann distribution. Still, as shown in row 4, some characteristics
of the original function, as the discontinuity in the space of objectives, can hold for the
univariate factorization.
An open question is under which conditions will the univariate approximation keep the
7

dominance relationship between the solutions. One conjecture is that if the factorized
approximation keeps the ranking of the original functions for all the objectives then
the dominance relationship will be kept, but this condition may not be necessary. The
answer to this question is beyond the scope of this paper. Nevertheless, we include the
discussion to emphasize why explicit modeling of interactions by means of the mNM
landscape together with the use of the Boltzmann distribution is relevant for the study
of MOPs.

5.4

Interactions and dependencies in the mNM landscape

By computing bivariate and univariate marginals from the Boltzmann distribution and
computing the mutual information for every pair of variables we can assess which are the
strongest pair-wise interactions.
−5

12

x 10

σ=1
σ=3
σ=5
σ=7
σ=9
σ=11
σ=13
σ=15
σ=17
σ=19

10

Mutual information

8

6

4

2

0

−2
1

2

3

4
5
6
7
Maximum order of interactions

8

9

Figure 3: Influence of the maximum order of the interactions and σ in the mutual information.
In this section we analyze how the maximum order of the interactions and the σ
parameter affect the dependencies in the Boltzmann distribution. A reference NM model
with (N = 10) was generated and by varying the parameters M ∈ {1, . . . , 9} and σ =
2i + 1, i ∈ {0, 1, . . . , 9}) we generated different mNM landscapes. The results presented
in this section are the average of 10 models for each combination of parameters. We focus
on the analysis of the dependencies in only one of the objectives.
Figure 3 shows the values of the mutual information for the combinations of the
maximum order of the interactions and σ. When the maximum order of the interactions is
1, the approximation given by the univariate factorization is exact, therefore, the mutual
information between the variables are 0 for all values of σ. The mutual information is
maximized when the maximum number of interactions is 2. For these mNM landscapes
we would expect the univariate approximation to considerably distort the shape of the
Pareto front, as shown in Figure 1, column 3, rows 3 and 4.

8

Figure 3 shows that σ can be used to tune the strength of the interactions between
the variables. As σ increases the mutual information also increases. This fact would
allow us to define objectives that have interactions of the same maximum order but with
different strength.

5.5

Discussion

We summarize some of the findings from the experiments:
• Univariate factorizations are poor approximations for mNM models of maximum
order two and higher.
• The mutual information between the variables of the NM landscape is maximized
for problems with maximum order of interaction 2.
• The σ parameter can be used for changing the shape of the Pareto fronts and
increasing the strength of the interactions in the objectives. In particular, there is
a direct effect of σ in the discontinuity of the Pareto front and the emergence of
clusters.

6

Conclusions

We have shown how the mNM landscape can be used to investigate the effect that interactions between the variables have in the shapes of the fronts and in the emergence of
dependencies between the variables of the problem. We have shown that the Boltzmann
distribution can be used in conjunction with the mNM model to investigate how interactions are translated into dependencies. A limitation of the Boltzmann distribution is
that is can be computed exactly only for problems of limited size.
The idea of using the Boltzmann distribution to modify the Pareto shape of the
functions can be related to previous work by Okabe et al. [13] on the application of deformations, rotations, and shift operators to generate test functions with difficult Pareto
sets. However, by using the Boltzmann distribution we explicitly relate the changes
in the shape of the Pareto to the relationship interactions-dependencies determined by
the Boltzmann distribution. This can be considered as an alternative path to other approaches to creation of benchmarks for MOPs, like the combination of single-objectives
functions of known difficulty [4] or the maximization of problem difficulty by applying
direct optimization approaches [19].
Our results can be useful for the conception and validation of MOEAs that use probabilistic modeling. In this direction, we have advanced the idea that the effectiveness of a
factorized approximation in the context of MOPs may be related to the way it preserves
the original dominance relationships between solutions. We have shown that the Boltzmann distribution changes the shape of the fronts but does not change which solutions
belong to the Pareto front.

9

Acknowledgment
This work has been partially supported by IT-609-13 program (Basque Government) and
the TIN2013-41272P (Spanish Ministry of Science and Innovation) project. R. Santana
acknowledges support from the Program Science Without Borders No. : 400125/2014-5).

References
[1] H. Aguirre and K. Tanaka. Insights and properties of multiobjective MNKlandscapes. In Proceedings of the 2004 Congress on Evolutionary Computation CEC2004, pages 196–203, Portland, Oregon, 2004. IEEE Press.
[2] H. E. Aguirre and K. Tanaka. Working principles, behavior, and performance
of MOEAs on MNK-landscapes. European Journal of Operational Research,
181(3):1670–1690, 2007.
[3] P. A. Bosman and D. Thierens. Multi-objective optimization with diversity preserving mixture-based iterated density estimation evolutionary algorithms. International
Journal of Approximate Reasoning, 31(3):259–289, 2002.
[4] D. Brockhoff, T.-D. Tran, and N. Hansen. Benchmarking numerical multiobjective
optimizers revisited. In Proceedings of the Companion Publication of the 2015 on
Genetic and Evolutionary Computation Conference, pages 639–646, Madrid, Spain,
2015.
[5] S. Kauffman. Origins of Order. Oxford University Press, 1993.
[6] P. Larrañaga, H. Karshenas, C. Bielza, and R. Santana. A review on probabilistic
graphical models in evolutionary computation. Journal of Heuristics, 18(5):795–819,
2012.
[7] M. López-Ibánez, A. Liefooghe, and S. Verel. Local optimal sets and bounded
archiving on multi-objective NK-landscapes with correlated objectives. In Parallel
Problem Solving from Nature–PPSN XIII, pages 621–630. Springer, 2014.
[8] J. A. Lozano, P. Larrañaga, I. Inza, and E. Bengoetxea, editors. Towards a New
Evolutionary Computation: Advances on Estimation of Distribution Algorithms.
Springer, 2006.
[9] N. Manukyan, M. J. Eppstein, and J. S. Buzas. Tunably rugged landscapes with
known maximum and minimum. arXiv.org, arXiv:1409.1143, 2014.
[10] L. Marti, J. Garcia, A. Berlanga, C. A. Coello, and J. M. Molina. On current
model-building methods for multi-objective estimation of distribution algorithms:
Shortcommings and directions for improvement. Technical Report GIAA2010E001,
Department of Informatics of the Universidad Carlos III de Madrid, Madrid, Spain,
2010.

10

[11] H. Mühlenbein and T. Mahnig. Evolutionary algorithms and the Boltzmann distribution. In K. A. DeJong, R. Poli, and J. Rowe, editors, Foundation of Genetic
Algorithms 7, pages 133–150. Morgan Kaufmann, 2002.
[12] H. Mühlenbein, T. Mahnig, and A. Ochoa. Schemata, distributions and graphical
models in evolutionary optimization. Journal of Heuristics, 5(2):213–247, 1999.
[13] T. Okabe, Y. Jin, M. Olhofer, and B. Sendhoff. On test functions for evolutionary
multi-objective optimization. In Parallel Problem Solving from Nature-PPSN VIII,
pages 792–802. Springer, 2004.
[14] M. Pelikan, K. Sastry, and D. E. Goldberg. Multiobjective estimation of distribution algorithms. In M. Pelikan, K. Sastry, and E. Cantú-Paz, editors, Scalable
Optimization via Probabilistic Modeling: From Algorithms to Applications, Studies
in Computational Intelligence, pages 223–248. Springer, 2006.
[15] R. Santana, C. Bielza, and P. Larrañaga. Conductance interaction identification by
means of Boltzmann distribution and mutual information analysis in conductancebased neuron models. BMC Neuroscience, 13(Suppl 1):P100, 2012.
[16] R. Santana, P. Larrañaga, and J. A. Lozano. Interactions and dependencies in
estimation of distribution algorithms. In Proceedings of the 2005 Congress on Evolutionary Computation CEC-2005, pages 1418–1425, Edinburgh, U.K., 2005. IEEE
Press.
[17] R. Santana, P. Larrañaga, and J. A. Lozano. Protein folding in simplified models
with estimation of distribution algorithms. IEEE Transactions on Evolutionary
Computation, 12(4):418–438, 2008.
[18] R. Santana, R. B. McDonald, and H. G. Katzgraber. A probabilistic evolutionary
optimization approach to compute quasiparticle braids. In Proceedings of the 10th
International Conference Simulated Evolution and Learning (SEAL-2014), pages 13–
24. Springer, 2014.
[19] R. Santana, A. Mendiburu, and J. A. Lozano. Evolving MNK-landscapes with structural constraints. In Proceedings of the IEEE Congress on Evolutionary Computation
CEC 2015, pages 1364–1371, Sendai, Japan, 2015. IEEE press.
[20] R. Santana, A. Mendiburu, and J. A. Lozano. Multi-objective NM-landscapes. In
Proceedings of the Companion Publication of the 2015 on Genetic and Evolutionary
Computation Conference, pages 1477–1478, Madrid, Spain, 2015.
[21] S. Verel, A. Liefooghe, L. Jourdan, and C. Dhaenens. On the structure of multiobjective combinatorial search space:MNK-landscapes with correlated objectives.
European Journal of Operational Research, 227(2):331–342, 2013.

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