=Paper=
{{Paper
|id=Vol-2353/paper53
|storemode=property
|title=Applied Tasks Modeling by Combinator Problems with Square Target Function and Approach to Their Solution
|pdfUrl=https://ceur-ws.org/Vol-2353/paper52.pdf
|volume=Vol-2353
|authors=Liudmila Koliechkina,Alla Nahirna,Olena Dvirna
|dblpUrl=https://dblp.org/rec/conf/cmis/KoliechkinaND19
}}
==Applied Tasks Modeling by Combinator Problems with Square Target Function and Approach to Their Solution==
https://ceur-ws.org/Vol-2353/paper52.pdf
Quadratic Optimization Problem on Permutation Set
with Simulation of Applied Tasks
Liudmyla Koliechkina1[0000-0002-4079-1201], Alla Nahirna2[0000-0002-2863-8706],
Olena Dvirna3[0000-0002-0750-6958]
1
University of Lodz, Uniwersytecka Str. 3, 90-137 Lodz, Poland
ludaplt1971@gmail.com
2
Kyiv International University, Lvivska str., 49, Kyiv, 03179, Ukraine
naghirnaalla@ukr.net
3
Poltava University of Economics and Trade, Koval str., 3, Poltava, 36000, Ukraine
lenadvirna@gmail.com
Abstract. The article discusses the formulation of an optimization problem with
a quadratic target function and additional constraints on the permutation set,
which can be a model of many applied problems. An algorithm for solving an
optimization problem with a quadratic target function and additional constraints
on permutations is proposed.
During the implementation of the method the first reference plan is found and
additional restrictions for it are checked at the first stage. Thus, in the beginning
of the algorithm, the number of considered solutions decreases. This makes it
possible at the first stage to reduce the number of possible solutions and narrow
the area of the problem study. An example of solving a theoretical problem us-
ing this method, demonstrating its effectiveness, is proposed.
Such task can be used to modeling various technological processes. The reason
for this is the optimization of mathematical models and algorithms for the pro-
posed models.
Keywords: optimization problems, combinatorial set of permutations, model of
optimization problems, quadratic target function, optimal solutions.
1 Introduction
Computer simulation of technological processes consists in optimization of the study
object by a mathematical model, and further model study with the help of imple-
mented computational algorithms on personal computers. The computer simulation
process, as a single process of building and researching a model, is used for research,
analysis, design and optimization of technological objects and systems [1]. Today,
computer modeling approaches combines both theory and practice. Working with a
model that represents a research object gives you the opportunity to explore its prop-
erties and behavior in all situations.
Computer modeling is the process of building a real study object's model and set-
ting up computational experiments on this model in order to understand and evaluate
various strategies based on the use of algorithms that ensure the functioning of this
study object. Thus, the process of computer simulation includes the model design, and
its application to solve the problem of optimization and design of technological proc-
esses in production. All these problems are extremely complex and include numerous
elements, variables, parameters, constraints, etc. Such problems can be modeled by
combinatorial optimization tasks [1-8].
The combinatorial optimization problems study comprises a fairly wide range of
mathematical models associated with the need to solve various important practical
problems of optimal planning, management and design [1, 2, 5, 7, 9, 10-12, 20-29]. In
this regard, many papers have recently appeared which investigate the combinatorial
optimization problems and propose approaches to their solution [8, 11-34, 36-38].
The paper presents a model of optimization problems on a combinatorial set of
permutations, which is a model of many applied problems. It is proposed the algo-
rithm for solving an optimization problem with a quadratic target function and addi-
tional restrictions on permutations.
During the method's implementation the first reference plan is located and addi-
tional restrictions are checked, which reduces the number of considered solutions at
the beginning of the algorithm
It is offered an example of solving a theoretical problem by the given method and
demonstrating its efficiency.
2 Literature review
Computer models have become a common tool for mathematical modeling and are
used in various areas of electronics, engineering, industry, and so on. Computer
models are used to obtain new knowledge about the object or for an approximate
assessment of the system’s behavior that are too complex for analytical research.
Computer modeling is one of the effective methods for studying complex systems.
The construction of a computer model is based on abstracting from the specific nature
of the phenomena or the original object under study and consists of two stages – first,
the creation of a qualitative, and then a quantitative model. The more significant
properties will be identified and transferred to the computer model – the more
approximate it will be to the real model, the greater the capabilities that the system
using this model will have. Today the works of many scientists are devoted to this
research area [1-5, 8-10, 21].
An important computer simulation area is analytical and simulation modeling. In
analytical modeling, mathematical (abstract) models of a real object are studied in the
algebraic equations form, as well as providing for the implementation of an
unambiguous computational procedure leading to their exact solution. In simulation,
mathematical models are studied in the algorithms form, which reproduce the
functioning of the system under study by sequentially performing numerous
elementary operations. Very relevant today for simulation and analytical modeling are
discrete models, in particular, combinatorial optimization, their study is the subject of
a great number of papers [1–17], [22–31].
A significant contribution to the development of modern discrete optimization
theory, the development and implementation of its methods in the study and solving
of important applied problems have been made over the course of more than forty
years under the scientific guidance of Academician I.V. Sergienko scientists of the
Institute of Cybernetics named by V.M. Glushkov National Academy of Sciences of
Ukraine. They obtained important results, which became the basis for the theory
development and the creation of n lines in discrete optimization.
At present, intensive studies of the stability problems of vector discrete
optimization problems are carried out at the Institute of Cybernetics named by V.M.
Glushkov of the National Academy of Sciences of Ukraine (I.V. Sergienko, T.T.
Lebedev, N.V. Semenova, T.I. Sergienko) [16, 17], Belarusian State University
(V.O.Emelyichev, D.P. Podkopayev, Yu.V. Nikulin, etc.) [6, 7], the Joint Institute of
Computer Sciences of the National Academy of Sciences of Belarus (Yu.N.Sotskov),
the Computational Center of the Russian Academy of Sciences (E.M. Gordeyev, V.K.
Leontiev), Omsk Branch of the Institute of System Studies of the Polish Academy of
Sciences (M. Libura), a number of universities of the Federal Republic of Germany
(E. Girlich), the Netherlands (ES van der Poort, G. Sierkma, APM Wagelmans, JAA
van der Veen), in the uniquely Colorado US (H.J. Greenberg).
Belarusian scientists under the scientific guidance of Professor V.O. Yemelycheva
successfully develops a constructive approach to the problem of the vector discrete
optimization problems correctness associated with the reception of quantitative
characteristics of the stability of specified problem statements, a mathematical
apparatus for studying the stability of multicriteria discrete problems with different
types, vector criteria functions, principles of optimality, as well as generalized and
parameterized principles of optimality are developed.
It is important to study the relation between different classes of point
configurations triangulation (regular, weakly regular, deployed, symmetric, political,
etc.), in particular, the study of the structure of a partially ordered set constructed on
the family of triangulation classes considered in relation to the inclusion relation.
Recently, algorithms for solving convex polyhedral admissibility problems are
actively explored [11, 18, 19]. One of the interesting computational approaches is the
reduction of the admissibility problem for a polyhedron to the projection problem on a
normal cone generated by a dual system of inequalities [18, 29, 31], which is
sufficiently close to the projection problem for a binary polyhedron. Today, in the
research area of various classes of combinatorial models, the new methods
development for their solution, great attention is paid to methods based on the use of
structural properties of combinatorial sets. The properties of combinatorial sets study
is closely related to the theory of polyhedral and graphs. The use of information about
the structure of the convex shell of admissible multivariate solutions, which is the
basis for many methods, is one of the most successful approaches to solving
combinatorial optimization problems for today. But when solving such tasks there are
problems related to the complexity of mathematical models, large volume of
information, etc.
Applied problems simulated by extreme discrete tasks often have a high
dimensionality, so they are quite complex from a computational point of view. The
main task is to determine the value of an argument belonging to a certain
combinatorial configuration for which the target function acquires the global
optimum. So it is necessary to develop the most effective algorithm, which is based
on the specific properties of the combinatorial configurations.
For the extreme problems of combinatorial optimization, polynomial algorithms
for finding an optimal solution based on the properties of the input data structure have
been developed, but there are few such work compared to methods based on partial
overview of the options. One of the approaches to solving such problem is to carry
out research and analysis of the combinatorial configurations properties, in which the
target function, which reflects the combinatorial nature of the tasks, is determined.
Analysis and study of combinatorial configurations as a target function argument,
setting the change in the values of the target function, depending on the elements
ordering of the selected combinatorial configuration and the structure of the input data
specificity, does not pay sufficient attention in the literature. But it should be noted
that the study of the certain tasks properties in order to identify their characteristic
properties and their use for solving the problem, gives the possibility of constructing
new approaches and the development of known methods.
Hence, one of the important problems in the discrete optimization area is the
detection of the properties of combinatorial configurations in extreme problems, the
use of which would allow to establish the regularity of the change in the values of the
target functions, depending on the argument ordering and on the specificity and
structure of the combinatorial configurations sets.
Today, significant results have been obtained in the area of research of
combinatorial models' various classes and the development of new methods for their
solution. The following foreign scientists made a fundamental contribution to the
development of discrete, in particular, combinatorial optimization: M. Gary, S. Berge,
D. Johnson, H. Papadimitriou, P. Pardalos, K. Staiglich, R. Stanley, F. Harari, V.A.
Emelichev, V.M. Sachkov [1-8, 10, 11, 12].
In turn, the many Ukrainian scientists' works are devoted to the various classes of
combinatorial optimization problems' study: L.F. Gulyanitsky, P.I. Stetsyuka, I.V.
Sergienko, N.S. Shor, Yu.G. Stoyan, S.V. Yakovlev, A.O. Yemetsa, V.O. Perepelitsy
and many others.
In particular, in [18–21], the authors describe the convex extensions theory and its
applications in combinatorial optimization problems. Combinatorial models'
applications in practical problems of geometric design are presented in the works of
L.F. Gulyanitsky, I.V. Sergienko, Yu.G. Stoyan, S.V. Yakovlev, N.S. Shor [22-27,
38].
Quadratic optimization on the permutation set is reflected in [28–31].
The permutation set's representation as the intersection of a permutation
polyhedron and a hypersphere is interesting, as well as optimization methods on
permutation configurations using the intersection described in [32].
The development of an integrated approach to the analysis of the properties of
combinatorial optimization supplies covers a wide range of studies of combinatorial
functions, combinatorial polyhedral, combinatorial configurations as an argument of
the target function. The results give the opportunity to improve existing methods for
solving such problems and develop new methods for optimal solutions searching. The
problems based on the properties of combinatorial configurations are actual problems
of combinatorial optimization. Of particular importance in this aspect is the
consideration of extreme problems in combinatorial configurations using graph
theory.
The research of tasks in graphs deals with such scientists as F. Harari, O. Ore,
I.V. Sergienko, V. O. Yemelichev, A. O. Zikov, V. O. Perepelytsya, R. I. Tyshkevich
and others. Despite quite large achievements in the area of discrete optimization, in
the process of modeling, there are extreme problems classes for which a number of
issues have not yet been investigated. Principal difficulties that arise during modeling
are also related to various types of uncertainty. These include: the availability of
many criteria for evaluating the quality of solutions, interval setting of task
parameters, etc. In these conditions, classical methods are not sufficiently suitable for
solving problems. As you know, most combinatorial optimization tasks can be
reduced to integer programming tasks, but this is not always justified, since it
eliminates the possibility of taking into account the combinatorial properties of task
solutions.
This work is a continuation of research in the extreme discrete problem’s area, in
particular, combinatorial optimization, as well as optimization problems under the
conditions of multicriteria, which arise in the study of many theoretical and applied
problems.
3 Formal problem statement
We consider the permutation as an ordered sample of elements a (a i1 , a i2 ,..., a ik ) ,
where ai j A , i j N n , j N n is it if s t s N n , t N n from some
multi-set A {a1 , a2 ,..., aq } , which is characterized by the base S ( A) {e1 , e2 ,..., ek } ,
where e j R1 , j N k and the multiplicity of the elements k (e j ) r j ,
j N k , r1 r2 ... rk q j N k , according to [14, 17].
A set of permutations with repetitions of n real numbers, among which k different,
is called the general permutation set and is denoted as Pnk (A) . This is the set of or-
dered n-samples from the multiset A under the condition n q k .
If we have n k q set of permutations without repetition, we denote it as Pn .
Obviously, Pn ( A) Pnn ( A) . In cases where the form of the set of permutations is not
indicated, it will be possible to write down these sets as P(A) .
It is known [14] that the convex hull of the set of permutations is a permutation
polyhedron conv P ( A) , which vertex set P(A) is equal to the set of permuta-
tions: vert ( A) P ( A) .
Without loss of generality, we order the elements of the multiset A in non-
descending order: a1 a2 ... aq and the elements of its foundation –in descending
order: e1 e2 ... ek .
Then the convex hull of a common set of P(A) permutations is a generic polyhe-
dron ( A) conv P ( A) , that is described by a well-known system of linear inequali-
ties:
n n
xj
j 1
j 1
aj,
i (1)
i
j 1
x j
j 1
aj,
j N n , j t , j t , j , t N i , i N n and P ( A) vert ( A) .
Consider the optimization problem:
Z (, P ( A)) : max{ ( a ) | a P ( A)} (2)
D {x R n | Gx ()b} (3)
where G R mn , b R m
n n n
(a )
j 1
c j x 2j a x x
i 1 j 1
ij i j (4)
on a set of permutations Pn ( A) .
Additional linear constraints form a multifaceted set D R n .
Then, to every point a Pn (A) will match point x X , one that satisfies equality
F ( x) (a) :
Z ( F , X ) : max{F ( x) | x X } (5)
n n n
where, F ( x)
j 1
c j x 2j a x x
i 1 j 1
ij i j (6)
and additional constraints:
D {x R n | Gx ()b} (7)
mn m
where G R , bR
X - nonempty set in R n , which is defined as follows:
X vert ( A) , conv P( A) (8)
It is natural to assume that the maximum of a quadratic function will be one of the
vertices of the permutation polyhedron, and the vertices of the graph G ( Pn ) will
match to all points of the set of permutations Pn ( A) .
The adjacency of the vertices of a permutation polyhedron is determined by a one-
time transposition of two vertex elements. The number of transpositions in the graph
is determined by the formula [35]:
n( n 1)
C n2 (9)
2
4 The algorithm for solving an optimization problem with a
quadratic target function and additional constraints on the
set is rearranged
The algorithm for finding the problem optimal solution consists of four steps, which
in a few steps make it possible to obtain an optimal solution.
STEP 1. Finding the first support solution.
Consider the first transposition of the target function:
x1 x 2 , (i 2,..., n) :
f12 ( x1 x 2 )(a1 x1 a1 x 2 ... a n 2 x n )
We form the necessary conditions for finding the first solution:
x1 x 2 ,
(10)
xi max, (max ai ).
Variants of transpositions:
x1 x 2 , (i 2,..., n) :
f12 ( x1 x 2 )(a1 x1 a1 x 2 ... a n 2 x n )
x1 x 3 : f 13 ( x1 x 3 )(a13 x1 a 23 x 2 ... a n3 x n )
………
x1 x n : f 1n ( x1 x n )(a1n x1 a 2 n x 2 ... a nn x n ) (11)
x 2 xi , (i 3,..., n);
x1 a 22
x 2 x 3 : f 23 ( x 2 x 3 )(a12 x 2 ... a n 2 x n )
………
x 2 x n : f 2 n ( x 2 x n )(a1n x1 a 2 n x 2 ... a nn
xn )
………
x n 1 x n : f n 1n ( x n 1 x n )(a1n 1 x1 a 2 n 1 x 2 ... a nn 1 x n )
The first solution will be: ( x1 , x 2 ,..., x n ) . It should be noted that there may be sev-
eral first solutions.
STEP 2. Check constraints: ( g1 , g 2 ,..., g n ) .
When checking constraints, the following options are possible:
All constraints are satisfied, then go to step 3.
At least one of the restrictions is not satisfied, then the next solution found for the
given transposition is considered. If there are none, then we consider the next point in
ascending order and proceed to step 1.
In the case of consideration of all transpositions, it is necessary to consider the
point in ascending order by three transpositions, four, etc.
STEP 3. Formation of conditions for finding the optimal solution.
Initial conditions for finding the optimal solution:
f ( x1 , x2 ,..., xn ) b,
g1 ( x1 , x 2 ,..., x n ) ()b1 , b1 b1 g1 ( x1 , x 2 ,..., x n ),
g 2 ( x1 , x 2 ,..., x n ) ()b2 , b2 b2 g 2 ( x1 , x 2 ,..., x n ),
(12)
...............................................................................,
g n ( x1 , x 2 ,..., x n ) ()bn , bn bn g n ( x1 , x 2 ,..., x n ).
STEP 4. Improved support solution.
Choose the next point from the set of permutations, which is better than the first
support solution. Next, we transpose this point with respect to the first support solu-
tion and find the numerical value of the transposition of the target function:
f tr ( x1 , x 2 ,..., x n ) b (13)
Prerequisite:
f tr ( x1 , x 2 ,..., x n ) b – growth.
4.1. If this condition is true, then it is necessary to check the growth of restrictions:
g i g i2 g 1i x ig i * c j x gj i * c i x gj i * c j x ig i * c i
2 2 1 1
(14)
All increments satisfy conditions (12), then the found point is the optimal solution.
Otherwise, we return to the beginning of the step 4.
4.2. If condition (13) is not fulfilled, we return to the beginning of step 4.
It should be noted that conditions (12) are sufficient for finding the optimal solu-
tion, and the fulfillment of inequality (13) is necessary for finding the optimal solu-
tion.
5 Example
Find the maximum value of the target function:
1 1
max F ( x) (2 x1 x 2 x3 x 4 ) 2 ( x 2 x3 ) 2 ( x3 x 2 2 x 4 ) 2 2 x 42
2 2
on the permutations set A4 ,where A (1,2,3,4) , with the following restrictions:
g1 5 x1 2 x 2 3 x3 4 x 4 15,
g 2 3x1 6 x 2 8 x3 x 4 31.
Solving.
For formula (1), the number of possible transpositions is: С 42 6 .
Consider the first transposition x1 x 2 , presenting the target function as a prod-
uct:
f12 ( x1 x 2 )( x1 x 2 6 x3 x 4 ) ,
accordingly, when searching for the first solution, it is necessary to consider the fol-
lowing conditions:
x1 x 2 ,
x3 min .
Then the first solution may be the following points of the permutations set: (4,3,1,2),
(4,2,1,3), (3,2,1,4).
Consider the first point (4,3,1,2):
g1 (4,3,1,2) 25 15,
g 2 (4,3,1,2) 12 31.
Additional restrictions are enforced. Then the initial conditions for improving the first
support solution will be:
max f1 (4,3,1,2) 40,
g11 (4,3,1,2) 10,
1
g 2 (4,3,1,2) 19.
When considering point 2, it is necessary to fulfill the condition f i - increases:
f 2 (4,2,1,3) ( x 2 x 4 )(4 x1 0.5 x 2 0.5 x 4 7 x 3 ) 6,5.
Consequently, the target function increases by 6.5 units, so there is a need to check
the increment of additional restrictions:
g11 (4,2,1,3) 6 10,
1
g 2 (4,2,1,3) 7 19.
Constraints are met, so the support solution can be improved:
max f 2 (4,2,1,3) f1 (4,3,1,2) f 2 (4,2,1,3) 46,5.
g11 (4,2,1,3) 16,
1
g 2 (4,2,1,3) 26.
Consider the point (3,2,1,4) , then you need to transpose x1 x 4 regarding point
2 (4,2,1,3) :
1 1
f14 (3,2,1,4) ( x1 x 4 )(3 x1 6 x 2 2 x3 3x 4 ) (3 4)(9 12 2 12) 5,5.
2 2
Since the target function decreases by 5.5 units, there is no point in considering this
point of the set of permutations.
Consider ascending, point (4,2,3,1) , respectively, transposition, x 3 x 4 regarding
point 2 (4,2,1,3) :
1 1
f 34 (3,2,1,4) ( x3 x 4 )(6 x 2 8 x1 x3 x 4 ) (3 1)(12 32 3 1) 16.
2 2
The target function decreases by 16 units, respectively, this point is not considered.
Therefore, point 2 (4,2,1,3) is optimal and cannot be improved.
max f 2 (4,2,1,3) 46,5.
6 Conclusion
The article represents an optimization problem model with a quadratic target function
and additional constraints on the combinatorial set of permutations as a model of
many applied problems. An algorithm for solving the optimization problem is pro-
posed and a numerical example is demonstrated. It should be noted that the problem
are very complex from a computational point of view. Therefore, their solutions re-
quire a lot of time and resources. This method significantly simplifies the procedure
for finding the optimal solution of an optimization problem with a quadratic target
function and additional constraints on the combinatorial set of permutations, since the
inequality of restrictions increments allows you to immediately determine whether the
point of the permutation set is a support solution or not. There is not a necessity to do
complex calculations of all constraints and the target function; it suffices to find the
increment of constraint and functions in the case of an improvement the solution.
The further development of this study is going to be aimed at realizing and adapt-
ing the formulated method on other combinatorial constructions, as well as develop-
ing new methods for solving combinatorial optimization problems, taking into ac-
count the input data.
References
1. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, New
York (2018)
2. Pardalos, P.M., Du, D.-Z., Graham, R.L. (Eds.).: Handbook of combinatorial optimization.
Springer, New York (2013)
3. Bernhard, K., Jens, V.: Combinatorial Optimization, Theory and Algorithms. Springer-
Verlag, Berlin (2012)
4. Karin, R., Saoub, A.: Tour through Graph Theory. Chapman and Hall/CRC, New York
(2017)
5. Berge, C.: Principes de combinatoire. Dunod, Paris (1968)
6. Yemelichev, V.A., Kovalev, M.M., Kravtsov, M.K.: Polytopes, graphs and optimisation.
Cambridge University Press, Cambridge (1984)
7. Sachkov, V.N.: Combinatorial methods of discrete mathematics. Nauka, Moscow (1975)
8. Hulianytskyi, L., Riasna, I.: Formalization and classification of combinatorial
optimization problems. Springer Optimization and its Applications. 130: pp. 239-250
(2017)
9. Yakovlev, S.V., Grebennik, I.V.: Localization of solutions of some problems of nonlinear
integer optimization. Cybernetics and Systems Analysis. 29(5): pp. 727-734 (1993)
10. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer Science &
Business Media, Berlin (2012)
11. Balakrishnan, R., Sridharan, S.: Foundations of Discrete Mathematics with Algorithms and
Programming. Chapman and Hall/CRC, New York (2018)
12. Bona, M.: Combinatorics of Permutations, Second Edition. Chapman and Hall/CRC, New
York (2012)
13. Shor, N.Z., Stetsyuk, P.I.: Modified r-algorithm to find the global minimum of polynomial
functions Cybernetics and Systems Analysis. 33(4): pp.482-497 (1997)
14. Emets, O.A., Koliechkina, L. N.: Solution of optimization problems with fractional-linear
objective functions and additional linear constraints on permutations. Cybernetics and
Systems Analysis. 40(3): pp. 329-339 (2004)
15. Donec, G. A., Kolechkina, L. M.: Method of ordering the values of a linear function on a
set of permutations. Cybernetics and Systems Analysis. 45(2): 204-213 (2009)
16. Sergienko, I.V., Volkovich, O.V., Roshchin, V.A.: Models and methods of solution of
quadratic integer programming problems. Cybernetics. 23(3): 289-305 (1987)
17. Semenova, N.V., Kolechkina, L.N., Nagornaya, A.N.: On Approach to Solving Vector
Problems with Fractionally Linear Functions of the Criteria on the Combinatorial Set of
Arrangements. Journal of Automation and Information Sciences. 42(1): pp. 67-80 (2010)
18. Yakovlev, S.V.: The theory of convex continuations of functions on vertices of convex
polygons. Computational Mathematics and Mathematical Physics. 34: pp. 959-965 (1994)
19. Yakovlev, S.: Convex extensions in combinatorial optimization and their applications.
Springer Optimization and its Applications. 130: pp. 567-584 (2017)
20. Yakovlev, S.V.: Bounds on the minimum of convex functions on Euclidean combinatorial
sets. Cybernetics. 25(3): pp. 385-391 (1989)
21. Pichugina, O.S., Yakovlev, S.V.: Continuous representations and functional extensions in
combinatorial optimization. Cybernetics and Systems Analysis. 52(6): pp. 921-930 (2016)
22. Yakovlev, S.V.: Тhe method of artificial space dilation in problems of optimal packing of
geometric objects. Cybernetics and Systems Analysis. 53(5): pp. 725-732 (2017)
23. Stoyan, Y.G., Yakovlev, S.V.: Configuration space of geometric objects. Cybernetics and
Systems Analysis. 54(5): 716-726 (2018)
24. Yakovlev, S.V.: On some classes of spatial configurations of geometric objects and their
formalization. Journal of Autom and Information Sciences . 50(9): 38-50 (2018)
25. Stoyan, Yu.G., Sokolovskii, V.Z., Yakovlev, S.V.: Method of balancing rotating discretely
distributed masses. Energomashinostroenie. 2: pp. 4-5 (1982)
26. Stoyan, Y., Romanova, T., Pankratov, A., Kovalenko, A., Stetsyuk, P.: Balance layout
problems: mathematical modeling and nonlinear optimization. Springer Optimization and
its Applications. vol. 114, рр. 369-400 (2016)
27. Gulianitsky, L.F. Sergienko, I.V.: Meta-evolutionary method of deformed polyhedron in
combinatorial optimization. Cybernetics and system analysis. 44(6): pp. 70-79 (2007)
28. Stoyan, Y.G., Yakovlev, S.V., Parshin, O.V.: Quadratic optimization on combinatorial sets
in Rn. Cybernetics and Systems Analysis. 27(4): pp. 561–567 (1991)
29. Yakovlev, S.V., Valuiskaya, O.A.: Optimization of linear functions at the vertices of a
permutation polyhedron with additional linear constraints. Ukrainian Mathematical
Journal. 53(9): pp. 1535-1545 (2001)
30. Pichugina, O.S., Yakovlev, S.V.: Functional and analytic representations of the general
permutations. Eastern-European Journal of Enterprise Technologies. 1(4): pp. 27-38
(2016)
31. Pichugina, O., Yakovlev, S.: Convex extensions and continuous functional representations
in optimization, with their applications. Journal of Coupled Systems and Multiscale Dy-
namics. 4(2): pp. 129–152 (2016)
32. Yakovlev, S.V., Pichugina, O.S.: Properties of combinatorial optimization problems over
polyhedral-spherical sets. Cybernetics and Systems Analysis. 54(1): pp. 99-109 (2018)
33. Koliechkina, L. N., Dvirna, O. A., Nagornaya, A. N.: Modified Coordinate Method to
Solve Multicriteria Optimization Problems on Combinatorial Configurations. Cybernetics
and Systems Analysis. 59(4): pp. 620-626 (2014)
34. Koliechkina, L. N., Dvirna, O. A.: Solving Extremum Problems with Linear Fractional
Objective Functions on the Combinatorial Configuration of Permutations Under Multicri-
teriality. Cybernetics and Systems Analysis. 53(4): pp. 590-599 (2017)
35. Donec, G. A., Kolechkina, L. M.: Construction of hamiltonian paths in graphs of permuta-
tion polyhedra. Cybernetics and Systems Analysis. 46(1): pp.7-13 (2010)
36. Kolechkina, L. M., Dvirna, O. A.: Solving extremum problems with linear fractional ob-
jective functions on the combinatorial configuration of permutations under multicriterial-
ity. Cybernetics and Systems Analysis. 53(4): pp. 590-599 (2017)
37. Kolechkina, L. M., Dvirna, O. A., Nagornaya, A. N.: Modified coordinate method to solve
multicriteria optimization problems on combinatorial configurations. Cybernetics and Sys-
tems Analysis. 50(4): pp. 620-626 (2014)
38. Koliechkina, L., Pichugina, O.: Multiobjective Optimization on Permutations with Appli-
cations, In DEStech Transactions on Computer Science and Engineering, 2018 IX Interna-
tional Conference on Optimization and Applications (OPTIMA 2018) (Supplementary
Volume), pp. 61-75 (2018)