Ensemble of Multimodal Genetic Algorithms for Design and Decision
Making Support Problems
Evgenii Sopov, Eugene Semenkin and Ilia Panfilov
Department of Systems Analysis and Operations Research, Siberian State Aerospace University, Krasnoyarsk, Russia
Keywords: Genetic Algorithms, Ensemble Methods, Multimodal Optimization, Selective Hyper-Heuristic.
Abstract: Many problems of design and decision making support can be stated as optimization problems. For real-world
problems, sometimes it is necessary to obtain many alternative solutions to the problem. In this case
multimodal approach can be used. The goal of multimodal optimization (MMO) is to find all optima (global
and local) or a representative subset of all optima. In recent years many efficient nature-inspired techniques
have been proposed for real-valued MMO problems. At the same time, real-world design and decision making
support problems may contain variables of many different types, including integer, rank, binary and others.
In this case, the weakest representation (namely binary representation) is used. Unfortunately, there is a lack
of efficient approaches for problems with binary representation. In this study, a novel approach based on a
selective hyper-heuristic in a form of ensemble for designing multi-strategy genetic algorithm is proposed.
The approach controls the interactions of many search techniques (different genetic algorithms for MMO)
and leads to the self-configuring solving of problems with a priori unknown structure. The results of numerical
experiments for benchmark problems from the CEC competition on MMO and for some real-world problems
are presented and discussed.
1 INTRODUCTION
Decision making involves the choice of one or more
alternatives from a list of options. The list of options
usually contains both good and bad (more or less
acceptable) solutions. The aim of rational decision
making is to maximize some criterion that describes
quality of the choice (Sen and Yang, 2012). Design
problems can be formulated in the same way (Ray and
Liew, 2002), the only difference is that alternatives
are not defined beforehand. It is obviously that such
problems can be stated as optimization problems. In
this case, alternatives are considered as candidate-
solutions, and quality criteria are considered as
objectives. Additional requirements can be performed
by constraints.
Many real-world design and decision making
support problems are complex and bad-formalized,
thus the quality criterion is usually considered as the
“black-box” model. Moreover, alternatives are
represented by complex structures that contain
variable parameters of many different types,
including categorical, integer, rank, binary and
others. Such optimization problems require implying
more advanced optimization techniques like
evolutionary algorithms (EA).
The general EA scheme uses the conception of
collective search based on the natural selection and
nature-inspired (genetic and evolutionary) operations
(Holland, 1975; Goldberg, 1989). All EAs in this
study are assumed to be binary genetic algorithms
(GAs).
From many practical points of view, the only
solution to the problem can be not enough, even it is
the optimal solution. For example, we need fallback
solutions if the optimal solution can’t be realized.
Moreover, identification of many different (optimal
and suboptimal) solutions is useful for better
understanding of the problem.
Optimization problems that have more than one
optimal solution (or there exists only one global
optimum and several local optima in the feasible
solution space) are called multimodal. The goal of
multimodal optimization (MMO) is to find all optima
(global and local) or a representative subset of all
optima. EAs and GAs are efficient in the multimodal
environment as they use a stochastic population-
based search. At the same time, traditional EAs and
GAs have a tendency to converge to the best-found
160
Sopov, E., Semenkin, E. and Panfilov, I.
Ensemble of Multimodal Genetic Algorithms for Design and Decision Making Support Problems.
DOI: 10.5220/0005976401600167
In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 160-167
ISBN: 978-989-758-198-4
Copyright
c
 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

optimum losing population diversity. In recent years
MMO have become more popular, and many efficient
nature-inspired MMO techniques were proposed.
Almost all search algorithms are based on
maintaining the population diversity, but differ in
how the search space is explored and how optima
basins are located and identified over a landscape.
The majority of algorithms and the best results are
obtained for real-valued MMO problems (Das et al.,
2011). Unfortunately, there is a lack of efficient
approaches for problems with binary representation.
Existing techniques are usually based on general
ideas of niching and fitness sharing. Heuristics from
efficient real-valued MMO techniques cannot be
directly applied to binary MMO algorithms because
of dissimilar landscape features in the binary search
space.
In this study, a novel approach based on a
selective hyper-heuristic in a form of ensemble of
MMO GA is proposed. Its main idea is to use many
MMO techniques with different search strategies and
adaptively control their interactions. Such an
approach would lead to the self-configuring solving
of problems with a priori unknown structure.
The rest of the paper is organized as follows.
Section 2 describes related work. Section 3 describes
the proposed approach. In Section 4 the results of
numerical experiments are discussed. In the
Conclusion the results and further research are
discussed.
2 RELATED WORK
The problem of GA-based design and decision
making support is well-studied (Sen and Yang, 2012;
Kaklaukas, 2015). At the same time, many complex
real-world problems are still a challenge for GAs and
other nature-inspired techniques.
One of the ways for finding many efficient
alternatives is multi-objective problem statement (Li
et al., 2015). In this case, a set of Pareto-optimal
solutions is obtained instead of the only optimal. The
multi-objective statement needs more advanced GA-
based techniques, and subsequent analysis of the
obtained Pareto set approximation. The Pareto set can
also contain very contrast solutions that are
interesting from the mathematical point of view as
they are still Pareto-optimal, but are not acceptable
from the practical point of view.
Another way is MMO statement. Over the past
decade interest for this field has increased. The recent
approaches are focused on the goal of exploring the
search space and finding many optima to the problem.
Many efficient algorithms have been proposed. Good
survey of widespread MMO techniques can be found
in (Deb and Saha, 2010; Das et al., 2011; Liu et al.,
2011). As we can see from many studies, there is no
universal approach that is efficient for all MMO
problems. Many researches design hybrid algorithms,
which are generally based on a combination of search
algorithms and some heuristic for a niching
improvement. Another way is a combining many
basic MMO algorithms to run them in parallel,
migrate individuals and combine the results. In
(Bessaou et al., 2000) an island model is applied,
where islands are iteratively revised according to the
genetic likeness of individuals. In (Yu and Suganthan,
2010) four MMO niching algorithms run in parallel
to produce offspring, which are collected in a pool to
produce a replacement step. In (Qu et al., 2012) the
same scheme is realized using the clearing procedure.
The conception of designing MMO algorithms in
the form of an ensemble seems to be promising. A
selective hyper-heuristic (Burke et al., 2010) that
includes many different MMO approaches (different
search strategies) can deal with many different MMO
problems. And such a hyper-heuristic can provide
self-configuration due to the adaptive control of the
interaction of single algorithms during the problem
solving. This idea was implemented in (Sopov,
2015a). The approach has demonstrated good results
with respect to multi-objective and non-stationary
optimization. In this study, we will apply this concept
to the MMO problem.
3 PROPOSED APPROACH
Heuristic and meta-heuristic search algorithms for
complex optimization problems are well-studied and
widely discussed (Bianchi et al., 2009; Boussaida et
al., 2013). One of the applications of the heuristic
search algorithms is the design of EAs. In other
words, a heuristic is used to design a heuristic, and it
is called hyper-heuristic (Ross, 2005; Burke et al.,
2010; Maashi et al, 2015). There also exist examples
of hyper-hyper-heuristics, which are the extension of
the idea of hyper-heuristics to select or combine
hyper-heuristics and generate new hyper-heuristics
(Pillay, 2015).
In this study, we will present a hyper-heuristic for
the design and control of the GA ensemble. The most
important step of the multi-EA search is the
interaction of component EAs. The general approach
is the island model with random migrations of
individuals. The majority of the proposed techniques
are based on the “winner-take-all” concept. There
Ensemble of Multimodal Genetic Algorithms for Design and Decision Making Support Problems
161

also exist coevolutionary approaches that are usually
based on a problem decomposition. In this study, we
will combine many interaction methods.
The ensemble method can be also used in the field
of EA. The main idea is to include different search
strategies in the ensemble and to design effective
control of algorithm interaction. Our hypothesis is
that different EAs are able to deal with different
features of the optimization problem, and the
probability of all algorithms failing with the same
challenge in the optimization process is low.
Moreover, the interaction of algorithms can provide
the ensemble with new options for optimization,
which are absent in stand-alone algorithms.
The general structure of the self-configuring
multi-strategy genetic algorithm proposed in (Sopov,
2015a) is called Self*GA (the star sign corresponds
to the certain optimization problem.
The total population size (or the sum of
populations of all component algorithms) is called the
computational resource. The resource is distributed
between algorithms, which run in parallel and
independent over the predefined number of iterations
(called the adaptation period). All algorithms have the
same objective and use the same encoding (solution
representation). All populations are initialized at
random. After the distribution, each GA included in
Self*GA has its own population which does not
overlap with populations of other GAs. At the first
iteration, all algorithms get an equal portion of the
resource.
After the adaptation period, the performance of
individual algorithms is estimated with respect to the
objective of the optimization problem. After that,
algorithms are compared and ranked. Search
strategies with better performance increase their
computational resource (the size of their populations).
We will discuss the design of a Self*GA for
MMO problems that can be named SelfMMOGA.
At the first step, we need to define the set of
individual algorithms included in the SelfMMOGA.
In this study we use six basic techniques, which are
well-studied and discussed (Singh and Deb, 2006;
Das et al., 2011), and they can be used with binary
representation with no modification. We have
included the following component algorithms in the
SelfMMOGA: Clearing (Alg1), Sharing (Alg2),
Clustering (Alg3), Restricted Tournament Selection
or RTS (Alg4), Deterministic Crowding (Alg5) and
Probabilistic Crowding (Alg6).
The motivation of choosing certain algorithms is
that if the SelfMMOGA performs well with basic
techniques, we can develop the approach with more
complex algorithms in further works.
The adaptation period is a parameter of the
SelfMMOGA. Moreover, the value depends on the
limitation of the computational resource (total
number of fitness evaluations).
The key point of any coevolutionary scheme is the
performance evaluation of a single algorithm. For
MMO problems performance metrics should estimate
how many optima were found and how the population
is distributed over the search space. Unfortunately,
good performance measures exist only for benchmark
MMO problems, which contain knowledge of the
optima. Performance measures for black-box MMO
problems are still being discussed. Some good
recommendations can be found in (Preuss and
Wessing, 2013). In this study, the following criteria
are used.
The first measure is called Basin Ratio (BR). The
BR calculates the number of covered basins, which
have been discovered by the population. It does not
require knowledge of optima, but an approximation
of basins is used. The BR can be calculated as

(

)
=


=1,(,

)
∈





(1)

(
,
)
=
1,∈()
0,ℎ
where pop is the population, k is the number of
identified basins by the total population, l is the
indicator of basin coverage by a single algorithm, b is
a function that indicates if an individual is in basin z.
To use the metric (1), we need to define how to
identify basins in the search space and how to
construct the function b(x,z).
For continuous MMO problems, basins can be
identified using different clustering procedures. In
this study, for MMO problems with binary
representation we use the following approach. We use
the total population (the union of populations of all
individual algorithms in the SelfMMOGA). For each
solution, we consider a predefined number of its
nearest neighbours (with respect to the Hamming
distance). If the fitness of the solution is better than
its neighbours fitness, it is denoted as a local optima
and the centre of the basin. The number of neighbours
is a tunable parameter.
The function b(x,z) can be easily evaluated by
defining if individual x is in a predefined radius of
basin centre z. The radius is a tunable parameter. In
this study, we define it as
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
162

=


(2)
where k is the number of identified basins (=
|

|
).
The second measure is called Sum of Distances to
Nearest Neighbour (SDNN). The SDNN penalizes the
clustering of solutions. This indicator does not require
knowledge of optima and basins. The SDNN can be
calculated as

(

)
=

(

,)



(3)


(


,
)
=min
∈\{

}
{

(


,
)}
where d
nn
is the distance to the nearest neighbour, dist
is the Hamming distance.
Finally, we combine the BR and the SDNN in an
integrated criterion K:
=∙()+(1−)∙
()
(4)
where 
is a normalized value of , 
defines weights of the BR and the SDNN in the sum
(∈[0,1]).
At the coopearative stage, in many coevolutionary
schemes, all individual algorithms begin each new
adaptation period with the same starting points (such
a migration scheme is called “the best displaces the
worst”). For MMO problems, the best solutions are
defined by discovered basins in the search space. As
we already have evaluated the approximation of
basins (Z), the solutions from Z are introduced in all
populations replacing the most similar individuals.
4 EXPERIMENTAL RESULTS
To estimate the approach performance, we have used
the following list of benchmark and real-world
problems:
 Six binary MMO problems are from (Yu and
Suganthan, 2010). These test functions are
based on the unitation functions, and they are
massively multimodal and deceptive.
 Eight real-valued MMO problems are from
CEC’2013 Special Session and Competition on
Niching Methods for Multimodal Function
Optimization (Li et al., 2013a).
 Fuzzy rule base classification system design
using MMO GA.
 Designing loan portfolios for the Bank of
Moscow.
4.1 Benchmark Problems
We have denoted the functions as in the source
papers. Real-valued problems have been binarized
using the standard binary encoding with 5 accuracy
levels proposed in the CEC’13 competition rules. In
all comparisons, all algorithms have equal maximum
number of the objective evaluations, but may differ in
population sizes.
The following criteria for estimating the
performance of the SelfMMOGA over the benchmark
problems are used for continuous problems:
 Peak Ratio (PR) measures the percentage of all
optima found by the algorithm (5).
 Success Rate (SR) measures the percentage of
successful runs (a successful run is defined as a
run where all optima were found) out of all runs.
=
|{
∈|

(,)≤
}|

(5)
where =
{


,

,…,

}
is a set of known optima, 
is accuracy level.
The maximum number of function evaluation and
the accuracy level for the PR evaluation are the same
as in CEC completion rules (Li et al., 2013a). The
number of independent runs of the algorithm is 50.
In the case of binary problems, we cannot define
the accuracy level in the PR, thus the exact points in
the search space have to be found. This is a great
challenge for search algorithms, thus we have
substituted the SR measure with Peak Distance (PD).
The PD indicator (6) calculates the average distance
of known optima to the nearest individuals in the
population (Preuss and Wessing, 2013).
=
1



(

,)



(6)
The detailed results of estimating the performance
of the SelfMMOGA with the pack of binary problems
can be found in (Sopov, 2015b). We have compared
the results with Ensemble of niching algorithms
(ENA) proposed in (Yu and Suganthan, 2010). The
experiments have shown that binary problems are not
too complex for the SelfMMOGA and the ENA –
there is no statistical significant difference in the
results.
The results of estimating the performance of the
SelfMMOGA with the pack of continuous problems
are presented in Tables 1. Table 1 shows a
comparison of results averaged over all problems
with other techniques.
Ensemble of Multimodal Genetic Algorithms for Design and Decision Making Support Problems
163

Table 1: Average PR and SR for each algorithm.
ε
SelfMMOGA DE/nrand/1/bin cDE/rand/1/bin N-VMO dADE/nrand/1 PNA-NSGAII
PR SR PR SR PR SR PR SR PR SR PR SR
1e-01 0.962 0.885 0.850 0.750 0.963 0.875 1.000 1.000 0.998 0.938 0.945 0.875
1e-02 0.953 0.845 0.848 0.750 0.929 0.810 1.000 1.000 0.993 0.828 0.910 0.750
1e-03 0.943 0.773 0.848 0.748 0.847 0.718 0.986 0.813 0.984 0.788 0.906 0.748
1e-04 0.907 0.737 0.846 0.750 0.729 0.623 0.946 0.750 0.972 0.740 0.896 0.745
1e-05 0.816 0.662 0.792 0.750 0.642 0.505 0.847 0.708 0.835 0.628 0.811 0.678
Average 0.916 0.780 0.837 0.750 0.822 0.706 0.956 0.854 0.956 0.784 0.893 0.759
We have compared the results of the
SelfMMOGA runs with some efficient techniques
from the competition. The techniques are
DE/nrand/1/bin and Crowding DE/rand/1/bin (Li et
al., 2013a), N-VMO (Molina et al., 2013),
dADE/nrand/1 (Epitropakis et al., 2013), and PNA-
NSGAII (Bandaru and Deb, 2013).
The settings for the SelfMMOGA are:
 Maximum number of function evaluation is
50000 (for cecF1-cecF5) and 200000 (for
cecF6-cecF8);
 Total population size is 200;
 Adaptation period is 10 generations 25 times
(for cecF1-cecF5) and 25 generations 40 times
(cecF6-cecF8);
 All specific parameters of individual algorithms
are self-tunable.
As we can see from Tables 1, the SelfMMOGA
shows results comparable with popular and well-
studied techniques. It yields to dADE/nrand/1 and N-
VMO, but we should note that these algorithms are
specially designed for continuous MMO problems,
and have taken 2nd and 4th places (Li et al., 2013b),
respectively, in the CEC competition. At the same
time, the SelfMMOGA has very close average values
to the best two algorithms, and outperforms PNA-
NSGAII, CrowdingDE and DE, which have taken 7th,
8th and 9th places in the competition respectively.
In this study, we have included only basic MMO
search techniques in the SelfMMOGA. Nevertheless,
it performs well due to the effect of collective
decision making in the ensemble. The key feature of
the approach is that it operates in an automated, self-
configuring way. Thus, the SelfMMOGA can be a
good alternative for complex black-box MMO
problems.
4.2 Real-world Problems
4.2.1 Designing Loan Portfolios for the
Bank of Moscow
The problem of bank loan portfolio design is an
optimization problem of maximizing the profit of the
bank with some constraints on the amount of free
liabilities, the amount of credit requested, periods of
credits, credit interests and so on. Input data to the
problem is a set of credit requests from loan
borrowers. The bank portfolio is a subset of requests
that are approved by the bank.
In this paper, the loan portfolio based on data
presented by Krasnoyarsk department of the Bank of
Moscow is discussed. The following profit model
(optimization objective) is used (7):
Profit(X) =
∑


∙(1+

∙

)∙

→


Risk(X) =

∑




∙
∑


∙

≤




∙

≤


(7)
=
(


,

,…,

)
,

∈{0,1}
where F – the amount of free liabilities held by the
Bank at a given time; N – the number of borrowers; k
j
– the amount of credit requested by the j-th borrower
j=1,N; t
j
– the period for which the j-th borrower takes
a loan; x
j
– Boolean variable taking the value 1, if the
k
j
loan is issued, and 0 otherwise; d
j
– interest (%) on
j-th credit; P
j
– probability of non-payment of loan
and interest on the loan; ρ – limitation on the total
riskiness of the loan portfolio.
As a candidate solution is binary vector, there is
no need to encode it to chromosome. The fitness
function is defined as the sum of the Profit and
penalty functions for given constraints.
The initial information about credit requests and
their characteristics are presented in Table 2.
The length of the chromosome is 50. The search
space contains 2
50
(≈10
15
) different portfolios. The
maximum number of the fitness evaluation is set to
10
6
that is 10
-9
% of the cardinality of the search
space.
The results of the bank portfolio design (global
and three local solutions) are presented in Table 3.
ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics
164

Table 2: Initial data for the loan portfolio design problem.
Request
No.
Request amount
Loan rate
(%)
Period Riskiness
1 10 000 000 25 75 0.042
2 5 300 000 28 80 0.039
3 2400000 25 91 0.029
4 50 000 000 23 84 0.033
5 1 000 000 28 64 0.026
.........
48 9 000 000 27 86 0.024
49 22 000 000 29 91 0.016
50 350 000 27 69 0.026
Total sum of requests = 256 695 000
The amount of free liabilities = 188 500 000
Table 3: Results for the loan portfolio problem.
Solutions
(the structure of the loan portfolio)
Profit of
portfolio
Rest of free
liabilities
Total riskiness
01111011111111110111101000111110111010000101010111 199734518.9 30000 0.0292
11011100111110110110011101110101111010100101011110 199691164 15000 0.0286
01011110101111100011101110111111011001111011001110 199668728.9 15000 0.028
00110011010001010110111000110110101111110101101111 199593407.3 10000 0.0276
As we can see from Table 3, solutions obtained
with the SelfMMOGA have very close values of the
profit, but have very different structures. Thus these
portfolios can be used as alternative solutions or as
additional information for the portfolio analysis.
The problem has been also solved using the brute-
force search. The first best solution founded by the
SelfMMOGA is the exact global solution to the
problem.
4.2.2 Fuzzy Rule Base Classification System
Design using MMO GA
Modern machine learning methods often use
evolutionary computation techniques as a design tool,
which is universal and can be applied for various
structures. These evolutionary algorithms applied for
machine learning problems are often called genetics-
based machine learning algorithms. The fuzzy rule-
based classification systems (FRBCSs) are effective
approaches in machine learning, as they can provide
easy-to-understand models for the end users
(Ishibuchi, 2005).
Traditional GAs applied to the FRBCSs design
have a tendency to converge to the best-found
optimum losing population diversity. Such single
best-found solution usually has very good accuracy,
but may have a structure that is not convenient for
human understanding and analysis. Thus there is a
good idea to find many (or all) global and acceptable
local optima which represent different solutions to the
problem. In a case of the FRBCS, such optima, while
saving comparable accuracy, may contain different
rules in the rule base and/or different fuzzy term
structures.
The number of rules in computational
experiments was fixed and equal to 12. The FRBCS
method, which have been implemented, is based on a
simple rule base encoding into the GA chromosome.
The chromosome contains fuzzy sets assigned to
input variables in the premise part and class labels
assigned to output variables in the conclusion part of
each rule in the rule base. The number of fuzzy sets
for granulation was fixed and equal to 5+1.
Additional fuzzy term is the “Don’t care” condition
(corresponding input variable is ignored).
The fitness function includes two values: error on
the training set and the complexity of the rule base.
The complexity of the rule base was calculated as the
ratio of number of non-empty fuzzy sets to the total
number of possible fuzzy sets in the rule base.
Including complexity of the rule base into the fitness
function allows creating of simpler rule bases. The
distance between two rule bases for the MMO GA
was calculated as the number of different fuzzy sets
for these rule bases. More detailed information can be
found in (Sopov et al., 2015).
The computational experiments for the fuzzy
classification were performed on 7 datasets from UCI
and KEEL repositories (KEEL, 2015; ics.uci.edu,
2015).
Ensemble of Multimodal Genetic Algorithms for Design and Decision Making Support Problems
165

Table 4: Classification Results for Test Sample.
Dataset Standard GA
SelfMMOGA
Solution 1
SelfMMOGA
Solution 2
SelfMMOGA
Solution 3
Australian 0.839 0.862 0.867 0.816
Banknote 0.947 0.892 0.867 0.862
Column 2c 0.773 0.789 0.768 0.751
Column 3c 0.668 0.741 0.674 0.619
Ionosphere 0.747 0.680 0.656 0.665
Liver 0.567 0.586 0.597 0.598
Seeds 0.874 0.793 0.691 0.621
The Table 4 contains the classification results for
the test sample obtained with the standard GA and
three best solutions obtained with the SelfMMOGA.
As we can see, for three datasets the standard GA
allows finding most accurate solutions. However, the
SelfMMOGA outperforms the standard GA on 4
datasets out of 7. Moreover, the best solution is not
always the first one – for example, for datasets
Australian and Liver, the best solution was second or
even third. More detailed results and obtained rule
bases description can be found in (Sopov et al., 2015).
Thus, using this method, several local optima
have been found, and the researcher is able to select
one of them. We suggest that the results can help the
human experts in a field of the solving problem to
obtain better (or may be very new) information about
the problem features.
5 CONCLUSIONS
In this study, a selective hyper-heuristic for control of
MMO GA ensemble (called SelfMMOGA) is
proposed. It involves many different search strategies
in the process of MMO problem solving and
adaptively control their interactions.
The SelfMMOGA allows complex MMO
problems to be dealt with, which are the black-box
optimization problems (a priori information about the
objective and its features are absents or cannot be
introduces in the search process). We have included 6
basic MMO techniques in the SelfMMOGA
realization to demonstrate that it performs well even
with simple core algorithms. We have estimated the
SelfMMOGA performance with a set of binary
benchmark MMO problems and continuous
benchmark MMO problems from CEC’2013 Special
Session and Competition on Niching Methods for
Multimodal Function Optimization. The proposed
approach has demonstrated a performance
comparable with other well-studied techniques.
Experimental results show that the SelfMMOGA
outperforms the average performance of its stand-
alone algorithms. It means that it performs better on
average than a randomly chosen technique. This
feature is very important for complex black-box
optimization, where the researcher has no possibility
of defining a suitable search algorithm and of tuning
its parameters.
We have also applied the SelfMMOGA for
solving some real-world problems to demonstrate the
effect of identifying many optima to the problem of
design and decision making support.
In further works, we will investigate the
SelfMMOGA using more advanced component
techniques.
ACKNOWLEDGEMENTS
The research was supported by President of the
Russian Federation grant (MK-3285.2015.9) and the
Russian Foundation of Basic Research (contract №20
16-01-00767, dated 03.02.2016).
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