Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices
The problems of the optimization of investment, with the use of analysis of a portfolio of investments, should be classified as complex in the computational sense of problem. To solve such problems it is expedient to use parallel computing systems. Realization of more effective algorithms, based on...
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| Дата: | 2009 |
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2009
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| Цитувати: | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices / A. Katkow // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 109-117. — Бібліогр.: 5 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860009230621212672 |
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| author | Katkow, A. |
| author_facet | Katkow, A. |
| citation_txt | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices / A. Katkow // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 109-117. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Электронное моделирование |
| description | The problems of the optimization of investment, with the use of analysis of a portfolio of investments, should be classified as complex in the computational sense of problem. To solve such problems it is expedient to use parallel computing systems. Realization of more effective algorithms, based on the use of iterative process of the random search, is examined in the paper. The problems of optimization of investment portfolio in the parallel computational environment namely in the coupled map lattices are investigated
Проблемы оптимизации инвестиций с помощью анализа портфеля инвестиций следует классифицировать как сложные вычислительные задачи. Для решения таких проблем целесообразно использование параллельных вычислительных систем. Рассмотрена реализация эффективных алгоритмов, основанных на использовании итерационных процессов случайного поиска. Исследованы проблемы оптимизации портфеля инвестиций в параллельных вычислительных сетях, а именно в решетках связанных отображений.
Проблеми оптимізації інвестицій за допомогою аналізу портфеля інвестицій слід класифікувати як складні обчислювальні задачі. Для вирішення таких проблем є доцільним використання паралельних обчислювальних систем. Розглянуто реалізацію ефективних алгоритмів, базованих на використанні ітераційних процесів випадкового пошуку. Досліджено проблеми оптимізації портфеля інвестицій у паралельних обчислювальних мережах, а саме у решітках зв‘язаних відображень.
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A. Katkow, D.Sc.
Czestochowa University of Technology
(E-mail: katkow@zim.pcz.pl , 42-200 Czestochowa, Poland)
Algorithms of Random Search
of Optimum Portfolio of the Investment
in the Structure of Coupled Map Lattices
The problems of the optimization of investment, with the use of analysis of a portfolio of invest-
ments, should be classified as complex in the computational sense of problem. To solve such
problems it is expedient to use parallel computing systems. Realization of more effective algo-
rithms, based on the use of iterative process of the random search, is examined in the paper. The
problems of optimization of investment portfolio in the parallel computational environment
namely in the coupled map lattices are investigated.
Ïðîáëåìû îïòèìèçàöèè èíâåñòèöèé ñ ïîìîùüþ àíàëèçà ïîðòôåëÿ èíâåñòèöèé ñëåäóåò êëàñ-
ñèôèöèðîâàòü êàê ñëîæíûå âû÷èñëèòåëüíûå çàäà÷è. Äëÿ ðåøåíèÿ òàêèõ ïðîáëåì öåëå-
ñîîáðàçíî èñïîëüçîâàíèå ïàðàëëåëüíûõ âû÷èñëèòåëüíûõ ñèñòåì. Ðàññìîòðåíà ðåàëèçàöèÿ
ýôôåêòèâíûõ àëãîðèòìîâ, îñíîâàííûõ íà èñïîëüçîâàíèè èòåðàöèîííûõ ïðîöåññîâ ñëó÷àé-
íîãî ïîèñêà. Èññëåäîâàíû ïðîáëåìû îïòèìèçàöèè ïîðòôåëÿ èíâåñòèöèé â ïàðàëëåëüíûõ
âû÷èñëèòåëüíûõ ñåòÿõ, à èìåííî â ðåøåòêàõ ñâÿçàííûõ îòîáðàæåíèé.
K e y w o r d s: portfolio analysis, random search, coupled map lattices.
Optimization of the investment activity of enterprises or individual clients with
the great quantity of components of the investment portfolio and a significant
quantity of securities is connected with the performing of the great quantity of
computational operations. The innovation process, which purpose in this area of
investment activity is the acceleration and simplification of the computational
process, consists of the use of computing systems with parallel realization of cal-
culations processes. On the other hand this process consists of the development
and use of new effective algorithms for solution of the problem of control of the
investment process. The paper studies the use of contemporary computing sys-
tems, based on realization of computational operations with application of cou-
pled map lattices and algorithmic proposals based on the development of effec-
tive algorithms with realization of the process of risk minimization within the
desired level with the profitableness of investments on the basis of the random
search algorithms.
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 109
The constructing of the investment portfolio consists of two processes con-
nected together [1]. The first stage of the portfolio creation consists of searching
for an answer to the question: what firms can be included on as the investment
portfolio elements. The complexity of the first stages consists in selection of the
necessary quantity of firms which will enter in the created investment portfolio.
We will call a set of firms, which can form the portfolio, as a basic set of compa-
nies. In principle it is possible to compose a basic set of all firms, whose securi-
ties are located in the free to sale stock exchange quotation. It is also possible to
select the most reliable and thriving firms of their number. In any case their
quantity must compose a significant group, that can guarantee that the invest-
ment portfolio would have solid foundation. The quantity of firms, which should
be selected to be included in the portfolio of the total number of firms from the
basic set, is also an important stage of decision.
It is customary to assume that the investment portfolio, which consists of 10
parts, appears as sufficiently well formed, if it is created on the basis of a set of
several dozens or may be hundreds of firms, which take part in the process of
the stock exchange quotation. About 250 firms take part in the auction process at
the Warsaw Stock Exchange of securities. If we attempt to form the investment
portfolio of 10 parts, then the number of possible combinations of possible reali-
zation of the portfolio reaches the significant magnitude and it can be evaluated
by the formulaC n
m
, where n is the total quantity of the firms and m — quantity of
component parts of the portfolio. The number of possible combinations in the
portfolio composition reaches 2
1017
. Thus the total number of combinations
reaches an enormous value. If we consider that the task about the selection of the
portion of each portfolio component contribution still must be solved for each
portfolio, then solution of the problem about location of the investment portfolio
optimum requires applying the most contemporary computers. It is interesting in
this context to examine the application of parallel computing techniques to de-
cide the problem of the investment portfolio optimum.
The selection of firms for including in the investment portfolio begins from
analysis of financial condition of the firms. The financial state of the firm is de-
termined on the basis of the analysis of the profitableness of firm’s securities and
of the investments risk. The dependence of the price of the firm stocks on a price
change of securities of other participants in the financial processes on the stock
exchange is also of great significance. This influence is expressed mathemati-
cally as a coefficient of correlation between the price of this firm share and secu-
rities of other firms. The second stage of creation of the investments portfolio
consists in determination of the part of the portfolio composed by contribution of
each of the firms, whose securities were included in its contents. This problem is
connected with the determination of value of minimum risk and it is a task of
mathematical programming.
A. Katkow
110 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
The second stage of the problem of creation of the investments portfolio can
be recorded for the population of securities, which consists of N elements, in a
following manner for standard variation of portfolio as a goal function:
minS x S x x S S rp i i
i
N
i
N
j
N
i j i j ij
2 2 2
1 1 1
2� �
� � �
� � � , (1)
for given level rate of return
R x Rp i i
i
N
�
�
�
1
, (2)
and a sum of weight coefficient x i
10
1
. �
�
�x i
i
N
(3)
as boundary conditions [2]. In this equations Si — the standard deviation of
returns of the company shares; Sp — variance portfolio; Rp — expected rate of return
of the portfolio; ri — expected rate of return on shares of the company i; xi — part of
the company shares i in the portfolio; rij — the correlation coefficient yields
shares of the company i and company j. The goal function is a nonlinear function
in this case. To solve this problem (1) it is possible to use, for example, the
method of Lagrange’s coefficients.
As it was mentioned above, the process of creating the optimum portfolio of
investments can be divided in two stages, one of which was connected with se-
lection of the portfolio components from a set of securities, which is expedient to
invest. The second part of solution of the problem of creating the optimum in-
vestments portfolio consists of the search for the coefficients of the participation
of each selected security in the portfolio composition that leads to minimization
of the goal function (1) within the limitations (2), (3). Both these tasks can be
successfully solved by using the parallel computational environment of the cou-
pled map lattice, two algorithms of random search being realized in each of
them. The first algorithm of the random search realizes the selection of portfolio
components from the base of securities, and the second algorithm of the random
search achieves a search for the optimum coefficients of each security participa-
tion in the portfolio composition.
The idea of creation of the architecture of parallel computational environ-
ment, which by the principles of functioning is similar to the neuron network,
belongs to John von Neumann. It is well known architecture Cellular Automata
(CA). Coupled Map Lattices (CML) is a generalization of cellular automata [3].
CML is a cellular automaton with information representation in the form of real
numbers [4]. In CML structure the computing components in the lattice sites ope-
Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 111
rate with real numbers and their interaction is not limited to Neumann or Moore
local neighborhood. In these structures, computing components can initiate pro-
cesses of information exchange between other computing components that are
not located in close proximity. This means that there is an opportunity to influ-
ence the local process of a random search of an optimal portfolio by means of the
results of a random process of finding the optimal portfolio, implemented in any
computational element of CML structures. CML like CA is placed in the cate-
gory of architectures of the Single Instruction Multiply Data type, according to
classification by M. Flynn. In this architecture a single flow of instructions and
plural flow of data take place. According to characteristic feature of CML archi-
tecture they create computational environment of identical computational ele-
ments, connected in such a manner that they create the grid of the computational
structure. It is known that high computing speed of CML is caused by wide
parallelization of computational process. This circumstance permits optimiza-
tion both on the component parts of the portfolio, and the values of the portfolio
components contribution to one and the same computational environment. In this
structure each cell of lattice is connected with four adjoining the same computational
cells (Pij), forming the computational network with large computing power.
A very important feature of CML structures, unlike CA is the ability to im-
plement the interaction with computing components located anywhere in the
CML structure and not only with neighboring elements as in the case of CA
structures. Each of the processors in the computational cell has Random Access
Memory, where it has information about the base set of shares, and carries out
the same computational algorithm of random search. The essence of the first al-
gorithm of the random search consists in the random searching of the invest-
ments portfolio of components of the base of securities. The essence of the se-
cond algorithm consists in the fulfillment of the operation of random searching
of a value of weight coefficients of the compound parts of the portfolio. Thus the
strategy of the search for such investments portfolio is realized in each cell
which is nearer as to its parameters to the optimum than the investments portfo-
lios created in the neighboring cells. For this purpose the first algorithm creates
portfolio from the randomly selected composition of shares and the second algo-
rithm generates the random contribution of each security or stock to the portfolio
composition. In each cell on the basis of those random selected portfolio compo-
nents the random search for the portfolio parameters is achieved which corre-
spond to the goal function (1) and boundary conditions (2), (3). As a result of
comparison the portfolio is determined, which parameters are better and these
parameters are transferred to the local neighboring cells.
The portfolio parameters are memorized in each of the local neighboring
cells as locally optimum portfolio. Then in each cell a new portfolio composition
A. Katkow
112 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
is created again with the new coefficients of stocks participation in their compo-
sition. The parameters of the newly created portfolio are compared with those of
the local optimum portfolio. As a result of comparison two versions are possible:
1. A new portfolio has the parameters, which appear to be worse than in the
local optimum portfolio. In this case a new portfolio does not remain and the pro-
cess of creation of the new portfolio is initiated in the cell.
2. A new portfolio has the parameters, which appear to be better than in the
local optimum portfolio. In this case the portfolio version is represented for the
comparison by the portfolios parameters in the neighboring cells.
3. As a result the comparison of the investments portfolio parameters cre-
ated in this cell and four neighboring cells such parameters are selected from
them which minimize the goal function to the best degree (1).
The example of selection of the component parts of the portfolio from three
components with the use of random search method let us examine below. We will
simulate a task in the CML structure. Let us accept the following numerical values
of the rates of return and risk as variation of stock’s rate of return for characteristic of
the securities, included in the portfolio. This is test example from [2]:
S
1
025� . ; R
1
005� . ; r
12
015� . ;
S
2
021� . ; R
2
01� . ; r
13
017� . ;
S
3
028� . ; R
3
015� . ; r
23
009� . ;
(4)
R p �01. .
The algorithm of the calculation part of contribution of the components to
the composition of the briefcase of investments is represented below.
Input:
X[i] - array of initial value of unknowns
Rate[i] - array of rate of return (i=1,…,n)
Var[i] - array of variation rate of return
Cov[ij] - array of covariation (i=1,…,n,j=1,…,m,i�j)
Varpold - variation of the portfolio (old)
Varpnew - variation of the portfolio (new)
e - error of the solution (Varpnew-Varpold)
E - maximum of the solution error
Output:
X[i] - array of solution
Varpnew - variation of the portfolio
k - a number of random search iteration
Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 113
k=0
while e <= E do
k=k+1
X[1] = random from 0 to 1.0
X[2] = random from 1.0-X[1]
X[3] = 1.0 – X[1] – X[2]
calculate Varpnew
calculate e
end while
We have n �3, m �3for the test example (4). The algorithm work consists of
three stages. In the first stage the random search for weight coefficients X[i],
which satisfy the boundary conditions (3) is performed, then Varpnew is calcu-
lated, a variation in the rate of return of the portfolio with the assigned rates of
return of the corresponding stocks, including in the portfolio (1), and at the end
the value e is calculated, which is interpreted as an error and shows how much
the value of variation in the rate of return of the portfolio approaches the opti-
mum value of variation on this iteration. Iterative process is ended, if value e be-
comes less than E which means that the degree of variation approximation to its
optimum value becomes less than E. It is possible to finish the fulfilment of the
computational process after performing the specific number of iterations.
We simulated the process of solution of the problem about the selection of
the optimum portfolio of investments in the CML environment. The results of
the process of simulation for test problem (4) are presented in Figure.
Let us calculate the values
x
1
0245� . , x
2
0508� . , x
3
0245� . , S p
2
01668min .� .
These results were obtained after 20 iterations in the CML environment. In this
case on the diagram the values x1, x2, S p
2
scaled with the coefficient that is equal
to 100.
Let us examine possible types of interaction between CA cells with the
search for the optimum portfolio of investments. The least rapid method of com-
munication between the cells and the simplest of them is based on the idea of rea-
lization of synchronous operation similar to Jacobi iteration technique and the
below exchange of information between the cells is named that of Jacobi’s type.
The essence of this method of exchange consists in the complete realization of
the processes of exchange between all CML cells with the fulfillment of one ite-
ration. As a result of the exchange of information between the ÑA cells one ver-
sion of portfolio with the best parameters is selected. The new versions of the
portfolio of investments in the cells are not created until it will end the process of
exchange as a result of which the best portfolio version will be selected. This
A. Katkow
114 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
version of the portfolio of investments is memorized in each CML cell. On the
following iteration the new version of the portfolio of investments is created in
each cell, however for the exchange of information between the neighboring
cells it is represented only that version of the portfolio, which as to its parameters
is equal or exceeds that, which was obtained on the previous iteration. After the
new version of the portfolio of investments is created in each CML cell it begins
the process of information exchange between cells as a result of which the port-
folio of investments with the best parameters is determined.
The process described above is repeated to that moment, when as a result of
search the version of portfolio with the optimum parameters is obtained. This
situation occurs when as a result of iteration in each cell the portfolio is created,
which parameters equal or differ to insignificant degree from the portfolio of in-
vestments, created on the previous iteration. The exchange of information be-
tween the cells is accomplished locally asynchronously on the basis of signals,
which are established by each cell, when it is ready to information exchange.
Each cell after the selection of the new portfolio of investments, approaches the
interrogation of adjacent cells for the purpose of determination of their current
state. When a cell is in the process of scanning the state of adjacent cells fixes
their readiness for the realization of exchange, there occurs the transfer of the
portfolio parameters from each peripheral cell to the central cell, where the port-
folio of investments with the best parameters is selected as a result of compari-
son. This portfolio is written in the memory of central cell and transferred to the
peripheral cells, where it is also written in the memory of cells.
As an example the implementation of the iteration process of random search
for the type of Jacobi method with a random selection of component in CML is
presented.
Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 115
25
20
15
10
5
0
x20 40 60
1
Sp
2
0 50 100 150 x
2
a á
Diagram x1 and x2 values as results of random search of minimum goal function S p
2
()
Input:
X [i] - array of initial values
E [i] - array error of solution
E - the maximum error of solution
e - error of solution
F [i] - array of flags
Output:
X [i] - array of calculated values
while E [i]� e
for i from 1 to q do
F [i] = 0
end for
s = 0
while s <q do
i = random from 1 to q
calculate X [i]
F [i] = 1
for i from 1 to q do
s = s + F[i]
end for
end while
for i from 1 to q do
evaluate X [i]
end for
end while.
In this algorithm q = w * w, where w is a number of row and column in the
lattice structure, and in this case we investigate interaction of the computing ele-
ment only with neighboring computing elements as in CA structures. The opera-
tion calculate X[i] is performed with respect to algorithm, shown above. Perform-
ing of the operation evaluate X[i] means, that the results of calculation in the lat-
tice cells are transmitted to neighboring cell in each cell of the lattice .
Each element of CML architecture in the iteration does the following calcu-
lation:
MK = MS + MR + MP + MB,
where MK — the amount of arithmetic operation, performed in one step of itera-
tive process; MS — arithmetic operations performed in calculating the expected
rate of return on the portfolio; MR — arithmetic operations performed in calcu-
A. Katkow
116 ISSN 0204–3572. Electronic Modeling. 2009. V. 31. ¹ 6
lating the risk of the portfolio investment; MP — arithmetic operations, carried
out a comparison of the results, selected by characterizing the individual ele-
ments cellular automat on computing the investment portfolios; MB — arithme-
tic operation for choice of new version of component portfolio set.
The total amount of mathematical operations MC, carried out by cells from
CML during the search for the optimal portfolio on one step of iteration process
can be described as
MC = q * MK + MWE + MWY,
where q is quantity of CML cells and MWE, MWY, respectively, operations of the
opening and launch of input streams for the introduction of information for the
initialization memory of each calculation element and output stream for re-
moval results of computing process in CML.
Let us assume that with the sequential realization of computational process
the same quantity of operations with the use of one CML cell is carried out. It is
natural that in this case the total time of the problem solution would be approxi-
mately q times more.
Some investigations of network realization of investment management are
represented in [5].
Summary. The innovation process of using the algorithms of the random
search for the solution of the problem of creation of the optimum portfolio of invest-
ments is investigated on the basis of the classical Markowitz’s model. The algorithm
of solution of test problem with the use of a process of random search is proposed
and the results of its simulation are given. The estimation of the effectiveness of so-
lution of the problem in CML environment is produced. It shows that with solution
of the problem in CML environment the rate of obtaining the solution increases ap-
proximately q times, where q is quantity of CML cells.
Ïðîáëåìè îïòèì³çàö³¿ ³íâåñòèö³é çà äîïîìîãîþ àíàë³çó ïîðòôåëÿ ³íâåñòèö³é ñë³ä êëàñè-
ô³êóâàòè ÿê ñêëàäí³ îá÷èñëþâàëüí³ çàäà÷³. Äëÿ âèð³øåííÿ òàêèõ ïðîáëåì º äîö³ëüíèì
âèêîðèñòàííÿ ïàðàëåëüíèõ îá÷èñëþâàëüíèõ ñèñòåì. Ðîçãëÿíóòî ðåàë³çàö³þ åôåêòèâíèõ
àëãîðèòì³â, áàçîâàíèõ íà âèêîðèñòàíí³ ³òåðàö³éíèõ ïðîöåñ³â âèïàäêîâîãî ïîøóêó. Äî-
ñë³äæåíî ïðîáëåìè îïòèì³çàö³¿ ïîðòôåëÿ ³íâåñòèö³é ó ïàðàëåëüíèõ îá÷èñëþâàëüíèõ ìå-
ðåæàõ, à ñàìå ó ðåø³òêàõ çâ‘ÿçàíèõ â³äîáðàæåíü.
1. Markowitz H. Portfolio Selection: Efficient Diversification of Investments. — London :
John Wiley & Sons, 1959.
2. Naugen R. A. Teoria nowoczesnego inwestowania. — W-wa : WIGPRESS, 1996.
3. Kaneko K. Theory and Application of Coupled Map Lattices. — Chichester : J. Willey &
Sons, 1993.
4. Kulakowski K. Automaty komorkowe. — Krakow : Wydawnictwo AGH, 2000.
5. Katkow A., Ulfik A. Network implementation of investment management // Electronic mode-
ling. — 2007. — 29, ¹ 4.
Submitted on 04.11.09
Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2009. Ò. 31. ¹ 6 117
|
| id | nasplib_isofts_kiev_ua-123456789-101532 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-12-07T16:41:16Z |
| publishDate | 2009 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Katkow, A. 2016-06-04T13:21:10Z 2016-06-04T13:21:10Z 2009 Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices / A. Katkow // Электронное моделирование. — 2009. — Т. 31, № 6. — С. 109-117. — Бібліогр.: 5 назв. — англ. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101532 The problems of the optimization of investment, with the use of analysis of a portfolio of investments, should be classified as complex in the computational sense of problem. To solve such problems it is expedient to use parallel computing systems. Realization of more effective algorithms, based on the use of iterative process of the random search, is examined in the paper. The problems of optimization of investment portfolio in the parallel computational environment namely in the coupled map lattices are investigated Проблемы оптимизации инвестиций с помощью анализа портфеля инвестиций следует классифицировать как сложные вычислительные задачи. Для решения таких проблем целесообразно использование параллельных вычислительных систем. Рассмотрена реализация эффективных алгоритмов, основанных на использовании итерационных процессов случайного поиска. Исследованы проблемы оптимизации портфеля инвестиций в параллельных вычислительных сетях, а именно в решетках связанных отображений. Проблеми оптимізації інвестицій за допомогою аналізу портфеля інвестицій слід класифікувати як складні обчислювальні задачі. Для вирішення таких проблем є доцільним використання паралельних обчислювальних систем. Розглянуто реалізацію ефективних алгоритмів, базованих на використанні ітераційних процесів випадкового пошуку. Досліджено проблеми оптимізації портфеля інвестицій у паралельних обчислювальних мережах, а саме у решітках зв‘язаних відображень. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices Алгоритмы случайного поиска оптимального портфеля инвестиций в структуре решетки связанных отображений Article published earlier |
| spellingShingle | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices Katkow, A. |
| title | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices |
| title_alt | Алгоритмы случайного поиска оптимального портфеля инвестиций в структуре решетки связанных отображений |
| title_full | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices |
| title_fullStr | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices |
| title_full_unstemmed | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices |
| title_short | Algorithms of Random Search of Optimum Portfolio of the Investment in the Structure of Coupled Map Lattices |
| title_sort | algorithms of random search of optimum portfolio of the investment in the structure of coupled map lattices |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101532 |
| work_keys_str_mv | AT katkowa algorithmsofrandomsearchofoptimumportfoliooftheinvestmentinthestructureofcoupledmaplattices AT katkowa algoritmyslučainogopoiskaoptimalʹnogoportfelâinvesticiivstrukturerešetkisvâzannyhotobraženii |