On the Approximation of Vector Optimization Problems
В работе исследуются условия сходимости приближенного метода решения задач многокритериальной оптимизации, когда целевые функции и допустимое множество заменяются их приближениями. Доказано, что достаточным условием сходимости являются равномерная сходимость приближенных функций к исходной функции и...
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2015
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| Cite this: | On the Approximation of Vector Optimization Problems / B.V. Norkin // Кибернетика и вычислительная техника. — 2015. — Вип. 179. — С. 35-42. — Бібліогр.: 13 назв. — англ. |
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| citation_txt | On the Approximation of Vector Optimization Problems / B.V. Norkin // Кибернетика и вычислительная техника. — 2015. — Вип. 179. — С. 35-42. — Бібліогр.: 13 назв. — англ. |
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| description | В работе исследуются условия сходимости приближенного метода решения задач многокритериальной оптимизации, когда целевые функции и допустимое множество заменяются их приближениями. Доказано, что достаточным условием сходимости являются равномерная сходимость приближенных функций к исходной функции и сходимость допустимого множества приближенных задач к допустимому множеству исходной задачи, по крайней мере, в окрестности решения.
У роботі досліджено умови збіжності наближеного методу розв'язання задач багатокритеріальної оптимізації у випадку коли цільові функції і допустима область замінюються їх наближеннями. Доведено, що достатньою умовою збіжності є рівномірна збіжність наближених функцій до початкової функції та збіжність допустимої множини наближени
We consider an approximation approach to solving vector optimization problems. The standard approach to such problems is to optimize one criterion under constraints on the others or to scalarize the problem, i.e. to combine all criteria into one scalar criterion. This paper describes a completely different approach, where the feasible set is approximated by a discrete grid (deterministic or random) and the vector function is approximately calculated on this grid. The obtained discrete problem is exactly solved by Pareto type optimization. The paper studies conditions for convergence of the approximation method when the objective functions and the feasible set are replaced by their more and more fine approximations.
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35
Интеллектуальное управление и
системы
УДК 519.6
ON THE APPROXIMATION OF VECTOR
OPTIMIZATION PROBLEMS
B.V. Norkin
V.M. Glushkov Institute of Cybernetics of NAS of Ukraine
В работе исследуются условия сходимости
приближенного метода решения задач многокритериальной оптимизации,
когда целевые функции и допустимое множество заменяются их
приближениями. Доказано, что достаточным условием сходимости
являются равномерная сходимость приближенных функций к исходной
функции и сходимость допустимого множества приближенных задач к
допустимому множеству исходной задачи, по крайней мере, в окрестности
решения.
Ключевые слова: векторная оптимизация, стохастическая
многокритериальная оптимизация, парето-оптимальность, дискретная
аппроксимация, эпсилон-доминирование.
У роботі досліджено умови збіжності наближеного методу
розв'язання задач багатокритеріальної оптимізації у випадку коли цільові
функції і допустима область замінюються їх наближеннями. Доведено, що
достатньою умовою збіжності є рівномірна збіжність наближених функцій
до початкової функції та збіжність допустимої множини наближених
задач до допустимої множини початкової задачі хоча б в околі розв´язку.
Ключові слова: векторна оптимізація, стохастична
багатокритеріальна оптимізація, парето-оптимальність, дискретна
апроксимація, епсілон-домінування.
INTRODUCTION
This paper studies conditions of convergence of the successive approximations
method for solving deterministic and stochastic vector optimization problems.
A general form of a vector optimization problem reads as follows:
,max)}(),...,({)(
R1 nXxm xFxFxF
⊂∈
→=
r
(1)
where functions ,,...,1),( mixFi = are continuous on a compact set nX R⊂ . The
problem is (a) to find individual elements or (b) the whole set of weakly Pareto
optimal points XX ⊂* such that for any *Xx ∈ there is no Xx ∈′ and
)()( xFxF
rr
>′ (component-wise). There are numerous approaches for solving
problem (1) [1–3]. The most known of them consists in maximization of some
component function iF under constraints on other functions or in aggregation of
criteria iF by some linear or non-linear utility function and solving the resulting
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
36
optimization problems by nonlinear programming methods [1]. However, in case
of non-convex functions iF it is not always possible to find all Pareto-optimal
solutions in such a way. So, other approaches, that are not reduced to optimization
of a scalar criterion, were developed, in particular, the parameter space
investigation method [2] and methods of evolutionary programming [3]. The latter
in effect are variants of controlled iterative random search method: at each
iteration an approximate discrete solution *
NX consisting of a finite number of
points is constructed and then, using information about the objective function
)( *
NXF
r
at points *
NX a new generation of points NY is generated in some way
(randomly) and a new discrete approximate solution *
1+NX is selected from the set
( )N
N
N YXX ,*= , and so on. Finding out conditions of evolutionary programming
algorithm convergence to the set of Pareto-optimal points is a serious scientific
problem and is the subject of active research [4–6]. Even convergence of the
simplest algorithms of this type is studied only in case of discrete feasible set X
[6]. Let us, for example, sample points NY in the described approach uniformly in
X and let a new approximation *
1+NX be a Pareto-optimal subset of the discrete
pair ( )N
N
N YXX ,*= .
Does the sequence { }*
NX of approximations converge to the Pareto-optimal
set *X of problem (1)? This article, in particular, aims at finding answers to such
type of questions.
PROBLEM SETTING
In practice, a formulation of the vector optimization problem may be more
complex than (1). For example, in the case of the stochastic vector optimization,
functions F
r
are actually expectations, ),(E)( ωxfxF
rr
= , where the random
variable ω is defined on some probability space ( )P,,ΣΩ , symbol E denotes
expectation (integral) over measure P [7, 8]. Usually, in practical problems
expectations cannot be calculated analytically, so they are estimated numerically
using quadratures or Monte Carlo method. In the latter case empirical
approximations of functions F
r
have the form ( )∑ == N
k k
NN xfNxF 1 ),(1),( ωω
rr
,
where ),...,( 1 N
N ωωω = is the set of independent and identically distributed
observations kω of the random variable ω . In [9] one can find conditions of
uniform convergence of the empirical approximations ),( NN xF ω
r
to the
expectation ),(E)( ωxfxF
rr
= on a compact set X . Then, instead of (1), one has to
consider a sequence of approximate problems:
,max)}(),...,({)( R1 nNXx
N
m
NN xFxFxF
⊂∈
→=
r
,...2,1=N , (2)
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
37
where the set of feasible solutions NX , in general, can also vary from task to task,
for example, the set NX can be a discrete approximation of the initial feasible set
X [2]. In the latter case, the approximating functions )(),...,(1 xFxF N
m
N can be
defined only on this discrete set NX . Denote *
NX the set of Pareto-optimal
solutions of (2). The problem is to establish conditions under which the sets *
NX
approximate the set of solutions *X .
In (1) iα -quantiles of random variables ),( ωxFi can serve as components of
the vector objective function { }),(E),...,,(E)( 1 ωω xFxFxF m=
r
. It is known [10] that
these quantiles can be found by solving the following auxiliary optimization
problem:
( ) ( ){ } 1R
min),(,),()1(maxE),(
∈
→−−−=Φ
yiiiiiiii xFyyxFyx ωαωα . (3)
An approximate solution )(xy N
i of (3) for each fixed x can be found, for
example, by N iterations of the stochastic quasi-gradient method [11], or as a
corresponding term of the variational series of the random variable ),( ωxFi . Thus,
one again encounters with the approximate problem (2) with functions
)()( xyxF N
i
N
i = . Another approach to solving (3) is to use an empirical
approximation of components ),( ii yxΦ :
=Φ ),( i
N
i yx
( ) ( ) ( ){ } 1R1 min),(,),()1(max1
∈= →−−−= ∑
iy
N
k kiiiikii xFyyxFN ωαωα
(4)
and finding its approximate solution )(xy N
i for each x by the linear programming
method.
MAIN RESULTS
As noted in [2], the question of convergence of the discrete approximation of
problem (1) is not easy. In our case the problem is further complicated by the fact
that not only feasible set, but also the objective functions are approximated. To
study convergence of approximate solutions of problems (2) to the solution of the
original task (1), we’ll need some more definitions.
Definition 1 (ε
r
-nondominated solutions). Point Xx∈ is called
ε
r
-nondominated solution of (1), if there is no other point Xz ∈ such that
ε
rrr
+> )()( xFzF , where .RmN ∈ε
r
Definition 2 [12, Section 4A]. For a sequence of sets ,...}2,1,R{ =⊂ NZ n
N
let us define the following cluster sets:
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
38
}lim,:{suplim kkk NkNNNN zzZzzZ ∞→∞→ =∈∃= ,
}lim,:{inflim NNNNNN zzZzzZ ∞→∞→ =∈∃= ,
NNNNiN ZZZ ∞→∞→∞→ == supliminflimlim .
Denote )(* ε
r
X the set of all ε
r
- nondominated solutions of (1). In the same
manner define )(* N
NX ε
r
Nε
r -nondominated solutions of (2). Our goal is to
establish conditions under which sets )(* N
NX ε
r
approximate )(* ε
r
X .
Let us note some properties of the multivalued mapping )(* εε
rr
X→ .
Lemma 1. The mapping )(* εε
rr
X→ is monotone, i.e. for 21 εε
rr
≤ one has
)()( 2*1* εε
rr
XX ⊆ .
Proof. Assume the contrary that for some )( 1* ε
r
Xx ∈′ , )( 2* ε
r
Xx ∉′ there is
Xz ∈′ such that 12 )()()( εε
rrrrr
+′≥+′>′ xFxFzF . This inequality contradicts the
assumption )( 1* ε
r
Xx ∈′ .
The lemma is proved.
Lemma 2. For upper semi-continuous (component-wise) on the closed set
nX R⊂ vector-function )(xF
r
the mapping )(* εε
rr
X→ is upper semicontinuous,
i.e. )()(limsup ** εε
rr
XX N
N ⊆∞→ for any sequence εε
rr
→N .
Proof. Let εε
rr
→N , ∞→N and xxX kk NN ′→∋)(* ε
r , ∞→k . We must
show that )(* ε
r
Xx ∈′ . Assume the contrary, that )(* ε
r
Xx ∈′ . Then there is Xz ∈′
such that ε
rrr
+′>′ )()( xFzF . From upper semicontinuity of F
r
it follows
( )kk NN
k xFxFzF εε
rrrrr
+≥+′>′ →∞ )(limsup)()( and thus for sufficiently large k
relation kk NNxFzF ε
rrr
+>′ )()( is fulfilled. This contradicts the initial assumption
)(* kk NN Xx ε
r
∈ .
The lemma is proved.
Let us make the following assumptions on relations between problems (1)
and (2)
A1. For any sequence { }xxX NN →∋ it holds true )()( xFxF NN rr
→ ,
∞→N .
A2. The sequence of feasible sets { },...2,1, =NX N of (2) satisfies conditions:
XX N ⊆ and for some mR∈ε
r it holds N
N XX ∞→⊆ inflim)(* ε
r
, i.e. for each
point )(* ε
r
Xx ∈ there is a sequence of feasible points NN Xx ∈ convergent to this
point )(* ε
r
Xx ∈ .
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
39
Condition A1 is satisfied in particular if functions NF
r
are defined on the set
NXX ⊇ and uniformly converge to F
r
on X . Other possibilities are considered
in [13]. Assumption A2 is automatically satisfied if XX N
N =∞→lim , for
example, if NX discretely approximates, with increasing accuracy, the feasible set
X .
Theorem 1 (on convergence of solutions of approximate tasks (2) to the
solutions of the original problem (1)). Suppose that the vector function )(xF
r
is
continuous on a compact set X , conditions A1–A2 are fulfilled and
0lim >=∞→ εε
rrN
N . Then
1) )()(suplim ** εε
rr
XX N
NN ⊆∞→ ,
2) )(inflim)( ** N
NN XX εε
rr
∞→⊆′ for all εε
rr
<′ .
Proof. Let us prove the first assertion of the theorem. Assume the contrary,
that there are )()( ** εε
rr
XxxX kk
k
NN
N ∉′→∋ . Since the vector function )(xF
r
is
bounded from above on a compact set X , then every point x′ outside of )(* ε
r
X
is ε
r
-dominated by points from )(* ε
r
X . Indeed, if it is not true, then there is an
infinite sequence of points Xz s ∈ such that )()( 1zFxF
rrr
<+′ ε ,
)()( 21 zFzF
rrr
<+ ε , …, i.e. )()( szFsxF
rrr
<+′←∞ ε , that contradicts boundedness
of the vector function F
r
on X . So, there is a point )(* ε
r
Xz ∈′ such that
ε
rrr
+′>′ )()( xFzF . By virtue of the condition A2 there is a sequence NN Xz ∈
such that zzN ′→ , ∞→N .
Thus, it holds true
( )kkkkk NNN
k
NN
k xFxFzFzF εε
rrrrrr
+=+′>=′ ∞→∞→ )(lim)()(lim)( .
Then kkkkk NNNNN xFzF ε
rrr
+> )()( for all sufficiently large k , that
contradicts the assumption )(* k
k
k N
N
N Xx ε
r
∈ . The first assertion is proved.
Now let us prove the second statement of the theorem. Assume the contrary,
that there exists )()( ** εε
rr
XXx ⊆′∈′ (see Lemma 1) such that
)(inflim * N
NN Xx ε
r
∞→∉′ . By condition A2 there exists a sequence
xxX NN ′→∋ . Then there exist its subsequence { },...2,1, =kx kN such that
)(* k
k
k N
N
N Xx ε
r
∉ for all sufficiently large k , so there are points k
k N
N Xz ∈ such
that kkkkk NNNNN xFzF ε
rrr
+> )()( . By compactness of k
k
N
N zXX ∋⊇ , without
loss of generality, we can consider that zz kN ′→ and xx kN ′→ , thus, by
assumption A1,
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
40
( ) εεε
rrrrrrrr
′+′>+′=+≥=′ ∞→∞→ )()()(lim)(lim)( xFxFxFzFzF kkkkk NNN
k
NN
k .
Thus, the point x′ is ε
r′ -dominated, that contradicts the assumption
)(* ε
r
′∈′ Xx .
The second statement is proved.
Remark 1. In [13], an analog of Theorem 1 was proved under a stronger
assumption than A2: XX N
N =∞→lim . If the set NX is a discrete approximation
of the feasible set X , than condition A2 shows that for the validity of the theorem
on convergence of solutions )(* N
NX ε
r то )(* ε
r
X it is enough to improve
approximations of the feasible set only in the vicinity of approximate Pareto-
optimal points )(* ε
r
X .
Remark 2. Theorem 1, in particular, means that
)()(limsup)(liminf)0( ***** εεε
rrrr
XXXXX N
NN
N
NN ⊆⊆⊆= ∞→∞→ , (5)
where )0(** r
XX = is the set of weakly Pareto-optimal solutions of problem (1).
And since the mapping )(* εε
rr
X→ is upper semicontinuous, also at 0=ε
r
, then
relation (5) means that ε
r
-approximate solutions of problem (2) for sufficiently
small ε
r
approximate the set of weakly Pareto optimal solutions of problem (1).
CONCLUSIONS
The paper studies a general approximation scheme for solving vector
optimization problems. The objective vector function and the feasible set of the
problem are substituted by their approximations. Accurate calculating of the
objective functions or constraints of the problem is often impossible for finite (or
reasonable) time and, therefore, the problem needs to be approximated. This
situation is typical for stochastic multiobjective optimization. Approximate
problems themselves are solved approximately with some accuracy, i.e. their
approximately nondominated solutions are found. It is shown that under natural
conditions, uniform convergence of approximation functions and set convergence
of feasible domains, the found solutions approximate from above and from below
approximately nondominated solutions of the original problem.
1. Miettinen K. Nonlinear multiobjective optimization. — Boston/London/Dordrecht: Kluwer
Academic Publishers, 1999. — 298 p.
2. Sobol I.M., Statnikov R.B. Vybor optimalnyh parametrov v zadachah so mnogimi
kriteriyami (Selection of optimal parameters in problems with multiple criteria). 2-nd ed,
revised and supplemented. — Moscow: Drofa, 2006. — 176 p. (In Russian).
3. Deb K. Multi-objective optimization using evolutionary algorithms. — Chichester: John
Willey & Sons, 2001. — 497 p.
4. Hanne T. On the convergence of multiobjective evolutionary algorithms // European J. of
Operational Research. — 1999. — 117. — P. 553–564.
5. Li Z., Li Zhe, Rudolph G. On the convergence properties of quantum-inspired multi-
objective evolutionary algorithms. In: Advanced intelligent computing theories and
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
41
applications. With aspects of contemporary intelligent computing techniques. — Berlin,
Heidelberg: Springer, 2007. — P. 245–255.
6. Laumanns M., Zenklusen R. Stochastic convergence of random search methods to fixed size
Pareto front approximations // European J. of Operational Research. — 2011. — 213. —
P. 414–421.
7. Ben Abdelaziz F. Solution approaches for the multiobjective stochastic programming //
European J. of Operational Research. — 2012. — 216. — P. 1–16.
8. Gutjahr W., Pichler A. Stochastic multi-objective optimization: a survey on non-scalarizing
methods // Annals of Operations Research. — 2013. — P. 1–25.
9. Shapiro A., Dentcheva D., Ruszczyсński A. Lectures on stochastic programming: Modeling
and theory. Second Edition. — Philadelphia: SIAM, 2014. — 494 p.
10. Koenker R. Quantile Regression. — Cambridge, New York: Cambridge University Press,
2005.
11. Ermoliev Y.M. Metody stochasticheskogo programmirovaniya (Methods of stochastic
programming). — Moscow: Nauka, 1976. — 240 p. (in Russian).
12. Rockafellar R.T., Wets R.J-B. Variational Analysis. — Berlin: Springer, 1998 (3rd Printing
in 2009). — 734 p.
13. Norkin B.V. Sample approximations of multiobjective stochastic optimization problems //
www://optimization-online.org. Electronic preprint. — November 2014. – Access:
http://www.optimization-online.org/DB_HTML/2014/11/4655.html
UDC 519.6
ON THE APPROXIMATION OF VECTOR
OPTIMIZATION PROBLEMS
B.V. Norkin
Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine
Vector optimization has a great variety of applications. Such problems
naturally appear in stochastic optimization, where the optimization problem
contains random parameters. In the latter case the vector objective function may
include mean value, median, variance, quantiles and other characteristics of the
random objective function. The difficulty is that these quantities usually cannot be
calculated exactly and are non-convex as functions of variable parameters. These
circumstances set additional difficulties for solving corresponding vector
optimization problems.
We consider an approximation approach to solving vector optimization
problems. The standard approach to such problems is to optimize one criterion
under constraints on the others or to scalarize the problem, i.e. to combine all
criteria into one scalar criterion. This paper describes a completely different
approach, where the feasible set is approximated by a discrete grid (deterministic
or random) and the vector function is approximately calculated on this grid. The
obtained discrete problem is exactly solved by Pareto type optimization.
The paper studies conditions for convergence of the approximation method
when the objective functions and the feasible set are replaced by their more and
more fine approximations.
Sufficient conditions are established for Pareto-optimal solutions of the
approximate problems to converge in set convergence sense to the Pareto optimal
solution of the original problem (with some accuracy). Namely, it is required for
the approximate functions to converge uniformly to the original function and for
the feasible set approximations (possibly discrete) to converge to elements of the
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
42
original feasible set, at least, in the vicinity of the solution. The result confirms a
natural hypothesis that the approximation accuracy should increase when
approaching to the solution.
Keywords: vector optimization, stochastic multicriteria optimization, Pareto
optimality, discrete approximation, epsilon-dominance.
1. Miettinen K. Nonlinear multiobjective optimization. Boston/London/Dordrecht: Kluwer
Academic Publishers, 1999. 298 p.
2. Sobol I.M., Statnikov R.B. Vybor optimalnyh parametrov v zadachah so mnogimi
kriteriyami (Selection of optimal parameters in problems with multiple criteria). 2-nd ed,
revised and supplemented. Moscow: Drofa, 2006. 176 p. (In Russian).
3. Deb K. Multi-objective optimization using evolutionary algorithms. Chichester: John Willey
& Sons, 2001. 497 p.
4. Hanne T. On the convergence of multiobjective evolutionary algorithms. European J. of
Operational Research. 1999. 117. P. 553–564.
5. Li Z., Li Zhe, Rudolph G. On the convergence properties of quantum-inspired multi-
objective evolutionary algorithms. In: Advanced intelligent computing theories and
applications. With aspects of contemporary intelligent computing techniques. Berlin,
Heidelberg: Springer, 2007. P. 245–255.
6. Laumanns M., Zenklusen R. Stochastic convergence of random search methods to fixed size
Pareto front approximations. European J. of Operational Research. 2011. 213. P. 414–421.
7. Ben Abdelaziz F. Solution approaches for the multiobjective stochastic programming.
European J. of Operational Research. 2012. 216. P. 1–16.
8. Gutjahr W., Pichler A. Stochastic multi-objective optimization: a survey on non-scalarizing
methods. Annals of Operations Research. 2013. P. 1–25.
9. Shapiro A., Dentcheva D., Ruszczyсński A. Lectures on stochastic programming: Modeling
and theory. Second Edition. Philadelphia: SIAM, 2014. 494 p.
10. Koenker R. Quantile Regression. Cambridge, New York: Cambridge University Press, 2005.
11. Ermoliev Y.M. Metody stochasticheskogo programmirovaniya (Methods of stochastic
programming). Moscow: Nauka, 1976. 240 p. (in Russian).
12. Rockafellar R.T., Wets R.J-B. Variational Analysis. Berlin: Springer, 1998 (3rd Printing in
2009). 734 p.
13. Norkin B.V. Sample approximations of multiobjective stochastic optimization problems.
www://optimization-online.org. Electronic preprint. November 2014. Access:
http://www.optimization-online.org/DB_HTML/2014/11/4655.html
Получено 12.12.2014
B.V. Norkin, 2015
ISSN 0452-9910. Кибернетика и вычисл. техника. 2015. Вып. 179
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| id | nasplib_isofts_kiev_ua-123456789-86145 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0452-9910 |
| language | English |
| last_indexed | 2025-11-28T03:20:13Z |
| publishDate | 2015 |
| publisher | Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України |
| record_format | dspace |
| spelling | Norkin, B.V. 2015-09-08T12:43:06Z 2015-09-08T12:43:06Z 2015 On the Approximation of Vector Optimization Problems / B.V. Norkin // Кибернетика и вычислительная техника. — 2015. — Вип. 179. — С. 35-42. — Бібліогр.: 13 назв. — англ. 0452-9910 https://nasplib.isofts.kiev.ua/handle/123456789/86145 519.6 В работе исследуются условия сходимости приближенного метода решения задач многокритериальной оптимизации, когда целевые функции и допустимое множество заменяются их приближениями. Доказано, что достаточным условием сходимости являются равномерная сходимость приближенных функций к исходной функции и сходимость допустимого множества приближенных задач к допустимому множеству исходной задачи, по крайней мере, в окрестности решения. У роботі досліджено умови збіжності наближеного методу розв'язання задач багатокритеріальної оптимізації у випадку коли цільові функції і допустима область замінюються їх наближеннями. Доведено, що достатньою умовою збіжності є рівномірна збіжність наближених функцій до початкової функції та збіжність допустимої множини наближени We consider an approximation approach to solving vector optimization problems. The standard approach to such problems is to optimize one criterion under constraints on the others or to scalarize the problem, i.e. to combine all criteria into one scalar criterion. This paper describes a completely different approach, where the feasible set is approximated by a discrete grid (deterministic or random) and the vector function is approximately calculated on this grid. The obtained discrete problem is exactly solved by Pareto type optimization. The paper studies conditions for convergence of the approximation method when the objective functions and the feasible set are replaced by their more and more fine approximations. en Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України Кибернетика и вычислительная техника Интеллектуальное управление и системы On the Approximation of Vector Optimization Problems Про апроксимацію задач векторної оптимізації Об аппроксимации задач векторной оптимизации Article published earlier |
| spellingShingle | On the Approximation of Vector Optimization Problems Norkin, B.V. Интеллектуальное управление и системы |
| title | On the Approximation of Vector Optimization Problems |
| title_alt | Про апроксимацію задач векторної оптимізації Об аппроксимации задач векторной оптимизации |
| title_full | On the Approximation of Vector Optimization Problems |
| title_fullStr | On the Approximation of Vector Optimization Problems |
| title_full_unstemmed | On the Approximation of Vector Optimization Problems |
| title_short | On the Approximation of Vector Optimization Problems |
| title_sort | on the approximation of vector optimization problems |
| topic | Интеллектуальное управление и системы |
| topic_facet | Интеллектуальное управление и системы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/86145 |
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