METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS
The development of high-speed methods and algorithms for global multidimensional optimization and their modifications in various fields of science, technology, and economics is an urgent problem that involves reducing computing costs, accelerating, and effectively searching for solutions to such pro...
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| Дата: | 2024 |
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General Energy Institute of the National Academy of Sciences of Ukraine
2024
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System Research in Energy| _version_ | 1871104371406667776 |
|---|---|
| author | Khaidurov, Vladyslav Tatenko, Vadym Lytovchenko, Mykyta Tsiupii, Tamara Zhovnovach, Tetiana |
| author_facet | Khaidurov, Vladyslav Tatenko, Vadym Lytovchenko, Mykyta Tsiupii, Tamara Zhovnovach, Tetiana |
| author_institution_txt_mv | [
{
"author": "Vladyslav Khaidurov",
"institution": null
},
{
"author": "Vadym Tatenko",
"institution": null
},
{
"author": "Mykyta Lytovchenko",
"institution": null
},
{
"author": "Tamara Tsiupii",
"institution": null
},
{
"author": "Tetiana Zhovnovach",
"institution": null
}
] |
| author_sort | Khaidurov, Vladyslav |
| baseUrl_str | https://systemre.org/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T12:57:48Z |
| description | The development of high-speed methods and algorithms for global multidimensional optimization and their modifications in various fields of science, technology, and economics is an urgent problem that involves reducing computing costs, accelerating, and effectively searching for solutions to such problems. Since most serious problems involve the search for tens, hundreds, or thousands of optimal parameters of mathematical models, the search space for these parameters grows non-linearly. Currently, there are many modern methods and algorithms of swarm intelligence that solve today's scientific and applied problems, but they require modifications due to the large spaces of searching for optimal model parameters. Modern swarm intelligence has significant potential for application in the energy industry due to its ability to optimize and solve complex problems. It can be used to solve scientific and applied problems of optimizing energy consumption in buildings, industrial complexes, and urban systems, reducing energy losses, and increasing the efficiency of resource use, as well as for the construction of various elements of energy systems in general. Well-known methods and algorithms of swarm intelligence are also actively applied to forecast energy production from renewable sources, such as solar and wind energy. This allows better management of energy sources and planning of their use. The relevance of modifications of methods and algorithms is due to the issues of speeding up their work when solving machine learning problems, in particular, in nonlinear regression models, classification, and clustering problems, where the number of observed data can reach tens and hundreds of thousands or more. The work considers and modifies well-known effective methods and algorithms of swarm intelligence (particle swarm optimization algorithm, bee optimization algorithm, differential evolution method) for finding solutions to multidimensional extremal problems with and without restrictions, as well as problems of nonlinear regression analysis. The obtained modifications of the well-known classic effective methods and algorithms of swarm intelligence, which are present in the work, effectively solve complex scientific and applied tasks of designing complex objects and systems. A comparative analysis of methods and algorithms will be conducted in the next study on this topic. |
| doi_str_mv | 10.15407/srenergy2024.03.046 |
| first_indexed | 2026-03-24T02:03:24Z |
| format | Article |
| fulltext |
Системні дослідження в енергетиці. 2024. 3(79) 46
ІНФОРМАЦІЙНО-ВИМІРЮВАЛЬНІ ТЕХНОЛОГІЇ,
МОНІТОРИНГ ТА ДІАГНОСТИКА В ЕНЕРГЕТИЦІ
_____________________________________________________________________________
ISSN 2786-7102 (Online), ISSN 2786-7633 (Print)
https://doi.org/10.15407/srenergy2024.03.046
UDC 517.9:519.6
Vladyslav Khaidurov1,2*, PhD (Engin.), Senior Researcher, https://orcid.org/0000-0002-4805-8880
Vadym Tatenko1, https://orcid.org/0009-0008-4869-9689
Mykyta Lytovchenko1, https://orcid.org/0009-0001-6671-7763
Tamara Tsiupii3, PhD (Engin.), Associate Professor, https://orcid.org/0000-0003-2206-2897
Tetiana Zhovnovach4, https://orcid.org/0000-0003-1037-4383
1National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", 37,
Beresteiskyi Avenue., Kyiv, 03056, Ukraine;
2General Energy Institute of NAS of Ukraine, 172, Antonovycha St., Kyiv, 03150, Ukraine;
3National University of Life and Environmental Sciences of Ukraine, 15, Heroiv Oborony St., Kyiv, 03041,
Ukraine;
4Cherkasy branch of European University, 83, Smilyanska St., Cherkasy, 18008, Ukraine
*Corresponding author: allif0111@gmail.com
_______________________________________________________________________________________
METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE
PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND
OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS:
REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS
Abstract. The development of high-speed methods and algorithms for global multidimensional optimization
and their modifications in various fields of science, technology, and economics is an urgent problem that
involves reducing computing costs, accelerating, and effectively searching for solutions to such problems.
Since most serious problems involve the search for tens, hundreds, or thousands of optimal parameters of
mathematical models, the search space for these parameters grows non-linearly. Currently, there are many
modern methods and algorithms of swarm intelligence that solve today's scientific and applied problems, but
they require modifications due to the large spaces of searching for optimal model parameters. Modern swarm
intelligence has significant potential for application in the energy industry due to its ability to optimize and
solve complex problems. It can be used to solve scientific and applied problems of optimizing energy
consumption in buildings, industrial complexes, and urban systems, reducing energy losses, and increasing
the efficiency of resource use, as well as for the construction of various elements of energy systems in general.
Well-known methods and algorithms of swarm intelligence are also actively applied to forecast energy
production from renewable sources, such as solar and wind energy. This allows better management of energy
sources and planning of their use. The relevance of modifications of methods and algorithms is due to the
issues of speeding up their work when solving machine learning problems, in particular, in nonlinear
regression models, classification, and clustering problems, where the number of observed data can reach
tens and hundreds of thousands or more. The work considers and modifies well-known effective methods and
algorithms of swarm intelligence (particle swarm optimization algorithm, bee optimization algorithm,
differential evolution method) for finding solutions to multidimensional extremal problems with and without
restrictions, as well as problems of nonlinear regression analysis. The obtained modifications of the well-
known classic effective methods and algorithms of swarm intelligence, which are present in the work,
effectively solve complex scientific and applied tasks of designing complex objects and systems. A
comparative analysis of methods and algorithms will be conducted in the next study on this topic.
Keywords: optimization, swarm intelligence, mathematical modelling, nonlinear regression, complex
objects and systems.
Системні дослідження в енергетиці. 2024. 3(79) 47
1. Introduction
With the development of modern computing systems, its application for solving optimization problems,
there is a need for the design of complex systems in thermal power engineering [1–3], mathematical modeling.
The modern theory of optimization methods, taking into account its new applications, has undergone significant
changes, which consist in the development of new and modification of existing methods and algorithms for
finding solutions to optimization, inverse problems and problems in incorrect
formulation [4–6].
Considerable interest is now focused on algorithms and methods of swarm intelligence, which are finding
more and more applied applications in modern science and technology [7, 8]. Swarm intelligence is a concept
that is inspired by observations of the behavior of animal colonies in nature, such as ants, bees, and swarms of
birds [9, 10]. This approach to artificial intelligence and optimization is based on modeling the behavior of
individual agents that interact with each other and with the environment. The collective behavior of these agents
is actively used to solve complex problems and find optimal solutions without centralized management.
The applications of swarm intelligence in science and technology include various fields such as
optimization, robotics, data networks, and even machine learning [11–13]. For example, optimization algorithms
based on swarm intelligence can be used to solve routing problems, find optimal solutions in complex parameter
spaces, or manage distributed systems [14–16].
In the field of robotics and the management of groups of robots, swarm intelligence allows for the creation
of efficient algorithms to coordinate the actions of many robots without centralized control. This is especially
useful in scenarios where adaptability to changing environmental conditions is required. In data networks, swarm
algorithms can be used to optimize routing and traffic management, taking into account dynamic network
conditions and changes in load. In the field of machine learning, swarm methods and algorithms can be applied
to create effective deep learning methods, in particular, in the problems of optimizing loss functions and finding
hyperparameters [17].
2. Methods and materials
2.1. Problem statement
Most of the applied optimization problems are reduced to finding the global extrema of functions in the
classical formulation, which is presented as follows [18, 19]:
𝑦 = 𝐹(𝑿) → min(max) 𝑿 ∈ 𝐺, 𝐺 ⊂ ℝ𝑛,
where 1 2 ,, , , nX X XX G is the search space for the values of the arguments X, n is the dimension of the
search space of the global extremum of the function. , 1;jX j n – factor (independent) variables that establish
a causal relationship with the dependent (resultant) variable y. The function F(X) is often subject to functional
constraints, which are presented in the form of the following functional dependencies:
, 1; ,
, 1; .
i i
i i
g S i K
g S i K R
X
X
The complexity of the objective function F(X) and the corresponding constraints gi(X) is determined by
a specific technical problems [20; 21]. The appearance of F(X) and gi(X) determines the choice of the
optimization method that will be used to search for * * * *
1 2 ,, , , nX X XX F(X*).
It should be noted that the dependence of the species:
min,y F X
Системні дослідження в енергетиці. 2024. 3(79) 48
In machine learning problems, it is often presented as a summary (or average) error, which consists of a deviation
between real data and model data, which is skipped due to functional dependence. Such functional dependencies
arise in regression (linear and nonlinear) analysis, in the problems of classification, data clustering, in the
construction of effective mathematical models of process control, forecasting of time series, modernization of
objects and systems, as well as in the development of complex energy complexes in today's conditions.
In this paper, we will consider the problems of searching for a global minimum of problems without
functional constraints on the objective function / functional, problems with constraints in the form of functions,
as well as problems of searching for nonlinear regression parameters, as well as methods for their
search.
To begin with, let's consider a one-factor (factor (independent) variable – x, dependent variable – y)
regression model. Let us assume that we know the general form of a mathematical model that describes a certain
process. It looks like , ,jy F x ; .1j N In the problem of regression analysis, it is assumed that there are
statistical data that are presented in the form of observations: , ,i ix y ; ,1i M .M N Knowing the
appearance , ,jy F x 1;j N of the model and observational data , ,i ix y 1; , i M ,M N it is necessary
to find the parameters
* ,j ; ,1j N under which the mathematical model is formed *,jy F x It most
accurately describes a certain process that takes place in a particular system.
That is, for such a problem, we can write a system of equations in the form:
2
3
1 1 2 1 2 1 2
1 2 3 1 2
, , , , , , , , , ,
, , , , , , , , , , .
N
M
N
N NM
y F x y F x
y F x y F x
(1)
But, obviously, the statistics are obtained with some error. This means that in reality we do not have a
system of form (1), but a system of the form:
1 1 2 1 1 2 1 2 2
1
2
3 2 3 3 1 2 ,
, , , , , , , , , ,
, , , , , , , , , ,M
N N
N N M M
y F x y F x
y F x y F x
(2)
where ,i 1;i M is the measurement error on the i-th observation. In this case, for (2) it is necessary to find
such values of the parameters in order to minimize the total error that was obtained as a result of measurements.
To do this, the well-known least-squares method is used:
* ,j 1; ,j N
1 2 3
1
1 2 3
1
min,
1 1
min
M
M j
j
M
M j
j
E
E
M M
(3)
or
2 2 2 2 2
1 2 3
1
2 2 2 2 2
1 2 3
1
min,
1 1
min ,
M
M j
j
M
M j
j
E
E
M M
(4)
or
max min.j
j
E (5)
Системні дослідження в енергетиці. 2024. 3(79) 49
For gradient methods, the first expression (3) is used for .E For swarm intelligence methods, it does not
matter which of the formulas (3), (4) or (5) is used as a criterion for the minimum error. It should be noted that
option (3) for E (through the sum of the modules) is less commonly used in gradient methods, since the
derivative of y t t the function at the point of the extremum does not exist. In this case, we have a value E
based on formula (3):
2
1 1 2 1
2 2
2 1 2 2 1 2
, , , ,
, , , , , , , , .
N
N M N M
E y F x
y F x y F x
(6)
Let's take the classic approach. Let us find the gradient vector for (6) and equate it to zero (a necessary
condition for the extremum of the function / functional):
1 1
1
1
1 1 2 1 1 2 1
2 1 2 2 1 2 2
1 2 1 2
1 1 2 1 1 2 1
2 1 2 2 1 2
2 , , , , , , , ,
2 , , , , , , , ,
2 , , , , , , , , 0.
2 , , , , , , , ,
2 , , , , , , , ,
N N
N
N N
N N
M N M N M
N N
N N
E y F x F x
y F x F x
y F x F x
E y F x F x
y F x F x
2
1 2 1 22 , , , , , , , , 0.
NM N M N My F x F x
(7)
From dependencies (7) we obtain a system of nonlinear (in the general case) equations of the form:
1
2
1 2 1 2
1
1 2 1 2
1
1 2 1 2
1
, , , , , , , , 0,
, , , , , , , , 0,
, , , , , , , , 0.
N
M
N i N i i
i
M
N i N i i
i
M
N i N i i
i
F x F x y
F x F x y
F x F x y
(8)
Obviously, system (8) has N equations and N unknowns. Iterative methods and algorithms are often
applied to this kind of system, for example, the Newtonian method, the Levenberg-Marquardt method, quasi-
Newtonian methods, etc.
For example, Newton's method for a system of the form:
1 1 2
2 1 2
1 2
, , , 0,
, , , 0,
, , , 0
N
N
N N
f
f
f
(9)
will look like this:
Системні дослідження в енергетиці. 2024. 3(79) 50
1 1 1 2
1 1
1
2 1 22 2
1
1
2
1
1 2
, , ,
, , ,
, , , ,
, , ,
k k k
k k N
k k kk k
k k k N
N
k k
k k k
N N
N N
f
J
f
f
(10)
where J is the Jacobian at the point 1 2, , ,
k k k
N , which is of the form:
1 2
1 1
1 2
2
1
1 2
1
1 2 1
,
, , ,
, , , , , ,
, , , , , ,
k k k
N
k k k k k k
N N
k k k k k
N
N N
N
k
N
N
J
f f
f f
(11)
where k is the iteration number. To start such an iterative process, (9)–(11) specify an initial approximation
0 0 0
1 2, , , .N The stopping criterion can be considered the fulfillment of one of the following conditions:
2
1 1
1
, 0,
N
k k k k
i i
i
(12)
2
1 2
1
1
, , , , 0,
N
k k k
i N
i
f
N
(13)
1 2
1
1
, , , , 0,
N
k k k
i N
i
f
N
(14)
1 2max , , , , 0.
k k k
i N
i
f (15)
Several conditions for stopping an iterative process can be applied, and not only (12), (13), (14), and (15).
Taking into account the above, it is possible to rewrite the iterative formula of Newton's classical method
(10), (11) for a system (8) of nonlinear (in the general case) equations:
1
2
1
1 1
1
2 2
1 2
1
1 2 1 2
1
1 2
1
1 2
1
1 2
, , ,
, , , , , , , ,
, , , , , , , ,
, , , ,
N
k k
k k
k k k
N
k k
N N
M
k k k k k k
N i N i i
i
M
k k k k k k
N i N i i
i
k k k
N i
F x F x y
F x F x y
F x
J
1 2
1
,
,
, , ,
M
k k k
N i i
i
F x y
(16)
Системні дослідження в енергетиці. 2024. 3(79) 51
where J is the Jacobian at the point 1 2, , ,
k k k
N , which is of the form:
2
1 2 1 2
1 2
1
1 2 1 2
,
, , , , , , , ,
, , ,
, , , , , , , ,
k k
k
k k k k
N i N i i
k k k
N
k k k k k k
N i
M
k l
i
l
N i
J
F
x F x y
F F
x x
(17)
where k is the iteration number. To start such an iterative process, (16)–(17) specify an initial approximation
0 0 0
1 2, , , .N The stopping criterion can be considered the fulfillment of one of the following conditions:
2
1 1
1
, 0,
N
k k k k
i i
i
(18)
2
1 2
1 1
1 2
, , , ,
1
, 0,
, , , ,
j
k k k
MN N i
k k k
j i
N i i
F x
N F x y
(19)
1 2
1 1
1 2
, , , ,
1
, 0,
, , , ,
j
k k k
MN N i
k k k
j i
N i i
F x
N F x y
(20)
1 2
1
1 2
, , , ,
max , 0.
, , , ,
j
k k k
M N i
j k k k
i
N i i
F x
F x y
(21)
Here, too, any of the conditions (18), (19), (20) and (21) can be used to stop the iterative process of finding
the optimal parameters of the nonlinear regression model.
To simplify the calculations of the first derivatives in (16), (17), numerical approximations are used, which
are derived on the basis of Taylor series, for example:
1 2
1 2
1 2
, , , Δ , , ,
1
, , , , ,
Δ , , , , , ,
j
k k k k
j j N i
k k k
N i
k k k k
j
j N i
F x
F x
F x
(22)
or
1 2
1 2
1 2
, , , , , ,
1
, , , , ,
Δ , , , Δ , , ,
j
k k k k
j N i
k k k
N i
k k k k
j
j j N i
F x
F x
F x
(23)
or
1 2
1 2
1 2
, , , Δ , , ,
1
, , , , ,
2Δ , , , Δ , , ,
j
k k k k
j j N i
k k k
N i
k k k k
j
j j N i
F x
F x
F x
(24)
Системні дослідження в енергетиці. 2024. 3(79) 52
or
1 2
1 2
1 2
Δ
, , , , , ,
21
, , , , .
Δ Δ
, , , , , ,
2
j
jk k k k
j N i
k k k
N i
j jk k k k
j N i
F x
F x
F x
(25)
It should be noted that each of the formulas (22), (23), (24) and (25) has its own error, which is obtained
from the decomposition of the function/functional into the Taylor series according to the corresponding
parameters.
To simplify the calculations of the second derivatives in (16) and (17), numerical approximations are also
used, which are derived from the Taylor series:
1 2
2
1 2 1 222
1 2
, , , , , ,
1
, , , , 2 , , , , , , ,
, , , , , ,
k k k k
j j N i
k k k k k k k
N i j N i
j
j
k k k k
j j N i
F x
F
x F x
F x
(26)
1 2
2 1 2
1
1 2
1 2
,
,
,
,, , , , , ,
, , , , , , ,
1
, , ,
4 , , , , , , ,
, , , , , , ,,
k k k k k
k l N i
k k k k k
k l N i
k k
N i
k k k k k
k l
k l N i
k
k
k k k k
k l N i
k l
k l
l
k l
k l
F x
F x
F
x
F x
F x
,
1; , 1; , .k N l N k l
(27)
It is also known here that (26) and (27) have their own error (also obtained on the basis of the Taylor series).
Similarly, it is possible to build a multivariate nonlinear regression model, that is, when statistical data are
presented in the form of:
11 12 1
21 22 2
1 2
,
s
s
M M Ms
x x x
x x x
x
x x x
(28)
where s is the number of factors that affect the observed quantity.
Obviously, the problems in solving a regression-type problem for a generalized nonlinear case with respect
to the sought unknown parameter values are as follows:
– calculation of the first derivatives for the functional dependence of a species 1 2, , , ,
k k k
N iF x on
unknown parameters (although this can be done not only analytically, but also numerically);
– calculation of the second derivatives for the functional dependence of the species 1 2, , , ,
k k k
N iF x
on unknown parameters (although this can be done not only analytically, but also numerically);
– Choosing an initial approximation 0 0 0
1 2, , , N is difficult;
– At each iteration, k you need to find 1
1 2, , , .
k k k
NJ
Системні дослідження в енергетиці. 2024. 3(79) 53
In this case, further research will focus on methods and algorithms that have a stochastic component. The
stochastic component makes it possible to find solutions to various applied problems quite accurately and
quickly. Three well-known methods and algorithms were taken as a basis: the particle swarm optimization
algorithm, the bee optimization algorithm, and the differential evolution method. The criteria for selecting basic
methods and algorithms are based on the following well-known facts:
– The chosen methods and algorithms are relatively easy to understand, making them accessible to a wide
range of software developers and engineers;
– The chosen methods and algorithms are able to adapt to changes in the environment or optimization
problems by making changes to the parameters or search strategy.
– the selected methods and algorithms are easily parallelized to perform simultaneous data processing;
– the chosen methods and algorithms do not use any information about the derivatives of the objective
function and the corresponding constraints on it;
– there are comparatively few parameters in the selected methods and algorithms;
– The chosen methods and algorithms are very effective for finding the global extremum of a function.
2.2. Problem solving methods
2.2.1. PSO algorithm
PSO uses a swarm of particles, where each particle represents a potential solution to a problem. Initially,
all the particles of the swarm occupy a random position in the space of the search for the solution of the problem
and have small random velocities. In the final iterations, the set of particles converges to one or more optimums
that are global (if there are several rather than one). The behavior of a particle in the solution-seeking hyperspace
is constantly adjusting to its experience and that of its neighbors. In addition, each particle remembers its best
position with the achieved local best value of the objective (fitness) function and knows the best position of the
particles of its neighbors, where the global optimum of the function was reached at the moment. In the search
process, the swarm particles exchange information about the best results achieved and change their positions
and speeds according to certain rules based on the currently available information about local and global
achievements. In this case, the global best result is known to all particles and it is corrected in the case when
some particle of the swarm finds a better position with a result that exceeds the current global optimum. Each
particle of the swarm is subject to fairly simple rules of behavior that take into account the local success of each
individual and the global optimum of all individuals (or some set of neighbors) of the swarm.
Each i-th particle is characterized by an iteration n of its position ix n in hyperspace and its velocity of
motion .iv n The velocity of the i-th particle is calculated as
*
1 1 2 21 ,best
i i i i iv n v n x n x n r x n x n r (29)
The position of the i-th particle is calculated as
1 1 ,i i ix n x n v n (30)
where: 1 2; ; ;i i i iMx n x n x n x n – the position of the i-th particle in the iteration n ;
1 2; ; ;best best best best
i i i iMx n x n x n x n – the best position of the i-th particle (personal best position);
* * * *
1 2; ; ; Mx n x n x n x n – the best position for the entire population (global best position);
1 2; , ;i i i iMv n v n v n v n is the velocity vector of the i-th particle in the iteration ;n
1 2, – positive
acceleration coefficients that regulate the contribution of the cognitive and social components;
Системні дослідження в енергетиці. 2024. 3(79) 54
1 11 12 1; ; ; Mr r r r , 1 21 22 2; ; ; Mr r r r are random number vectors that introduce an element of randomness
into the search process.
Let us consider the influence of various constituents in calculating the velocity of a particle according to
(29). The first term in (29) iv n preserves the previous direction of the velocity of the i-th parts and can be
considered as the moment that prevents a sharp change in the direction of the velocity and acts as an inertia
velocity. Cognitive velocity 1 1
best
i ix n x n r determines the characteristics of a particle with respect to its
prehistory, which maintains a better position for a given particle. The effect of this term is that it tries to bring
the particle back to a better achieved position. The third term *
2 2ix n x n r defines the social velocity,
which characterizes the particle in relation to its neighbors. The effect of the social component is that it tries to
direct each particle towards the global optimum achieved by the swarm (or some of its immediate environment).
The displacement of the particle's position is carried out on the basis of (30).
Algorithm for Optimizing Numerical Functions
1. Initializing
1.1. Specifying Parameters
1 2, , and 1 2, 0;4 .
1.2. Set the maximum number of iterations ,N population size ,K the length of the particle position
vector M minimum and maximum values of the position vector , , 1, ,min max
j jx x j M minimum and
maximum values for the velocity vector , , 1, ,min max
j jv v j M And 0.max
jv
1.3. Defining the Cost Function (Goal Function)
1, , , ,MF min x x x x
where x is the position vector of the particle.
1.4. Creating the Original Population P
1.4.1. Particle Number 1, k P
1.4.2. Randomly create a vector position
kx
1, , , ,min max min
k kM kj j j jx x x x x x rand kx
where ()rand is a function that returns a uniformly distributed random number in the range [0; 1]
1.4.3. Creating a Better Position Vector : best best
k k kx x x
1.4.4. Randomly Generate a Velocity Vector
kv
1, , , min max min
k k kM kj j j jv v v v v v v rand
or
0kjv
1.4.5. If , , ,best
k k kx x v P then , , , 1best
k k kP P x x v k k
1.4.6. If ,k K then go to step 1.4.2
1.4.7. Define a particle of the current population with the best position
*
* *argmin , k k
k
k F x x x
2. Iteration Number 1n
3. Particle Number 1k
4. Velocity vector modification
Системні дослідження в енергетиці. 2024. 3(79) 55
4.1. 1 r rand , 2 r rand
4.2. *
1 1 2 2
best
k k k k kv v x x r x x r
4.3. Speed limits
kv present, i.e.
max , , min , , 1,min max
kj j kj kj j kjv v v v v v j M
or absent
5. Position Modification
5.1. k k kx x v ; 5.2. 1j
5.3. If ,min
kj jx x then , min min
kj j kj j kj kjx x x x v v
5.4. If ,max
kj jx x then , max max
kj j kj j kj kjx x x x v v
5.5. If ,j M then 1,j j Go to step 5.3
6. Definition personal (local) best Position: if ,best
k kF x F x then
best
k kx x
7. If ,k K then 1k k
8. Determine the particle of the current population that is best in terms of the function of the target
* argmin k
k
k F x
9. Determining the Global Best Position: if *
* ,
k
F x F x then *
*
k
x x
10. Stop condition: If ,n N then 1,n n go to step 3
The result is x*.
2.2.2. Bees algorithm
The bee algorithm is based on the behavior of honey bees. It is based on the behavior of foraging bees and
is an extension of the bee system. There is a phase of the worker bee (busy foraging) and the scout bee. The
purpose of the algorithm is to determine the location of good areas in the search space. Scout bees perform a
random search. The found good (in terms of the value of the objective function / functionality) areas are
investigated with the help of local search. The solution corresponds to the position of the bee located in a certain
area.
Algorithm for Optimizing Numerical Functions
1. Initializing
1.1. Specifying Parameters , ,max to create a circle, and 0;1 , 0;1max 0;1 .
1.2. Set the maximum number of iterations ,N population size ,K the number of plots (and the bees of
these plots) ,sL number of elite plots ,esL The number of bee departures in an elite area ,eZ The
number of departures of the worker bee in a regular area ,oZ Length of Bee Position Vector M
(dimension of the search space), minimum and maximum values for the position vector
, , 1, .min max
j jx x j M
1.3. Defining the Cost Function (Goal Function)
1, , , ,MF min x x x x
where x is the vector of the bee's position.
1.4. Randomly create a vector of a better position
* * * *
1 2, , , , (),min max min
M kj j j jx x x x x x x x rand
where ()rand is a function that returns a uniformly distributed random number in the range 0;1
1.5. Creating the Original Population
Системні дослідження в енергетиці. 2024. 3(79) 56
1.5.1. Bee Number 1, k P
1.5.2. Randomly create a vector position
kx
1, , , ()min max min
k kM kj j j jx x x x x x rand kx
1.5.3. If ,kx P then , 1kP P x k k
1.5.4. If ,k K then go to step 1.5.2
2. Iteration Number 1n
3. Identify the bee with the best position
* argmin k
k
k F x
3. Particle Number 1k
4. If *
* ,
k
F x F x then *
*
k
x x
5. Arrange P by the value of the objective function, i.e. 1F F k kx x
6. Worker Bee Phase (Local Search)
6.1. Plot number 1l
6.2. Determine the size of the circle
,1
,
e es
o es s
Z l L
Z
Z L l L
6.3. Create a Circle for a Position l-(a) To the extent permitted
6.3.1. n
maxn
6.3.2. 1 2 ,max min
zj lj j jx x n x x rand
1; , 1;j M z Z
6.3.3. max ; , min ; ,min max
zj j zj zj j zjx x x x x x
1; , 1;j M z Z
6.4. * argmin z
z
z F x
6.5. * lz
Якщо f x f x then * .l z
x x
6.6. If ,sl L then 1,l l Go to step 6.2
7. Scout Bee Phase (Random Search):
,min max min
lj j j jx x x x rand
1; , 1;sj M l L K
8. Stop Condition
If ,n N then 1,n n go to step 3
9. Identify the bee with the best position
* argmin , k
k
k F x *
*
k
x x
The result is x*.
2.2.3. Differential evolution method
The main idea of the method of differential evolution is the combination of mutation and crossing to
efficiently find optimal solutions in multidimensional parameter spaces. The method uses some ideas of genetic
algorithms, but, unlike the latter, does not involve working with binary code.
Problem formulation: 1 2min, , , , .MF X X X X X
Системні дослідження в енергетиці. 2024. 3(79) 57
The main stages of the method are as follows:
1) For each chromosome in a population ,ix 1, , , 1;i i iMx x x i N (N is the dimension of the
population), three other random members of that population are selected
1 2 3, , ,x x x
1 2 3 1 2 3, , , i i ix x x x x x x x x .
2) A mutant vector is generated: 1 2 3 , 0;2v x F x x F .
3) The vector difference
2 3x x is scaled by a user-defined hyperparameter , 0;2 .F F
A visual illustration of the method is shown in Figure 1.
Fig. 1. Illustrative illustration of the method of differential evolution
4) We form a test vector based on crossing ix with a mutant vector v : for each coordinate of the
chromosome, a number is generated 0;1r according to the normal distribution.
If r P , P is a given constant (another parameter of the algorithm
0;1P ), then the corresponding coordinate of the chromosome is replaced by
, 1; ,j ijv x j M
M – the dimension of the search space (the number of independent variables of the fitness
function).
5) If ,i if x f v x v .
All of the above steps are performed up to the method stop criterion (classical, as in other algorithms and
swarm intelligence methods).
2.3. Combination of deterministic and stochastic methods
In complex scientific and technical applied problems, the objective function or functional can be computed
for a relatively long time, even on modern computing systems. Stochastic methods and algorithms of global
Системні дослідження в енергетиці. 2024. 3(79) 58
optimization can find global extremes of objective functions without requiring additional information about
them, in particular, the differentiability of objective functions. Of course, the number of calls to the procedure
for finding the value of the target function in this case increases. In this case, the total number of calculations
increases. This means that the processor time of searching for the global extremum of the objective function also
increases. In this case, methods and algorithms are developed that provide for a deterministic component, as well
as a stochastic component. The deterministic component works quickly and efficiently refines the solution found,
and the stochastic component prevents the method/algorithm from getting stuck in local extremums.
3. Practical results. Modification of the described methods and algorithms of swarm intelligence
In computational mathematics, each modification of an optimization method/algorithm involves a robot
in several ways:
a) reducing the total number of iterations to achieve a certain error. The positive effect of reducing the
total number of iterations leads to a decrease in the total processor time for solving a specific problem or solving
a specific problem. This, in turn, reduces the overall load on the system;
b) improving the accuracy of calculations. The positive effect of this is to accelerate the convergence of
the optimization method/algorithm. This, in turn, again leads to a decrease in the number of iterations, which
means a decrease in the processor time required to solve a problem or a specific task.
This paper proposes several modifications of the algorithms described above.
1) Modification of the position of the worst element of the population in the algorithms and methods
of swarm intelligence described above. Replace the worst element of
t
j
worsx the population with the average
position of the element throughout the population
Average
jx 1;j M according to the following formula:
1
1
1; ,
N
i
Average
j
i
jx x j M
N
(31)
where N is the total number of items in the population, and M is the dimension of the search space. This approach
can be performed on a per-iteration or periodically every K iterations.
2) Modification of the position of the elements of the population to the best in increments h. The
renewal of each element of the population is carried out according to the formula:
, -
,
-
, 1; , 1;
Best j ij
ij ij Best i
Best i
i N j M
x x
x x h x x
x x
(32)
where
Bestx is the best vector of the population in terms of the value of the objective function. It should be noted
that the step of motion h can be determined proportionally (according to the linear law) to the circumference of
the population, which will coincide to the global optimum over time.
3) Changing the hyperparameters of optimization methods/algorithms. This approach involves
changing one or more hyperparameters during the operation of the program. For example, in a differential
evolution algorithm, the parameter F can be replaced by a random real number in the range from 0 to 2 through
K iterations.
In the bee algorithm, there is a dependence of the parameter change n on the number of iterations n.
In this case, in order to optimize functions of large dimensions, a problem arises, which is to reduce this
parameter in advance. In order to prevent this from happening before finding the global extremum of the function
or functional, such a case is assumed to be
/
,
n K
maxn (33)
where [⋅] is the integer part of the number, K is the period that determines how many iterations the value of the
η parameter will be updated.
In the method of differential evolution, it is proposed to take the hyperparameter F as a random number
from 0 to 2 every K iterations. This is due to the fact that at each iteration, the method finds values closer to the
global optimum of the function or functional.
Системні дослідження в енергетиці. 2024. 3(79) 59
4. Discussion
The main advantage of modern methods and algorithms of swarm intelligence is the intuitive structure of
the natural origin of the swarm, which means that it is a relatively simple software implementation. Therefore,
it makes swarm intelligence tools popular in various fields of research.
Most processes are structurally nonlinear. Swarm intelligence demonstrates high efficiency in finding
global optima for a wide range of functions, including nonlinear and non-convex functions, and works effectively
with nonlinear functions through the appropriate mechanism of evolution, which allows you to move in the
parameter space to find the global optimum. Another of the main advantages of the methods and algorithms
under consideration is the small number of hyperparameters that need to be configured. This simplifies the use
of methods and algorithms and customization compared to other algorithms.
5. Conclusions
The development of computing technologies makes it possible to simulate the behavior of complex objects
and systems. The development of new and modification of existing systems involves the construction of
mathematical models in the form of optimization in order to study various parameters of such systems. Such
models make it possible to have an idea of the modes of operation of various objects and systems, their optimal
parameters (settings), geometric properties of such systems. The use of swarm intelligence contributes to the
study and development of such objects and systems, since the usual approximate methods of global multivariate
optimization do not have such an opportunity due to the complexity and nonlinearity of functional dependencies
that describe systems and their operation.
In this work, three relevant methods and algorithms of swarm intelligence are studied: the algorithm of
global optimization by a swarm of particles, the bee algorithm for finding global solutions to problems that are
presented in extreme statements, and the method of differential evolution. These methods have been tested on
various scientific and applied problems, including problems that boil down to the search for global extremes of
multivariate functions with and without constraints, as well as on problems that reduce to the use of nonlinear
multivariate regression models and forecasting time series that arise during the study of the work of various
system complexes.
Three modifications of the considered methods and algorithms have been obtained. The first modification
consists in replacing the worst element of the population with the average position of the element throughout the
population. This approach works effectively when applied at each iteration or periodically at every K iteration.
The second modification of these methods and algorithms is to apply the principle of deterministic movement
towards the best element of the population. Movement step h It is determined in proportion (according to the
linear law) to the circumference of the population, which will coincide to the global optimum over time. The
third modification consists in changing the hyperparameters of optimization methods/algorithms. This involves
changing one or more hyperparameters in the course of the program's operation.
The practical results of the article are that the complex application of three modifications for each
method/algorithm gives advantages in increasing the dimension of search in an extreme problem. For
multivariate problems, modifications affect the elements of the population and allow for a more accurate
definition of the solution. This reduces the total number of iterations of the method/algorithm, which means that
it reduces the time to find the optimum of the problem with a given accuracy.
References
1. Yavuz, G., Durmuş, B., & Aydın, D. (2022). Artificial bee colony algorithm with distant savants for constrained
optimization. Applied Soft Computing, 116.
2. Ewees, A. A., Gaheen, M. A., Yaseen, Z. M., & Ghoniem, R. M. (2022). Grasshopper Optimization Algorithm with
Crossover Operators for Feature Selection and Solving Engineering Problems. IEEE Access, 10, 23304–23320.
https://doi.org/10.1109/ACCESS.2022.3153038
3. Yu, Y. P., Liu, J. Ch., & Wei, Ch. (2022). Hawk and pigeon's intelligence for UAV swarm dynamic combat game via
competitive learning pigeon-inspired optimization. Science China Technological Sciences, 65(5), 1072–1086.
https://doi.org/10.1007/s11431-021-1951-9
Системні дослідження в енергетиці. 2024. 3(79) 60
4. Pradhan, C., Senapati, M. K., Ntiakoh, N. K., & Calay, R. K. (2022). Roach Infestation Optimization MPPT Algorithm
for Solar Photovoltaic System. Electronics, 11(6), 927. https://doi.org/10.3390/electronics11060927
5. Fowler, M., Abbott, A. J., Murray, G. P., & McCall, P. J. (2021). Flying In-formation: A computational method for
the classification of host seeking mosquito flight patterns using path segmentation and unsupervised machine learning.
bioRxiv. https://doi.org/10.1101/2021.11.24.469809
6. Nayak, M., Das, S., Bhanja, U., & Senapati, M. R. (2023). Predictive Analysis for Cancer and Diabetes Using Simplex
Method Based Social Spider Optimization Algorithm. IETE Journal of Research, 69(10), 7342–7356.
https://doi.org/10.1080/03772063.2022.2027276
7. Nadimi-Shahraki, M. H., Taghian, S., Mirjalili, S., Zamani, H., & Bahreinineja, A. (2022). Gaze cues learning-based
grey wolf optimizer and its applications for solving engineering problems. Journal of Computational Science, 61.
https://doi.org/10.1016/j.jocs.2022.101636
8. Yazdani, R., Mirmozaffari, M., Shadkam, E., & Taleghani, M. (2022). Minimizing total absolute deviation of job
completion times on a single machine with maintenance activities using a Lion Optimization Algorithm. Sustainable
Operations and Computers, 3(3), 10–16. https://doi.org/10.1016/j.susoc.2021.08.003
9. Niu, G., Li, X., Wan, X., He, X., Zhao, Y., Yi, X., Chen, C., Xujun, L., Ying, G., & Huang, M. (2022). Dynamic
optimization of wastewater treatment process based on novel multi-objective ant lion optimization and deep learning
algorithm. Journal of Cleaner Production, 345. https://doi.org/10.1016/j.jclepro.2022.131140
10. Wang, S. H., Zhou, J., & Zhang, Y. D. (2022). Community-acquired pneumonia recognition by wavelet entropy and
cat swarm optimization. Mobile Networks and Applications, 1–8. https://doi.org/10.1007/s11036-021-01897-0
11. Al-Dyani, W. Z., Ahmad, F. K., & Kamaruddin, S. S. (2022). Improvements of bat algorithm for optimal feature
selection: A systematic literature review. Intelligent Data Analysis, 26(1), 5–31. https://doi.org/10.3233/IDA-205455
12. Sun, B., Li, Y., Zeng, Y., Li, C., Shi, J., & Ma, X. (2021). Distribution transformer cluster flexible dispatching method
based on discrete monkey algorithm. Energy Reports, 7, 1930–1942. https://doi.org/10.1016/j.egyr.2021.03.041
13. Rahkar Farshi, T., & Orujpour, M. (2021). A multi-modal bacterial foraging optimization algorithm. Journal of
Ambient Intelligence and Humanized Computing, 12(11), 10035–10049. https://doi.org/10.1007/s12652-020-02755-
9
14. Altay, O. (2022). Chaotic slime mould optimization algorithm for global optimization. Artificial Intelligence Review,
55(5), 3979–4040. https://doi.org/10.1007/s10462-021-10100-5
15. Chakraborty, S., Saha, A. K., Sharma, S., Mirjalili, S., & Chakraborty, R. (2021). A novel enhanced whale
optimization algorithm for global optimization. Computers & Industrial Engineering, 153.
https://doi.org/10.1016/j.cie.2020.107086
16. Hassanien, A. E., & Emary, E. (2016). Swarm intelligence: principles, advances, and applications. CRC Press is an
imprint of Taylor & Francis Group, an Informa business. ISBN: 978-1-4987-4107-1.
17. Manjarres, A. V., Sandoval, L .G. M., & Suarez, M. J. S. (2018). Data Mining Techniques Applied in Educational
Environments: Literature Review. Digital Education Review, 33, 235–266. https://doi.org/10.1344/der.2018.33.235-
266
18. Prabha, S. L., & Dr. Shanavas, A. R. M. (2015). Application of Educational Data mining techniques in e-Learning A
Case Study. International Journal of Computer Science and Information Technologies, 6(5), 4440–4443.
19. Castro, F., Vellido, A., Nebot, À., & Mugica, F. (2007) Applying Data Mining Techniques to e-Learning Problems.
In L. C. Jain, R. A. Tedman & D. K. Tedman (Еds.), Evolution of Teaching and Learning Paradigms in Intelligent
Environment. Studies in Computational Intelligence. Heidelberg: Springer, 62, 183–221. https://doi.org/10.1007/978-
3-540-71974-8_8
20. Rahman, A., Sultan, K., Aldhafferi, N., & Alqahtani, A. (2018). Educational Data Mining for Enhanced teaching and
learning. Journal of Theoretical and Applied Information Technology, 96(14), 4417–4427.
21. Aldowah, H., Al-Samarraie, H., & Fauzy, W. M. (2019). Educational data mining and learning analytics for 21st
century higher education: A review and synthesis. Telematics and Informatics, 37, 13–49.
https://doi.org/10.1016/j.tele.2019.01.007
https://doi.org/10.1080/03772063.2022.2027276
http://dx.doi.org/10.1016/j.susoc.2021.08.003
https://doi.org/10.1016/j.jclepro.2022.131140
Системні дослідження в енергетиці. 2024. 3(79) 61
МЕТОДИ Й АЛГОРИТМИ РОЙОВОГО ІНТЕЛЕКТУ ДЛЯ ЗАДАЧ
НЕЛІНІЙНОГО РЕГРЕСІЙНОГО АНАЛІЗУ ТА ОПТИМІЗАЦІЇ
СКЛАДНИХ ПРОЦЕСІВ, ОБ’ЄКТІВ ТА СИСТЕМ: ОГЛЯД І
МОДИФІКАЦІЯ МЕТОДІВ Й АЛГОРИТМІВ
Владислав Хайдуров1,2*, канд. техн. наук, ст. досл., https://orcid.org/0000-0002-4805-8880
Вадим Татенко1, https://orcid.org/0009-0008-4869-9689
Микита Литовченко1, https://orcid.org/0009-0001-6671-7763
Тамара Цюпій3, канд. фіз.-мат. наук, доцент, https://orcid.org/0000-0003-2206-2897
Тетяна Жовновач4, https://orcid.org/0000-0003-1037-4383
1Національний технічний університет України «Київський політехнічний інститут імені Ігоря
Сікорського», Берестейський просп., 37, м. Київ, 03056, Україна;
2Інститут загальної енергетики НАН України, вул. Антоновича, 172, м. Київ, 03150, Україна;
3Національний університет біоресурсів і природокористування України, вул. Героїв Оборони, 15,
м. Київ, 03041, Україна;
4Черкаська філія ПВНЗ «Європейський університет», вул. Смілянська, 83, м. Черкаси, 18008, Україна
*Автор-кореспондент: allif0111@gmail.com
Анотація. Розробка швидкісних методів й алгоритмів глобальної багатовимірної оптимізації і їх
модифікацій у різних сферах науки, техніки, економіки є актуальним завданням, яке передбачає
зменшення обчислювальних затрат, прискорення і ефективний пошук розв’язків такого роду задач.
У зв’язку з тим, що більшість серйозних завдань передбачають пошук десятків, сотень або тисяч
оптимальних параметрів математичних моделей, простір пошуку цих параметрів зростає
нелінійно. Нині існує багато сучасних методів й алгоритмів ройового інтелекту, які вирішують
науково-прикладні завдання сьогодення, але вони потребують модифікацій у зв’язку з великими
просторами пошуку оптимальних параметрів моделей. Сучасний ройовий інтелект має значний
потенціал для застосування в енергетичній галузі через свою здатність до оптимізації та
розв’язання складних проблем. За допомогою нього можна вирішувати науково-прикладні задачі
оптимізації споживання енергії в будівлях, промислових комплексах та міських системах, зменшуючи
втрати енергії та підвищуючи ефективність використання ресурсів, а також задачі для побудови
різних елементів енергетичних систем загалом. Відомі методи й алгоритми ройового інтелекту
також активно застосовують для прогнозування виробництва енергії від відновлюваних джерел,
таких як сонячна та вітрова енергія. Це дозволяє краще управляти джерелами енергії та планувати
їхнє використання. Актуальність модифікацій методів й алгоритмів обумовлена питаннями
прискорення швидкості їх роботи під час розв’язання задач машинного навчання, зокрема у моделях
нелінійної регресії, задачах класифікації, кластеризації, де кількість спостережуваних даних може
сягати десяти і сотні тисяч або більше. У роботі розглянуті й модифіковані відомі ефективні
методи й алгоритми ройового інтелекту (алгоритм оптимізації роєм частинок, бджолиний
алгоритм оптимізації, метод диференціальної еволюції) для пошуку розв’язків багатовимірних
екстремальних задач з обмеженнями і без обмежень, а також задач нелінійного регресійного аналізу.
Отримані модифікації відомих класичних ефективних методів й алгоритмів ройового інтелекту, які
присутні у роботі, ефективно розв’язують складні науково-прикладні задачі конструювання
складних об’єктів і систем. Порівняльний аналіз методів й алгоритмів буде проведено у наступному
дослідженні за даною тематикою.
Ключові слова: оптимізація, ройовий інтелект, математичне моделювання, нелінійна регресія,
складні об’єкти та системи.
Надійшла до редколегії: 01.05.2024
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| id | systemreorg-article-858 |
| institution | System Research in Energy |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:23:02Z |
| publishDate | 2024 |
| publisher | General Energy Institute of the National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | systemreorg/b8/31d1f0f549fbd9b7a70c7bd6841749b8.pdf |
| spelling | systemreorg-article-8582026-07-18T12:57:48Z METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS Методи й алгоритми ройового інтелекту для задач нелінійного регресійного аналізу та оптимізації складних процесів, об’єктів та систем: огляд і модифікація методів й алгоритмів Khaidurov, Vladyslav Tatenko, Vadym Lytovchenko, Mykyta Tsiupii, Tamara Zhovnovach, Tetiana optimization, swarm intelligence, mathematical modelling, nonlinear regression, complex objects and systems. оптимізація, ройовий інтелект, математичне моделювання, нелінійна регресія, складні об’єкти та системи. The development of high-speed methods and algorithms for global multidimensional optimization and their modifications in various fields of science, technology, and economics is an urgent problem that involves reducing computing costs, accelerating, and effectively searching for solutions to such problems. Since most serious problems involve the search for tens, hundreds, or thousands of optimal parameters of mathematical models, the search space for these parameters grows non-linearly. Currently, there are many modern methods and algorithms of swarm intelligence that solve today's scientific and applied problems, but they require modifications due to the large spaces of searching for optimal model parameters. Modern swarm intelligence has significant potential for application in the energy industry due to its ability to optimize and solve complex problems. It can be used to solve scientific and applied problems of optimizing energy consumption in buildings, industrial complexes, and urban systems, reducing energy losses, and increasing the efficiency of resource use, as well as for the construction of various elements of energy systems in general. Well-known methods and algorithms of swarm intelligence are also actively applied to forecast energy production from renewable sources, such as solar and wind energy. This allows better management of energy sources and planning of their use. The relevance of modifications of methods and algorithms is due to the issues of speeding up their work when solving machine learning problems, in particular, in nonlinear regression models, classification, and clustering problems, where the number of observed data can reach tens and hundreds of thousands or more. The work considers and modifies well-known effective methods and algorithms of swarm intelligence (particle swarm optimization algorithm, bee optimization algorithm, differential evolution method) for finding solutions to multidimensional extremal problems with and without restrictions, as well as problems of nonlinear regression analysis. The obtained modifications of the well-known classic effective methods and algorithms of swarm intelligence, which are present in the work, effectively solve complex scientific and applied tasks of designing complex objects and systems. A comparative analysis of methods and algorithms will be conducted in the next study on this topic. Розробка швидкісних методів й алгоритмів глобальної багатовимірної оптимізації і їх модифікацій у різних сферах науки, техніки, економіки є актуальним завданням, яке передбачає зменшення обчислювальних затрат, прискорення і ефективний пошук розв’язків такого роду задач. У зв’язку з тим, що більшість серйозних завдань передбачають пошук десятків, сотень або тисяч оптимальних параметрів математичних моделей, простір пошуку цих параметрів зростає нелінійно. Нині існує багато сучасних методів й алгоритмів ройового інтелекту, які вирішують науково-прикладні завдання сьогодення, але вони потребують модифікацій у зв’язку з великими просторами пошуку оптимальних параметрів моделей. Сучасний ройовий інтелект має значний потенціал для застосування в енергетичній галузі через свою здатність до оптимізації та розв’язання складних проблем. За допомогою нього можна вирішувати науково-прикладні задачі оптимізації споживання енергії в будівлях, промислових комплексах та міських системах, зменшуючи втрати енергії та підвищуючи ефективність використання ресурсів, а також задачі для побудови різних елементів енергетичних систем загалом. Відомі методи й алгоритми ройового інтелекту також активно застосовують для прогнозування виробництва енергії від відновлюваних джерел, таких як сонячна та вітрова енергія. Це дозволяє краще управляти джерелами енергії та планувати їхнє використання. Актуальність модифікацій методів й алгоритмів обумовлена питаннями прискорення швидкості їх роботи під час розв’язання задач машинного навчання, зокрема у моделях нелінійної регресії, задачах класифікації, кластеризації, де кількість спостережуваних даних може сягати десяти і сотні тисяч або більше. У роботі розглянуті й модифіковані відомі ефективні методи й алгоритми ройового інтелекту (алгоритм оптимізації роєм частинок, бджолиний алгоритм оптимізації, метод диференціальної еволюції) для пошуку розв’язків багатовимірних екстремальних задач з обмеженнями і без обмежень, а також задач нелінійного регресійного аналізу. Отримані модифікації відомих класичних ефективних методів й алгоритмів ройового інтелекту, які присутні у роботі, ефективно розв’язують складні науково-прикладні задачі конструювання складних об’єктів і систем. Порівняльний аналіз методів General Energy Institute of the National Academy of Sciences of Ukraine 2024-07-01 Article Article application/pdf https://systemre.org/index.php/journal/article/view/858 10.15407/srenergy2024.03.046 System Research in Energy; No. 3 (79) (2024): System Research in Energy; 46-61 Системні дослідження в енергетиці; № 3 (79) (2024): Системні дослідження в енергетиці; 46-61 2786-7102 2786-7633 en https://systemre.org/index.php/journal/article/view/858/768 Copyright (c) 2024 Vladyslav Khaidurov, Vadym Tatenko, Mykyta Lytovchenko, Tamara Tsiupii, Tetiana Zhovnovach https://creativecommons.org/publicdomain/zero/1.0 |
| spellingShingle | optimization swarm intelligence mathematical modelling nonlinear regression complex objects and systems. Khaidurov, Vladyslav Tatenko, Vadym Lytovchenko, Mykyta Tsiupii, Tamara Zhovnovach, Tetiana METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS |
| title | METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS |
| title_alt | Методи й алгоритми ройового інтелекту для задач нелінійного регресійного аналізу та оптимізації складних процесів, об’єктів та систем: огляд і модифікація методів й алгоритмів |
| title_full | METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS |
| title_fullStr | METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS |
| title_full_unstemmed | METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS |
| title_short | METHODS AND ALGORITHMS OF SWARM INTELLIGENCE FOR THE PROBLEMS OF NONLINEAR REGRESSION ANALYSIS AND OPTIMIZATION OF COMPLEX PROCESSES, OBJECTS, AND SYSTEMS: REVIEW AND MODIFICATION OF METHODS AND ALGORITHMS |
| title_sort | methods and algorithms of swarm intelligence for the problems of nonlinear regression analysis and optimization of complex processes, objects, and systems: review and modification of methods and algorithms |
| topic | optimization swarm intelligence mathematical modelling nonlinear regression complex objects and systems. |
| topic_facet | optimization swarm intelligence mathematical modelling nonlinear regression complex objects and systems. оптимізація ройовий інтелект математичне моделювання нелінійна регресія складні об’єкти та системи. |
| url | https://systemre.org/index.php/journal/article/view/858 |
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