Review of Business and Economics Studies, 2018, том 6, № 1

Review of  
Business and 
Economics  
Studies

EDITOR-IN-CHIEF
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Faculty, Financial University, Moscow, 
Russia
ailyinsky@fa.ru 

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Adam Smith Business School, 
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ISSN 2308-944X

Вестник 
исследований 
бизнеса  
и экономики

ГЛАВНЫЙ РЕДАКТОР
А.И. Ильинский, профессор, декан 
Международного финансо вого факультета Финансового университета 

ВЫПУСКАЮЩИЙ РЕДАКТОР
Збигнев Межва, д-р экон. наук

РЕДАКЦИОННЫЙ СОВЕТ

М.М. Алексанян, профессор Бизнесшколы им. Адама Смита, Университет 
Глазго (Великобритания)

К. Вонг, профессор, директор Института азиатско-тихоокеанского бизнеса 
Университета штата Калифорния, 
Лос-Анджелес (США)

К.П. Глущенко, профессор экономического факультета Новосибирского 
госуниверситета

С. Джеимангал, профессор Департамента статистики и математических финансов Университета Торонто 
(Канада)

Д. Дикинсон, профессор Департамента экономики Бирмингемской бизнесшколы, Бирмингемский университет 
(Великобритания)

В.Л. Квинт, заведующий кафедрой 
финансовой стратегии Московской 
школы экономики МГУ, профессор 
Школы бизнеса Лассальского университета (США)

Г. Б. Клейнер, профессор, член-корреспондент РАН, заместитель директора Центрального экономико-математического института РАН

Э. Крочи, профессор, директор по 
научной работе Центра исследований 
в области энергетики и экономики 
окружающей среды Университета 
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Д. Мавракис, профессор, 
директор Центра политики 
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Национального университета  
Афин (Греция)

С. Макгвайр, профессор, директор Института предпринимательства 
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Депар та мента математических 
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Ворчестерского политехнического 
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Бизнес-кол леджа Университета 
Стони Брук (США) 

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Школа Бизнеса Даремского 
университета,
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Сассекский университет 
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университета

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16+

CONTENTS

Optimization of the Shape of the Pareto Set  

in the Problems of Multi-criterial Programming

Bula Katendi Axel, E. A. Umnov, A. E. Umnov  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Does Enterprise Value Really Depend on WACC and Free Cash Flow? 

The Evidence of Irrationality from the Oil and Gas Sector

Pavel E. Zhukov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

Weather Derivatives in Russia: Farmers’ Insurance  

against Temperature Fluctuations

Eric Carkin, Stanislav Chekirov, Anastasia Echimova, Caroline Johnston, Congshan Li, 

Vladislav Secrieru, Alyona Strelnikova, Marshall Trier, Vladislav Trubnikov . . . . . . . .29

Does Social Inequality Stimulate the Economic Growth? 

(On the examples of the chosen developing countries)

Alina Pukhaeva, Elena Miroshina (Silantieva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

Country Risk in International Investment 

Its’ Structure and Methods of Estimation

Alexey Ivkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56

Business Valuation of Nike

Elena Miroshina (Silantieva), Egor Romanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

“Uniform Subsidy” and New Trends in Financing  

of Agricultural Insurance in Russian Federation

L.Yu. Piterskaya, N. A. Tlisheva, A. V. Piterskaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83

Review of  
Business and  
Economics  
Studies

Volume 6, Number 1, 2018

Вестник 
исследований 
бизнеса  
и экономики

№ 1, 2018

CОДЕРЖАНИЕ

Оптимизация формы множества Парето  

в задачах многокритериального программирования

Була Аксел Катенди, Е. А. Умнов, А. Е. Умнов . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Разве стоимость компании действительно зависит от средневзвешенной 

стоимости капитала и свободного денежного потока?  

Свидетельства иррациональности в нефтегазовом секторе

Павел Е. Жуков . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Погодные деривативы в России:  

страхование фермеров от колебаний температуры

Eric Carkin, Станислав Чекиров, Анастасия Екимова, Caroline Johnston, Congshan Li, 

Vladislav Secrieru, Алена Стрельникова, Marshall Trier, Владислав Трубников . . . . . 29

Стимулирует ли социальное неравенство экономический рост? 

(на примерах выбранных развивающихся стран)

Алина Пухаева, Елена Мирошина (Силантьева) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Страновой риск в международных инвестициях:  

структура и методы расчета

Алексей Ивкин . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Оценка бизнеса корпорации Nike

Елена Мирошина (Силантьева), Егор Романов . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

«Единая субсидия» и новые веяния  

в субсидировании агрострахования в Российской Федерации

Людмила Питерская, Нафсэт Тлишева, Анастасия Питерская . . . . . . . . . . . . . . . . 83

Review of Business and Economics Studies  
doi: 10.26794/2308-944X-2018-6–1-5–16
2018, Vol. 6, No. 1, 5–16
Optimization of the Shape of the Pareto 
Set in the Problems of Multi-criterial 
Programming

Bula Katendi Axel1, E. A. Umnov2, A. E. Umnov2

Moscow institute of Physics and Technology (State University)

1axelbula@gmail.com, 2mail@umnov.ru

Abstract
in this paper, a scheme for using the method of smooth penalty functions for the dependence of solutions of 
multi-criterial optimization problems on parameters is being considered. in particular, algorithms based on the 
method of smooth penalty functions are given to solve problems of optimization by the parameters of the level 
of consistency of the objective functions and to find the corresponding shape of the Pareto’s set.
Keywords: multi-criterial parametrical programming tasks; set of Pareto; method of smooth penalty functions; 
optimization problem in terms of parameters.
JEL classification: C61, C65
I

n mathematical modeling, it is often necessary to formalize preferences for states of the modeled 
object that generates several independent target functions. According to the historically established tradition, in this case, it is customary to talk about of multi-criterial optimization problems.
A finite-dimensional multi-criterial model is a mathematical model with N objective functions:

 
 
 
   
       
(
)
[
]
,
max1,
k
x
f
x u
k
N
→
=
,  
(1)

subject to maximization possessing at interior points of the set of elements
n
x
E
∈
, and satisfying 
the following conditions:

 
 
 
 
 
(
)
[
]
,
01,
iy
x u
i
m
≤
=
  
(2)

where 
r
u
E
∈Θ ⊆
— vector of parameters of the model. It is assumed that the functions 
(
)
,
kf
x u  and 

(
)
,
iy
x u  are sufficiently smooth, i. e. they have continuous derivatives of a desired order in all their 
arguments.
The incorrectness in the general case of such a statement is obvious, since the element x  that is 
extremal for one of objective functions, in general, is not such for others.
However, useful information can be obtained by successively solving the following problems with 
a criterion for finding an extremum on the set (2) of each of the functions (1) separately for 
[
]
1,
k
N
=
:

(
)
,
max
k
x
f
x u →

 
 
 
 
 
subject to 
(
)
[
]
,
01,
iy
x u
i
m
≤
=
  
(3)

The objective function 
(
)
,
kf
x u  is called improvable in the feasible point 
0
x  (i. e., satisfying the 
condition (3)) if there is another feasible point 
1x , for which 
(
)
(
)
1
0
,
,
k
k
f
x u
f
x u
>
.
It is clear that the solution of problem (3) for any k = [1, N] is un-improvable, or “non-ideal” at the 
point of view of the other objective function 
(
)
,
kf
x u .

The concept of improving the multi-criterial objective function allows the feasible points to be 
divided into two subsets: for the first, all feasible points improve all objective functions and for the 
second, there are points for which the improvement of one function causes the deterioration of at 
least one other function.
The second subset is called a Pareto-type set or, simply, a Pareto set.
A general universal approach to the solution of multi-criterial optimization problems has not been 
proposed yet, but numerous approaches have been developed (see Fiacco & McCormick, 1968; Lotov 
& Pospelov, 2008), which limit the number of solutions.
For example, in the practical use of multi-criterial mathematical models, the set of independent 
objective functions is often replaced by a single one, thus passing to the standard problem of mathematical programming, allowing finding consistent or compromising solutions on the Pareto set in 
a certain sense.

Statement of the problem
In this article, the problem of finding an element on the set (2) that minimizes the gap between the 
objective functions will be considered as a compromise. In other words, this is a mathematical programming problem of the following form:
 
 
 
 
 
 
 
 
 
 
minimize ρ, subject to ρ≥ 0,

 
 
 
 
 
(
)
[
]
,
01,
,
iy
x u
i
m
≤
=
  
(4)

 
 
 
 
 
(
)
( )
[
]

*
,
1,k
k
f
x u
F
u
k
N
≥
−ρ
=

whose solution will be denoted by 
( )

** u
ρ
 and 
( )

**
x
u . Here 
( )
( )
(
)
*
*
,
k
k
k
F
u
f
x
u u
=
 and ρ  is called 
“mismatch value”.
The problem (4) is naturally called a two-level parametric problem since in its formulation it 
contains solutions 
( )
[
]

*
1,k
F
u
k
N
=
 of problems with single criterion (3), which we call first level 
problems. In this case, both in the problems of the first and the second level, it is assumed that the 
vector of the parameters u∈Θ  is fixed.
It is clear that the extreme value of the mismatch between the criteria in the general case is determined by the properties of the Pareto set and depends on the parameters vector u . Therefore, it 
is natural to indicate the third level optimization problem for the models (1)–(2) as follows:

 
 
 
 
optimize the expression 
( )

** u
ρ
 subject to u∈Θ   
(5)

which solution will be the vector of parameters 
***
u
∈Θ  and the number 
(
)

***
**
*** .
u
ρ
= ρ
 In the present 
paper, possible solutions to problem (5) will be considered.

Solution method
Let us consider the problem of finding in the parameter space a standard method (for example, 
gradient) of finding the extremum of the mismatch value of the objective functions of the multicriterial model (3)–(4)–(5).
The specificity of this problem is based on the fact that the formulation of the problem (5) (the 
upper level or third level) includes the dependence 
( )

** u
ρ
, the solution of the problem (4) (the second 
level) which in turn, depends on 
( )
( )
(
)
[
]

*
*
,
1,k
k
k
F
u
f
x
u u
k
N
=
∀
=
 —  the solutions of the problem (3) 
(the lower level or first level).
The functions 
( )

** u
ρ
 and 
( )
[
]

*
1,k
F
u
k
N
∀
=
 in the general case (even for smooth functions 
(
)
,
kf
x u  
and 
(
)
,
iy
x u ) may be not differentiable, that why the use of any numerical method based on Taylor 
approximations is not possible.

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming

It is proposed the use of the method of smooth penalty function to overcome this difficulty 
(see Umnov, 1975) and obtain a sufficiently smooth approximation dependences of 
( )

** u
ρ
 and 

( )
[
]

*
1,k
F
u
k
N
∀
=
.
It’s assumed that the penalty function 
(
)
,
P
s
τ
, which penalizes the restriction 0
s ≤
, satisfies the 
following conditions:
1 
0
∀τ >
 and 
s
∀ , the function 
(
)
,
P
s
τ
 has continuous derivatives with respect to all its arguments 
up to the second order
2 
0
∀τ >
 and 
s
∀ ,

 
 
 
 
 
 

2

2
0;0P
P

s
s

∂
∂
>
>
∂
∂
 
 (6)

3 
(
)
,
0P
s
s
τ
>
∀  and 
0
∀τ >
, and,

 
 
 
 
 
      

(
)
0
,0
lim
,
0,0
s
P
s
s
τ→+
+∞
>

τ
= 
<


  
(7)

When solving the third-level problem by an iterative method, for each step of the method, it is 
necessary preliminary to solve the problems of the second and first levels for a fixed vector of parameters u . Let us first consider a possible scheme for solving first-level problems. In fact, we will 
use an auxiliary function for the one-criterion problems (3), as follows:

 
 
 
(
)
(
)
(
)
(
)
[
]

1
, ,
,
,
,
1,m

k
k
i
i
A
x u
f
x u
P
y
x u
k
N

=
τ
=
−
τ
∀ ∈
∑
 
(8)

while a sufficiently smooth penalty function 
(
)
,
P
s
τ
 satisfies conditions (6) and (7).
As shown in Zhadan (2014), instead of the smooth approximations 
( )

*
k
x
u  solutions of each task 
of problem (3), we can take 
( )
k
x
u  stationary points of the auxiliary function (8), defined like:

 
 
 
 
 
  
[
]
( )
01,9
k

j

A
j
n
x
∂
=
∀ ∈
∂
 
(9)

or

[
]

1
01,m
k
i

j
i
j
i

f
y
P
j
n
x
y
x
=

∂
∂
∂
−
=
∀ ∈
∂
∂
∂
∑

Since the condition of the second-level problem (4) includes the dependencies 
( )
( )
(
)
[
]

*
*
,
1,k
k
k
F
u
f
x
u u
k
N
=
∀
=
 which are not differentiable functions for all their arguments, 
then for these dependencies it is also necessary to choose a smoothed approximation.
As an approximation, the auxiliary function calculated at a stationary point 
( )
( )
(
)
,
,
k
k
k
F
u
A
x
u u
=
τ
 
can be used, because (due to the properties of the penalty function method) its value for small positive τ  is close to the optimal value of the objective function of the k -th problem (3).
Standard optimization methods used for lower-level tasks, based on the use of continuous gradients or other differential characteristics, suggest that in addition to the solving system (9), these 
characteristics themselves can be found.
Let us demonstrate this using the example of calculating the derivatives of the function 
( )
k
F
u  
with respect to the components of the vector u of parameters.
As 
( )
( )
(
)
,
,
k
k
k
F
u
A
x
u u
=
τ
, then according to the rule for differentiating a composite functionof 
several variables, we have:

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming

[
]

1
1,n
j
k
k
k

p
p
j
p
j

x
F
A
A
p
r
u
u
x
u
=

∂
∂
∂
∂
=
+
∀ ∈
∂
∂
∂
∂
∑

Using (9), we have:

 
 
 
 
      
( )
(
)
[
]
,
,
1,k
k
k
p
p

F
A
x
u u
p
r
u
u
∂
∂
=
τ
∀ ∈
∂
∂
 
(10)

Note that the last simplification would be impossible if for 
( )

*
k
F
u  a more natural approximation 

( )
(
)

*
,
k
k
f
x
u u  is used instead of the smoothing approximation 
( )
(
)
,
,
k
k
A
x
u u
τ
.
Let us now look into the solution to the second-level problem. To make application of the penalty 
function method more convenient, the problem (4) is expressed as:
maximize −ρ , subject to −ρ  ≥ 0,

 
 
 
 
         
(
)
[
]
,
01,
,
iy
x u
i
m
≤
=
  
(11)

(
)
[
]
, ,
01,k
Y
x u
k
N
ρ
≤
=

where 
(
)
( )
(
)

*
, ,
,
k
k
k
Y
x u
F
u
f
x u
ρ
=
−ρ−

The solution to this problem will be denoted by 
( )

** u
ρ
 and 
( )

**
x
u .
Let us define the auxiliary function for the problem (10) as follows:

 
                    
(
)
(
)
(
)
(
)
(
)
(
)

1
1

, , ,
,
,
, ,
,
,

N
m

k
i
k
i

E
x u
P
P
Y
x u
P
y
x u

=
=
τ ρ
= −ρ−
τ −ρ −
τ
ρ
−
τ
∑
∑
 
(12)

replacing previously in 
(
)
, ,
k
Y
x u
ρ
 the dependency 
( )

*
k
F
u  by its smoothed approximation 
( )
k
F
u .
For the set of variables {
}
1
2
,,,,n
x x
x
ρ
…
, the conditions for the stationarity of the auxiliary function (12) will be:

 
 
 
      

[
]

1

1
1

1
0                                       

 

0       
1, 

=

=
=

∂
∂
∂
= − +
+
=
∂
∂
∂

∂
∂
∂
∂
∂

=
−
=
∀ =
 ∂
∂
∂
∂
∂


∑

∑
∑

N

k
k

N
m
k
i

k
i
j
k
j
i
j

E
P
P
Y

f
y
E
P
P
j
N
x
Y
x
y
x

ρ
ρ

 
(13)

Let the solutions of system (13) be ( )
u
ρ
 and ( )
x u , then, as a smoothed approximation of the  
dependency 
( )

** u
ρ
, we can use the function 
( )
( )
( )
(
)
,
,
,
E u
E
u
x u u
= −
ρ ρ
. The derivatives of this function by the components of the vector u  ofparameters and the rule for differentiating a composite 
function of several variables give us:

[
]

1
1,n
j

p
p
j
p
p
j

x
E
E
E
E
p
r
u
u
x
u
u
=

∂
∂
∂
∂
∂
∂ρ
=
+
+
∀ ∈
∂
∂
∂
∂
∂ρ ∂
∑
.

From (13) we know that 
0
E
∂
=
∂ρ

 and 
[
]
01,j

E
j
N
x
∂
=
∀ =
∂

.

Then the last expression can be written simply:

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming

( )
( )
(
)
[
]
,
,
,
1,p
p

E
E
u
x u u
p
r
u
u
∂
∂
=
ρ ρ
∀ ∈
∂
∂
 
(14)

Finally, we obtain formulas for the gradient components of 
( )
E u  in terms of the functions usedin 
the formulation of the multicriterial model (4)–(5) and the method of smooth penalty functions. 
From (12) it is obtained:

1
1

N
N
k
i

p
k
p
i
p
k
k

Y
y
E
P
P

u
Y
u
y
u
=
=

∂
∂
∂
∂
∂
= −
−
∂
∂
∂
∂
∂
∑
∑

where 
k
k
k

p
p
p

Y
F
f
u
u
u
∂
∂
∂
=
−
∂
∂
∂

 and the value of 
k

p

F
u
∂
∂

 can be found from (10).

Formulas (14) allow us to solve the third-level problem by applying any of the first-order methods, 
for example, conjugate directions. Note that second-order methods should also be considered here. 
However, this will be done at the end of the article, while now let us illustrate an example.

Proposed method in use
Let us consider multi-criterial mathematical model in which 
3
1
2
3
T
x
x x x
E
=
∈


 is a vector of independent variables and 
2
1 2
T
u
u u
E
=
∈


 is a vector of parameters.
The problem is to maximize for x  and u∈Θ  the functions:

(
)
(
)
(
)
1
1
2
2
2
3
,
,,
,,
f
x u
x
f
x u
x
f
x u
x
=
=
=

subject to 
(
)
(
)
(
)
(
)
1
2
3
1
1, 2
1
2
1, 2
2
3
1, 2
3
1, 2
0,0,0,x
x
x
a u u
x
a
u u
x
a
u u
x
b u u
≥
≥
≥
+
+
≤

and where the functions 
(
)
(
)
(
)
(
)
1
1, 2
2
1, 2
3
1, 2
1, 2
,,a u u
a
u u
a
u u
and b u u
 are given by the condition below.

A valid region of the model (with an allowable fixed u ) is a rectangular pyramid OABC. The Pareto 
set coincides with the face of ABC or is a part of it. 
100
T
A
u
= 
 and 
2
00
T
B
u
= 
 .
We assume that the set Θ in the parameter space is given by the condition that the sum of the 
lengths of the segments OA, OB and OC is constant and equal 3.

Figure 1. Geometric interpretation of models (6)–(7).

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming

Applying the standard methods of analytic geometry, we find that for the compatibility of the 
system of model constraints, the existence of 
0
r ≥
 is necessary such that:

(
)
1
1
2
2
,
a u u
u r
=

(
)
2
1
2
1
,
a
u u
u r
=

(
)
3
1
2
1 2
,
a
u u
u u
=

(
)
1
2
1 2
,
b u u
u u r
=

By choosing r  such that 
1
2
3
r
u
u
=
−
−
, we assure that the set Θ  will not be empty when 
1
0.1
2.5
u
≤
≤
 
and 
2
0.1
2.5
u
≤
≤
.
The minimum value of the discrepancy between the criteria in this example depends on the form 
of the Pareto set, which is the triangle ABC, or a part of it. A graphic representation of the dependence of the error value of the objective functions on the parameters 
1u  and 
2u is shown in pictures 
2 and 3. Let us see with more details the properties of this dependence.
Clearly, the solutions of the first-level problems (3) for fixed 
1u  and 
2
u  are:

( )
(
)
( )
(
)
( )
(
)

*
*
*
1
1
2
2
3
1
2
,,3
.
f
x
u
u
f
x
u
u
f
x
u
u
u
=
=
=
−
−

Consequently, the task of the second level (4) —  minimizing the discrepancy of the criteria, will 
have the form:
minimize ρ according to {
}
1
2
3
,
,,
x x
x ρ

subject to 
0
ρ ≥
,

1
2
3
0,0,0,x
x
x
≥
≥
≥

2
1
1
2
1 2
3
1 2 ,
u rx
u rx
u u x
u u r
+
+
≤

1
1
,
x
u
≥
−ρ

2
2
,
x
u
≥
−ρ

3x
r
≥
−ρ ,

1
2
3
r
u
u
=
−
−

His solution will be designed by 
(
)

**
1
2
,
.
u u
ρ

Finally, the task of the third level (5) for our example will be:
minimize 
(
)

**

1
2
,
u u
ρ
 by {
}
1
2
,
u u
 when 
1
0.1
2.5
u
≤
≤
 and 
2
0.1
2.5
u
≤
≤
.
It is known from the theory of mathematical programming that the properties of the dependence 
(
)

**

1
2
,
u u
ρ
 are primarily determined by how the set of constraints of a model of the «inequality» type 
is divided into active and inactive ones, that is, the first of which are satisfied as equalities, and the 
second —  as strict inequalities.
This separation depends on the values of the parameters of the model and its optimal variant 
determines the solution of the second-level problem.
First, suppose that the values of the model parameters initiate a conflict of all three criteria simultaneously. In other words, the improvement of the value of any one of the objective functions of 
the model is possible only if the values of all the others deteriorate.
In this case, the last five constraints of the second-level problem must be active, and we obtain 

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming

the following system of equations, which allows us to find the analytical form of the dependence 

(
)

**

1
2
,
u u
ρ
.

2
1
1
2
1 2
3
1 2

1
1

2
2

3

1
2

3
u rx
u rx
u u x
u u r

x
u

x
u

x
r

r
u
u

+
+
=


=
−ρ

=
−ρ


=
−ρ

 =
−
−


Solving this system above, we have the analytic form of ρwhich depends on 
1u  and 
2
u :

(
)

**
1
2

1
2
1
2

2
,
1
1
1
3

u u

u
u
u
u

ρ
=
+
+
−
−

The stationary points of 
(
)

**

1
2
,
u u
ρ
 are 11,33
,33
T
T
T
−
−





 and 33 T

  can be easily found, 
and for the first point the function has a local maximum with the value 2/3 according to the Sylvester 
criterion, not for the others because they don’t satisfy the condition of non-negativity of the variables 
1
2
3
,,x
x
x  and r .
The formula obtained is valid only in a certain area contained in Θ . An analysis of the isoline 
system shown in Picture 3, allows selecting five areas with different sets of active restrictions. Light 
lines determine the boundaries between the areas. The formula obtained above is valid only in area 
4. In this area, the Pareto set of the model under consideration consists of the interior points of the 
triangle ABC.
Outside area 4, the formula for 
(
)

**

1
2
,
u u
ρ
 is different. For the area 1, for example, 
(
)

**

1
2
,
u u
ρ
 is 
found from the system of equations:

2
1
1
2
1 2
3
1 2

1

2
2

3

1
2

03

u rx
u rx
u u x
u u r

x

x
u

x
r

r
u
u

+
+
=

=

=
−ρ




=
−ρ

=
−
−





Figure 2. The quantity OA
OB
OC
+
+
 is constant (graphic —  3D).

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming

since the set of active restrictions in it is different: it contains the condition 
1
0
x =
, instead of 

1
1
x
u
=
−ρ . We can easily prove that in the area 1

(
)

**
1
2

2
1
2

2
,
1
1

3

u u

u
u
u

ρ
=
+
−
−

There are not stationary points for this dependency.
For areas 2 and 3, the arguments and results are similar. The Pareto sets in areas 1, 2, and 3 are 
the sides of the triangle ABC: BC, AC, and AB, respectively. Finally, we note that in area 5 the system 
of conditions (2) is contradictory.
In this case, the exact solution to the problem of the upper (third) level has the form:

**
**
**
1
2
2
1,1,3
u
u
=
=
ρ
=

Let us now describe the method for solving the third-level problem for the variant of the multicriterial model. We have:
objective functions: maximize by 
3
1
2
3
T
x
x x x
E
=
∈



(
)
1
1
,
f
x u
x
=

(
)
2
2
,
f
x u
x
=

(
)
3
3
,
f
x u
x
=

subject to 
1
2
3
0,0,0,
x
x
x
≥
≥
≥
( )
( )
( )
( )
1
1
2
2
3
3
a u x
a
u x
a
u x
b u
+
+
≤

where

( )
(
)

( )
(
)
( )
( )
(
)

1
2
1
2

2
1
1
2

3
1 2

1 2
1
2

3
3
3

a u
u
u
u
a
u
u
u
u
a
u
u u
b u
u u
u
u


=
−
−

=
−
−

=


=
−
−


Figure 3. The quantity OA
OB
OC
+
+
 is constant (isoline view).

Optimization of the Shape of the Pareto Set in the Problems of Multi-criterial Programming