ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
274
2181-3187
OPTIMAL LOSS METHOD.
A.I.Ismoilov
Senior Lecturer of the Department of
Applied Mathematics and Informatics
, Fergana State University. Doctor
of Philosophy (PhD) in Physics and Mathematics
E-mail: ismoilovaxrorjon@yandex.com
Abduhalilova Sohiba Abdurasul qizi
Fergana State University Applied Mathematics
direction, 3rd year student, 22-08 group
sohibaabduhalilova159@gmail.com
Annotation:
This article discusses the theoretical foundations and practical
application of the optimal loss method. The essence of the method is that it implies
decision-making aimed at minimizing losses while managing the activity of an object
or system. The article analyzes the optimal loss criteria, the methods of their calculation
and their application to real economic and technical questions. It also outlines strategies
for determining optimal losses based on statistical and probability methods. The results
of the study serve to increase the effectiveness of the optimal loss method and improve
decision-making processes.
Keywords:
optimal loss, loss function, decision theory, probability model, cost-
effectiveness, management strategy, statistical estimation, minimization method.
Log in
Most of the modern scientific and practical problems are closely related to optimal
decision-making. Any decision necessarily comes with a certain degree of risk and the
possibility of loss. Therefore, one of the most important tasks in decision-making is to
minimize losses – that is, choose the optimal path.
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
275
2181-3187
The optimal loss method is a technique that serves to find the optimal solution by
identifying the loss function in complex systems, processes, or management problems
and reducing it to a minimum value. This method is widely used in economics,
management, engineering, artificial intelligence, medicine and many other fields.
The method is theoretically based on decision theory and probability analysis and
makes it possible to identify losses in real conditions, model them, as well as to develop
optimal solutions. This article analyzes the concept of the optimal loss method, its main
types and the possibilities of applying them to practical issues.
The first steps of this method are similar to the Gaussian method. Assuming
0
a
that the lead element is,
11 1
12
2
1
1,
1
21 1
22
2
2
2,
1
1 1
2
2
m,
......................................
n
n
n
n
n
n
m
m
mn
n
n m
a x
a x
a x
a
a x
a x
a x
a
a x
a x
a x
a
+
+
+
+
+ +
=
+
+ +
=
+
+ +
=
(1)
The first equation of the system
1
1
1
1
12
2
1
1,n 1
...
n
n
x
b x
b x
b
+
+
+ +
=
(2)
to the appearance of a new species. Then we lose only the second equation of
(1):
1
x
(1)
(1)
(1)
22
2
2
2,n 1
...
n
n
a x
a x
c
+
+ +
=
Now
(1)
22
0
a
, supposing that, let's put this equation into (3) as follows:
(2)
(2)
2
2
23
3
2
2,n 1
...
n
n
x
b x
b x
b
+
+
+ +
=
(3)
With this equation, we get rid of equation (2
2
x
). As a result
(2)
(2)
(2)
1
13
3
1
1,n 1
...
n
n
x
c x
c x
c
+
+
+ +
=
( )
2
(2)
(2)
2
23
3
2
2,n 1
...
n
n
x
c
x
c
x
c
+
+
+ +
=
is formed. here
(2)
(1)
(1)
(2)
1
2
2
ij
j
j
j
c
b
b b
=
−
,
(2)
(2)
2
2
j
j
c
b
=
(j 3)
.
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
276
2181-3187
k
Suppose substitutions on the previous equations
As a result of the implementation of (1)system, the following equally strong
system-
To be cited:
( )
( )
( )
1
1,
1
1
1
1,
1
( )
( )
( )
,
1
1
,
1
1,1 1
1,
1
1
1,
1,
1
...
...............................................
...
...
...
............................
k
k
k
k
k
n
n
n
k
k
k
k
k k
k
kn
n
k k
k
k
k
k
k
n
n
k
n
x
c
x
c
x
c
x
c
x
c
x
c
a
x
a
x
a
x
a
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ +
=
+
+ +
=
+ +
+ +
=
1 1
,
1
1
,
1
...................
...
...
n
n k
k
nn
n
n n
a x
a
x
a x
a
+
+
+
+ +
+ +
=
(4)
We multiply
k
the previous equation of this system
1,
1,2
1,
,
,...,
k
k
k
k
k
a
a
a
+
+
+
by the
corresponding ones,
(
1)
k
+
subtract the results from the equation and divide the
resulting equation
1
k
x
+
by the coefficient before the infinite. The resulting
(
1)
k
+
equation would look like this:
( )
( )
( )
1
1,
2
2
1,
1,
1
...
k
k
k
k
k
k
k
k
n
n
k
n
x
c
x
c
x
c
+
+
+
+
+
+
+
+
+ +
=
Now, with the help of this equation, if we lose the previous equation of the
system (4),
k
1
k
x
+
then we have again the system of form (4), only
k
the form of .
(
1)
k
+
. . At the same time, if
( )
1,
1
,
1
1,
1
0
k
k
k
k
r k
k
r
r
a
c
a
+
+
+
+
=
−
In this case, we get the following formulas:
( )
1,
1,
,
(
1)
1
1,
( )
1,
1
1,
,
1
1
(
1)
( )
( )
(
1)
,
1
1,
,
(
1, 2,..., ;
2,
3,...,
1).
k
k
k
p
k
r r p
k
r
k
p
k
k
k
k
k
r r k
r
k
k
k
k
ip
ip
i k
k
p
a
a
c
c
a
a
c
c
c
c
c
i
k p
k
k
n
+
+
+
=
+
+
+
+
+
=
+
+
+
+
−
=
−
=
−
=
= +
+
+
After the -step of substitutions
n
is also determined, the following formulas are
generated for the solution of (1) systemapping:
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
277
2181-3187
( )
,
1
n
i
i n
x
c
+
=
(
1, 2,..., )
i
n
=
.
Here, too, the control of the computational process is similar to the Gaussian
method. All leading even in the optimal method of loss
It is necessary that the elements must be different from zero. If this fact is not
known in advance, then it is more expedient to change the calculation scheme and
eliminate the unknowns by selecting the main elements line by line. To do this, after
(
1)
k
+
eliminating unknowns
1
2
,
,...,
k
x x
x
from the equation,
(
1)
1,
1
1,
1
k
k
k
k
k
s sp
s
a
a
c
+
+
+
+
=
−
(
1)
p
k
+
is the largest element by modulus, then it is necessary to redefine the variables:
1
k
p
x
x
+
=
and
1
p
k
x
x
+
=
, and then continue to eliminate the unknowns according to the
optimal loss rule.
The advantage of the optimal loss method is that
n
Although the number of arithmetic operations required to solve an ordinal
system is the same as in the Gaussian method, this method allows efficient use of
computer memory, i.e., doubling the order of the system. (4) It can be seen from the
system that once the step of optimal loss is
k
completed, the last equation of a given
system
(
)
n k
−
remains unchanged. Taking this into account, we will introduce one line
before each step, without fully entering all the elements of the matrix into the
memory. In this case
(
1)
k
+
, one memory cell will be enough to perform the step
( )
(
1)
1
f k
k n k
n
=
− + + +
, which
( )
( )
1,
1
1,
1
( )
( )
,
1
,
1
............
........................
............
k
k
k
n
k
k
k k
k n
c
c
c
c
+
+
+
+
It serves to accommodate the matrix and the coefficients of the equation in (4).
(
1)
k
+
Now
( )
f k
that we have found the maximum of , we
n
make sure that the area
with the cell is sufficient to solve the ordered system
(
1)(
5)
4
n
n
+
+
. For example,
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
278
2181-3187
solving a system of 122-order equations on a computer with 4095 operating memory
without using peripherals or calculating the determinant of an optional matrix of this
order
It is possible.
As an example
1
2
3
4
1
2
3
1
2
3
4
1
2
3
4
2
4, 2
1, 6
3
3, 2
0, 4
3
2, 4
1, 6
1, 6
0,8
1
2
1, 5
0
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
+
+
−
=
−
+
−
= −
−
+ −
= −
−
− +
=
Let's solve the system with the optimal method of elimination. First Equation
1
2
3
4
2,1
0,8
1,5
16
x
x
x
x
+
+
−
=
(5)
and multiply this by —0,4 and subtract from the second equation of the system:
2
3
4
3,84
2, 08
0, 60
0,96
x
x
x
−
−
= −
We divide this by 3.84 and form the required equation:
2
3
4
0,54167
0,15625
0, 2500
x
x
x
−
−
= −
(6)
Where's (2.13) and
2
x
Yo'qotsak,
1
2
4
1,93750
1,17182
2,12501
x
x
x
+
−
=
(7)
If we divide (7) by 1.6 and (6) by −0.8 from the third equation of the system,
and divide the resulting equation
3
x
by the coefficient before it,
3
4
0, 29611
1,81556
x
x
−
=
(8)
origin.
Using this equation, we eliminate (6) and (7) dai
3
x
,
1
4
2
4
0, 59811
1, 39322
0, 31664
0, 73343
x
x
x
x
−
= −
−
=
(9)
is formed.
Now we lose the
1
2
3
,
,
x x x
fourth equation of the system using equations (8) -
(9):
4
1,11872
1, 67564
x
=
. From this and from (5)-(9) we find the unknowns in sequence:
4
3
2
1
4, 00065;
3, 0019;
1,99999;
0,99922
x
x
x
x
=
=
=
=
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
279
2181-3187
Calculating the determinant.
Both the Gaussian method and the optimal loss method can be used to
calculate the determinant. Following
11
12
1
2
21
22
1
1
...
.......
...
n
n
n
n
nn
a
a
a
a
a
a
A
a
a
a
=
Be required to find the determinant of the matrix. For this,homogeneous, linear
Ax
o
=
(10)
To solve the system we apply the Gaussian method. The resulting
A
matrix
(1)
(1)
12
1
(2)
2
1
0
1
...
.......
...
0
0
1
n
n
b
b
b
B
=
The triangle is replaced by a matrix, and the system (10) is equivalent to it
0
Bx
=
system.
If we pay attention, the
B
elements of the matrix
A
are formed as a result of the
following two elementary substitutions from
1
2
1
,
,...,
n
A A
A
−
the matrix and the
subsequent auxiliary matrices:
1) the division into hypothetical
(1)
(
1)
11
22
,
,...,
n
nn
a
a
a
−
leading elements as different
from "zero";
2) separating rows from rows of matrix and helper
1
2
1
,
,...,
n
A A
A
−
rows that are
proportional to the leading rows, respectively.
As a result of the first substitution, the determinant of the matrix is also divided
into a corresponding leading element, while the second substitution leaves the
determinant unchanged. That's why
(1)
(
1)
11
22
det
1 det
,
,...,
n
nn
A
B
a
a
a
−
=
=
From this place ESA
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
280
2181-3187
(1)
(
1)
11
22
det
,
,...,
n
nn
A
a
a
a
−
=
(11)
So the determinant is equal to the multiplication of the leading elements in the
Gausian compact scheme.
The matrix determinant can also be calculated using the optimal loss method.
Here, too, the determinant leads all
( )
1,
2
1,
,
1
1
k
k
k
k
k
k
r r k
r
a
a
a
c
+
+
+
+
=
=
−
is equal to the multiplicity of elements:
1
det
n
k
k
A
a
=
=
(12)
If any of the leading elements is zero, then the scheme for selecting the head
element by line should be used. But in this case
k
a
, it
1
( 1)
k
l
+
−
would be necessary to
multiply the elements by in order to preserve the determinant's gesture. The number
here
k
l
refers to the number of unknowns lost in the step if
k
all unknowns that were
not lost in the previous step are
1, 2,...,
n k
−
numbered in
1
k
+
successive sequences
from left to right. But when the calculation is performed with the formulas (11) or
(12) as usual, the multiplication of the former factors
det
A
for
i
n
something can
be equal to or multiplied by the machine zero, unless
i
it is small (large).
To get rid of such a defect, it is necessary to calculate according to the formula
(11)
det
A
as follows:
1
1
det
(
( 1)
)(
( 1)
)
j
k
l
l
j
k
j
k
A
q
a
r
a
+
+
=
−
−
Here it is
q
close to the largest possible number in the exposition, close
r
to the
smallest number, and at the same time
1
q r
=
;
j
a
Among the leading elements, the
ones that are smaller in modulus are the
k
a
remaining leading elements.
Now let's look at a case study of this method
Example: Solve the following system of equations using the optimal loss
method.
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
281
2181-3187
2
8
3
2
11
2
2
3
x
y
z
x
y
z
x
y
z
+ − =
− − + = −
− + + = −
1.
Let's pick a basic equation and leave it accessible to others to work with.
2
8
x
y
z
+ − =
That's what we get.
2.
From the other equations, we get x by using Equation 1.
Equation 2:
3
2
11
x
y
z
− − +
= −
We multiply Equation 1 by 1.5:
1.5 (2 x y z)
3x 1.5 y 1.5z 12
+ − =
+
−
=
Now we add Equation 2 and this result:
( 3x y 2 z) (3x 1.5 y 1.5z)
11 12
− − +
+
+
−
= − +
Result:
2
y
z
+ =
(new Equation 2)
Now we add Equation 3 and Equation 1:
( 2 x y 2 z) (2 x y z)
3 8
− + +
+
+ − = − +
Result:
2 y z
5
+ =
(new Equation 3)
3.
We work with the 2 new equations in the second step
2
2
5
y
z
y
z
+ =
+ =
We multiply Equation 1 by -1 and add to Equation 2:
(2
) (y z)
5 2
3
y
z
y
+ − + = − =
Now we've found y. Let's put it in the first equation:
2
3
2
1
y
z
z
z
+ = + = = −
4.
X not even topamiz
We use Equation 1:
2
8
2
3 ( 1)
8
2 x
4
x
2
x
y
z
x
+ − =
+ − − =
= =
Final solution: x=2, y=3, z=-1
References:
1. Ahlberg J., Nilson E., Walsh J. Spline theory and "e applications. Moscow, Mir
Publ., 1972.
ОБРАЗОВАНИЕ НАУКА И ИННОВАЦИОННЫЕ ИДЕИ В МИРЕ
https://scientific-jl.org/obr
Выпуск журнала №-69
Часть–6_ Мая –2025
282
2181-3187
2. B a h v a l o v N. S. Chislennme metodm, vol. I, Moscow, "Nauka", 1973,
3. B a h v a l o v N. S. Ob optimalnykh otsenkakh velocity skosimosti quadrature
protsessov i metodov integratsii tipa Monte-Carlo na klassnh funktsii [On optimal
estimates of the rate of convergence of quadrature processes and methods of
integration of the Monte-Carlo type on class functions]. "Numerical method of
solving diff. and integral equations and quadrature formulas". Moscow, Nauka, 1964
(5-63-betlar).
4. Berezin I. S., Zhidkov N. P. Metodn vmchisleniy, vol. 1., izd. 3rd. Moscow,
Nauka, 1966.
5. B u s l e n k o N. P. et al. Metodn statisticheskikh ispntanii (metod Monte-Carlo).
Moscow, Fizmatgiz Publ., 1962.
6. V a r g a R. Funktsional'nysh analiz i teoriya approximatsii v chislennom analize
[Functional analysis and the theory of approximation in a chislennom analize].
Moscow, Mir Publ., 1974.
7. V o e v o d i n V. V. Chislenne metodn algebrm. Theory and algorithm. Moscow,
Nauka Publ., 1966.
8. G e l f o n d A. O. Calculus of finite differences, 3rd correction. Ed. Moscow,
Nauka Publ., 1967.
9. Goncharov V. L. Teoriya interpolirovaniya i approximation funktsii [Theory of
interpolation and approximation of functions]. Moscow, Gostekhizdat Publ., 1954.
10. Daugavet I. K. Vvedenie v teoriyu approximations funktsii [Introduction to the
theory of approximations of functions]. Leningrad State University, Leningrad, 1977.
