Statistical approximation of multicriteria problems of stochastic programming
The article validates an approximation technique for solving multiobjective stochastic optimization problems. As a generalized model of a stochastic system to be optimized, a vector “input–random output” system is considered. Random outputs are converted into a vector of
 deterministic perfo...
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| Опубліковано в: : | Доповіді НАН України |
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| Дата: | 2015 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Видавничий дім "Академперіодика" НАН України
2015
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Statistical approximation of multicriteria problems of stochastic programming / B.V. Norkin // Доповiдi Нацiональної академiї наук України. — 2015. — № 4. — С. 35-41. — Бібліогр.: 12 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860193216681213952 |
|---|---|
| author | Norkin, B.V. |
| author_facet | Norkin, B.V. |
| citation_txt | Statistical approximation of multicriteria problems of stochastic programming / B.V. Norkin // Доповiдi Нацiональної академiї наук України. — 2015. — № 4. — С. 35-41. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Доповіді НАН України |
| description | The article validates an approximation technique for solving multiobjective stochastic optimization problems. As a generalized model of a stochastic system to be optimized, a vector “input–random output” system is considered. Random outputs are converted into a vector of
deterministic performance/risk indicators. The problem is to find those inputs that correspond
to Pareto-optimal values of output indicators. The problem is approximated by a sequence
of deterministic multicriteria optimization problems, where, for example, the objective vector
function is a sample average approximation of the original one, and the feasible set is a discrete sample approximation of the feasible inputs. Approximate optimal solutions are defined as weakly Pareto efficient ones within some vector tolerance. Convergence analysis includes
establishing the convergence of the general approximation scheme and establishing the conditions of convergence with probability one under proper regulation of sampling parameters.
Обгрунтовано апроксимацiйний пiдхiд до розв’язання задач багатокритерiальної стохастичної оптимiзацiї. В якостi узагальненої моделi стохастичної системи, що оптимiзується, використовується векторна модель типу “вхiд–випадковий вихiд”. Випадковi виходи перетворюються в вектор детермiнованих показникiв ефективностi i ризику. Проблема полягає в тому, щоб знайти тi входи, якi вiдповiдають Парето-оптимальним значенням вихiдних показникiв. Ця задача наближається послiдовнiстю задач детермiнованої багатокритерiальної оптимiзацiї, де, наприклад, цiльова вектор-функцiя є вибiрковим середнiм наближенням вихiдної функцiї, а допустима множина є дискретним наближенням можливих входiв. Наближенi оптимальнi розв’язки визначаються як слабо ефективнi (з деякою точнiстю) за Парето. Аналiз збiжностi включає в себе обгрунтування збiжностi загальної
апроксимацiйної схеми i встановлення умов збiжностi з ймовiрнiстю одиниця при адекватному регулюваннi параметрiв вибiрки.
Обосновывается аппроксимационный подход к решению задач многокритериальной стохастической оптимизации. В качестве обобщенной модели оптимизируемой стохастической системы используется векторная модель типа “вход–случайный выход”. Случайные выходы
преобразуются в вектор детерминированных показателей эффективности и риска. Проблема состоит в том, чтобы найти те входы, которые соответствуют Парето-оптимальным значениям выходных показателей. Эта задача приближается последовательностью
задач детерминированной многокритериальной оптимизации, где, например, целевая вектор-функция является выборочным средним приближением исходной функции, а допустимое множество является дискретным приближением возможных входов. Приближенные
оптимальные решения определяются как слабо эффективеные (с некоторой точностью) по
Парето. Анализ сходимости включает в себя обоснование сходимости общей аппроксимационной схемы и установление условий сходимости с вероятностью единица при адекватном регулировании параметров выборки.
|
| first_indexed | 2025-12-07T18:06:49Z |
| format | Article |
| fulltext |
UDC 519.6
B.V. Norkin
Statistical approximation of multicriteria problems
of stochastic programming
(Presented by Academician of the NAS of Ukraine I.V. Sergienko)
The article validates an approximation technique for solving multiobjective stochastic opti-
mization problems. As a generalized model of a stochastic system to be optimized, a vector
“input–random output” system is considered. Random outputs are converted into a vector of
deterministic performance/risk indicators. The problem is to find those inputs that correspond
to Pareto-optimal values of output indicators. The problem is approximated by a sequence
of deterministic multicriteria optimization problems, where, for example, the objective vector
function is a sample average approximation of the original one, and the feasible set is a di-
screte sample approximation of the feasible inputs. Approximate optimal solutions are defined
as weakly Pareto efficient ones within some vector tolerance. Convergence analysis includes
establishing the convergence of the general approximation scheme and establishing the condi-
tions of convergence with probability one under proper regulation of sampling parameters.
Contemporary approach to the optimal decision making is based on the modeling and the optimi-
zation of systems. Any complex system can be described by an “input–output” model y = g(x, ω),
where x denotes the input parameter vector from some feasible set X, ω is a vector of uncertain
parameters from a set Ω, y is an output vector from a set Y, and g is some mapping of X × Ω
into Y. The model g may be given by mathematical relations, a simulation computer program
or as an output result of an optimization or other solver. The vector ω of uncertain parameters
can be either deterministic or random, with distribution P . In the first case, the optimization
problem reads: min
ω∈Ω
f(x, g(x, ω)) → max
x∈X
, which corresponds to the so-called minimax decision-
making approach. The second case is related to the stochastic programming [1, 2], where the
“input-output” pairs are evaluated by means of some utility functional f(x, y, ω), and a correspon-
ding optimization problem is formulated as F (x) = Ef(x, g(x, ω)) → max
x∈X
, where E denotes the
expectation over the distribution P . Note that the above stochastic programming problem already
contains a vector criterion ~f(x) = {f(x, g(x, ω)), ω ∈ Ω}, with a large number of components,
which are combined in many cases into one or more scalar indicators. The most commonly used
indicator is the average value F (x) = Ef(x, g(x, ω)), along with the variance functions, probabi-
lity, quantile (VaR), and other risk indicators [3, 4]. Optimization of such indicators requires
substantial computational resources, and, in particular, the usage of parallel computing.
However, the efficient and unambiguous choice of the utility function f is not always possible.
In this case, we have to deal directly with the vector model y = g(x, ω), which maps an input x
into a random vector output g(x, ω). To make rational decisions, we have to define a preference
relation on the set of random vectors g(x, ω). This can be done in various ways [3, 4], e. g.,
we can calculate an average output Eg(x, ω) and then to use a natural deterministic preference
relation on the space Y.
Unlike standard one-criterion stochastic programming problems [1, 2], the problem of output
vector optimization can contain nonconvex, nonsmooth, and discontinuous functions. So, the
© B. V. Norkin, 2015
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №4 35
traditional stochastic programming methods like the gradient-type procedures [1] might not
be applicable. In this case, the random search methods, for example, evolutionary or hybrid
algorithms should be applied [5]. In case of a small dimension n of the set X ⊂ Rn, a simple
multiobjective random search (MRS) method can appear to be competitive. First, this method
randomly generates a cloud of points in the feasible region, and then nondominated points are
selected. Next, chiefly in a vicinity of the chosen nondominated points, new random points
are generated, and again nondominated points are selected, and so on. Efficiency of the search
is boosted due to the fact that the new points are generated in perspective areas. Remark
that the MRS method naturally admits the parallel and interactive implementation, which is
essential for stochastic multiobjective problems. In case of stochastic multicriteria optimization,
the additional difficulty consists in proper numerical evaluation of the vector objective function
that contains multidimensional integrals (expectations). In the present paper, we assume that the
objective functions are estimated by statistical sampling and focus on the method convergence
analysis. The considered MRS method belongs to nonscalarizing approaches [4] and, in spite of
the used sample average approximations, is substantially different from the scalarization method
of [6]. The results of numerical experiments and applications of the MRS method to insurance
optimization problems are reported in [7, 8].
2. ~ǫ-dominance and ~ǫ-efficiency. The following concept appears to be useful for the control
over the accuracy, strength, and directing preferences in Rm.
Definition 1. (~ǫ-dominance and ~ǫ-efficiency/optimality). The vector ~z1 ∈ Rm ~ǫ dominates
a vector ~z2 ∈ Rm, if ~z1 > ~z2 + ~ǫ (componentwise), where ~ǫ ∈ Rm. The subset Z∗(~ǫ) of the set
Z ⊂ Rm is called ~ǫ-efficient/optimal if, for any ~z ∈ Z∗(~ǫ), there is no ~z ∗ ∈ Z, ~z ∗ 6= ~z, such
that ~z ∗ > ~z + ~ǫ.
The concept of ~ǫ-efficiency was introduced in [9]. In case of ~ǫ > 0, it generalizes the standard
notion of the ǫ-optimality of a scalar optimization. It also includes the notion of weak Pareto
optimality, which corresponds to ~ǫ = 0. Further various generalizations of the ~ǫ-efficiency are
discussed in [10]. By adding ~ǫ to a vector ~z, the importance of components of ~z can be controlled
(e. g., the importance of criteria in vector optimization). Namely, an increase of the ǫi component
decreases the importance of the zi component. Moreover, in contrast to [9, 10], we allow ~ǫ /∈ Rm
+ .
If ~ǫ contains negative components, then the ~ǫ-dominance of ~z1 over ~z2 admits that some compo-
nents of ~z1 can be somewhat smaller than the corresponding components of ~z2. Remark that if
the point ~z ∗ ∈ Z is ~ǫk-efficient for some sequence {Rm ∋ ~ǫk → 0, k = 1, 2, . . .}, then ~z ∗ is called
a generalized efficient point (see [11, Definition 5.53]).
Let us recall some notations and definitions [12, Section 4 A], which concern the convergence
of a sequence of sets {Zi ⊂ Rm, i = 1, 2, . . .}:
lim sup
i
Zi = {z : ∃ zik ∈ Zik , z = lim
k
zik},
lim inf
i
Zi = {z : ∃ zi ∈ Zi, z = lim
i
zi},
lim
i
Zi = lim inf
i
Zi = lim sup
i
Zi.
Lemma 1. (Properties of the ~ǫ-optimal mappings). Let the sequence of sets {Zi ∈ Rm}
converge to a compact set {Z ⊂ Rm}, lim
i
Zi = Z. Denote, by Z∗
i (~ǫ) and Z∗(~ǫ), the subsets of
~ǫ-nondominated points in Zi and Z, respectively. Let lim
i
~ǫi = ~ǫ. Then, for any ~ǫ′ 6 ~ǫ, ~ǫ′ 6= ~ǫ,
the relations
Z∗(~ǫ′) ⊆ lim inf
i
Z∗
i (~ǫi) ⊆ lim sup
i
Z∗
i (~ǫi) ⊆ Z∗(~ǫ)
36 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2015, №4
hold true, where the last inclusion, in particular, indicates that the mapping ~ǫ → Z∗(~ǫ) is upper
semicontinuous. Moreover, in case of a convex set Z, we have lim inf
i
Z∗
i (~ǫi) = lim sup
i
Z∗
i (~ǫ) =
= Z∗(~ǫ).
3. Approximation of multicriteria optimization problems. Consider a general multi-
criteria optimization problem,
~F (x) = {f1(x), . . . , fm(x)} → max
x∈X⊂Rn
, (1)
where the functions fi(x), i = 1, . . . ,m, are assumed to be (semi)continuous on a compact set
X ⊂ Rn, and the preference relation in the criteria space Rm is set out by the nonnegative cone
Rm
++ = {x ∈ Rm : xi > 0, i = 1, . . . ,m}. The problem is to find a weak Pareto-optimal set X∗
and a subset X∗(~ǫ) of ~ǫ-efficient points of the set ~ǫ ∈ Rm.
It is easy to see that the mapping ~ǫ → X∗(~ǫ) is upper semicontinuous for the upper semi-
continuous vector function ~F (·), i. e., lim sup
i
X∗(~ǫi) ⊆ X∗(~ǫ) for any sequence ~ǫi → ~ǫ.
Let us consider also the approximations for problem (1):
~F i(x) = {f i1(x), . . . , f im(x)} → max
x∈Xi⊂Rn
, i = 1, 2, . . . , (2)
where the sequence of sets {Xi} converges to the set X, lim
i
Xi = X, and the sequence of
vector functions {~F i(x), x ∈ Xi} converges to the vector function ~F (x), x ∈ X (in the sense
of Definition 2 or 3). Denote, by X∗
i (~ǫ, ) the set of ~ǫ-nondominated points of problem (2), i. e.,
X∗
i (~ǫ) is a ~ǫ-efficient subset of Xi.
Concerning ~F and {~F i}, we assume the certain continuity and convergence properties outlined
in the following definitions.
Definition 2. (Continuous convergence of a sequence of vector functions). A sequence of
vector functions {~F i(x), x ∈ Xi} is called continuously convergent to a vector function ~F (x),
x ∈ X, if a) lim
i
Xi = X, b) for any sequence Xi ∋ xi → x, lim
i
~F i(xi) = ~F (x) holds
(componentwise).
Definition 3. (Graphical convergence from below of a sequence of vector functions). A sequ-
ence of vector-functions {~F i(x), x ∈ Xi} is called graphically convergent from below to a vector
function ~F (x), x ∈ X, if a) lim
i
Xi = X, b) for each sequence Xi ∋ xi → x, lim sup
i
~F i(xi) 6 ~F (x)
holds (componentwise), and c) for any point x ∈ X, there is a sequence Xi ∋ xi → x such that
lim
i
~F i(xi) = ~F (x) (componentwise).
The concepts of continuous and graphical convergence of multivalued mappings and functions
(in the latter case, epi- and hypo-convergence) were comprehensively studied in [12, Sections 6E,
6G, 7B]. Definitions 2 and 3 differ from the corresponding notions in [12] by that the domains Xi
and X of functions ~F i and ~F in Definitions 2 and 3 are explicitly outlined, but the functions
in [12, Definition 5.41] are considered on a common domain X or Rn. In addition, Definition 3
extends the definition of the graphical convergence of scalar functions [12, 7(3), 7(9), Defini-
tion 7.1] to vector functions.
Examples of the sequences of vector functions that are graphically convergent from below.
E1. Obviously, if a sequence {~F i(x), x ∈ Xi} converges continuously to {~F (x), x ∈ X}, i. e.,
lim
i
Xi = X and lim
i
~F i(xi) = ~F (x) for any sequence Xi ∋ xi → x, then {~F i(·)} converges to
~F (·) graphically.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №4 37
E2. Obviously, if all scalar components, except the first one, of {~F i(·)} converge continuously
to the corresponding scalar components of {~F (·)}, and if the first component of {~F i(·)} hypo-
converges to the first component of ~F (·), then {~F i(·)} graphically converges from below to ~F (·).
E3. If ~F i(x) = ~F (x, yi), x ∈ X, Y ∋ yi → y, where ~F (x, y) is componentwise upper semi-
continuous on X × Y and is continuous at y ∈ Y for any x ∈ X, then {~F i(·, yi)} graphically
converges from below to ~F (·, y). In particular, this case includes a stationary sequence of upper
semicontinuous vector functions ~F i(x) = ~F (x), x ∈ Xi = X.
E4. Example of the construction of a continuously convergent sequence of functions. Let
a function ~F (x), x ∈ X be continuous on a closed set X, and let a sequence of functions
{~F i(x), x ∈ Xi ⊆ X} be such that ∆i : = sup
x∈Xi
‖~F i(x) − ~F (x)‖ → 0 with i → ∞. Then, for
any sequence (Xi ∋)xi → x, we obtain
‖~F i(xi)− ~F (x)‖ 6 ‖~F i(xi)− ~F (xi)‖+ ‖~F (xi)− ~F (x)‖ 6 ∆i + ‖~F (xi)− ~F (x)‖ → 0.
E5. If the objective vector function of problem (1) has the form of an expectation, ~F (x) =
= E~F (x, ω), x ∈ X, then the sample average approximations ~F i(x) = (1/Mi)
Mi∑
k=1
~F (x, ωk) can be
used instead of ~F (x), where {ωk, k = 1, 2, . . .} are i. i. d. observations of the random parameter ω.
Terms of the uniform and, therefore, continuous convergence of the empirical estimates ~F i(x) to
~F (x) on the set X can be found in [2, Section 7.2.5]. Next, we present the sufficient conditions of
continuous convergence of discretely defined empirical functions ~F i to a continuous expectation
function ~F . Assume that the functions ~F (x, ω) are uniformly bounded on X, ‖~F (x, ω)‖ 6 M ,
and, at each point x ∈ Xi of a discrete set Xi ⊂ X (with the number of elements Ni), an
empirical estimate ~F i(x) = (1/Mi)
Mi∑
k=1
~F (x, ωk) such that
Pr{‖~F i(x)− ~F (x)‖ > δ} 6 C exp
{
−2Miδ
2
M2
}
∀ δ > 0
is independently constructed. Such estimates follow, e. g., from the Hoeffding inequality [2, Secti-
on 7.2.8] with C = 2m. Then ∆i = max
x∈Xi
‖~F i(x)− ~F (x)‖ → 0 with probability one, if, for example,
Mi > αNi, α > 0, and the numerical sequence {Ni} strictly monotonically increases to infinity.
Indeed, the assertion follows from the fact that, for any δ > 0, the relations
∞∑
i=1
Pr{∆i > δ} 6
∞∑
i=1
CNi exp
{
−2
Miδ
2
M2
}
6
∞∑
i=1
CNi exp
{
−2
Niαδ
2
M2
}
< +∞
hold true.
Theorem 1 (Convergence of solutions of the approximate problems (2)). Let a sequence of
sets {Xi} converge to a compact set X, lim
i
Xi = X, and let a sequence of functions {~F i(x), x ∈
∈ Xi} graphically converge from below to a vector function ~F (x), x ∈ X. Let lim
i
~ǫi = ~ǫ. Then,
for each ~ǫ′ < ~ǫ,
X∗(~ǫ′) ⊆ lim inf
i
X∗
i (~ǫi) ⊆ lim sup
i
X∗
i (~ǫi) ⊆ X∗(~ǫ).
38 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2015, №4
4. Multicriteria random search (MRS) algorithm and its convergence. The next
multicriteria random search algorithm uses the random discrete approximations Xi =
i⋃
k=1
X̃k of
a feasible set X and estimates ~F i(x), x ∈ Xi, of the objective function of (1). The algorithm
generates a random sequence of approximate solutions X∗
i , i = 1, 2, . . ., of task (1) as follows.
At the first iteration, the first generation of N1 points X̃1 is randomly generated in the set X,
the estimates ~F 1(x) of the objective function ~F (x) are built for all points x ∈ X̃1, and, in the
set {~F 1(x), x ∈ X̃1}, a subset {~F 1(x), x ∈ X∗
1 (~ǫ)} of all ~ǫ1-nondominated points is chosen.
Suppose that, at iteration i, we have already built the set X∗
i . Then (preferably in a vicinity
of the set X∗
i ) a new generation of Ni random points X̃i is generated, the estimates ~F i(x) of the
objective function ~F (x) for all x ∈ Xi = ∪i
k=1X̃k are built, and, from the set {~F i(x), x ∈ Xi},
the subset {~F i(x), x ∈ X∗
i } of ~ǫi-nondominated points is chosen. Then we proceed to iteration
i+ 1. The process continues indefinitely long or ends at reaching the limit of iterations.
Below, we formulate the conditions of convergence of the MRS algorithm. The next statement
follows from the Borel–Cantelli lemma.
Lemma 2. Let a sequence of random sets {X̃i, i = 1, 2, . . .} be such that X̃i ⊆ X with
probability one. Let, with nonzero probability pi(x, δ) > 0, the set X̃i intersect with any δ-vicinity
of any point x ∈ X, and let
∑
i
pi(x, δ) = +∞ hold. Then, with probability one, lim sup
i
X̃i = X,
and, thus, lim
i
i⋃
k=1
X̃k = X.
Lemma 3. Let X̃i ⊆ X, lim sup
i
X̃i = X, lim
i
~ǫi = ~ǫ, and let ~F (x) be continuous on X and
∆i = sup
x∈Xi
‖~F i(x) − ~F (x)‖ → 0. Then, for each ~ǫ′ < ~ǫ, we have
X∗(~ǫ′) ⊆ lim inf
i
X∗
i(~ǫ) ⊆ lim sup
i
X̂i(~ǫ) ⊆ X∗(~ǫ).
The above lemma actually covers the case of the sample average approximation of a vector
objective function outlined in Example E5.
As a consequence of Lemmas 2 and 3, we obtain, by Theorem 1, the following result on the
convergence of a multicriteria random search algorithm to the ~ǫ-nondominated set of problem (1).
Theorem 2 (convergence of the MRS algorithm). Let the vector function ~F (x) be continuous
on a compact set X ⊂ Rm, let the random sets X̃i with positive probability pi(x, δ) > 0 intersect
with any δ-vicinity of each point x ∈ X, and let
∑
i
pi(x, δ) = +∞. Denote Xi =
i⋃
k=1
X̃k. Let
{~F i(x), x ∈ Xi ⊂ X} be a sequence of random vector functions such that, with probability one,
∆i = sup
x∈Xi
‖~F i(x)− ~F (x)‖ → 0. Then, with probability one, a) all cluster points of {X∗
i (~ǫi)} belong
to X∗(~ǫ) and b) for each point x∗ ∈ X∗(~ǫ′), ~ǫ′ < ǫ, there is a sequence of points {xi ∈ X∗
i (~ǫi)}
convergent to x∗.
In particular, if ~ǫ > 0, then, for each weakly Pareto optimal point x∗ ∈ X∗ ⊆ X, there is
a sequence of points {xi ∈ Xi(ǫ)} convergent to x∗. By virtue of the upper semicontinuity of
the mapping X∗(~ǫ) at ~ǫ = ~0 for a sufficiently small vector ~ǫ, the set X∗(~ǫ) will appear in an
arbitrarily small neighborhood of the weakly Pareto optimal points X∗ = X∗(~0).
5. Conclusions. The article describes an approximation technology for the multicriteria
stochastic optimization of “input–random output” systems. In practice, these models are highly
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №4 39
nonlinear and nonconvex. Their functioning can generally be evaluated by deterministic vector
indicators such as the means, quantiles, probabilities of reaching/exiting specified areas, etc.
Collapsing the vector indicator into a scalar one in view of its optimization is not always possible.
So, the task is to find such inputs that correspond to the Pareto-optimal performance vector
indicators. This paper validates a methodology of solving such problem by a random search with
the selection of Pareto-optimal points.
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Received 15.01.2015V.M. Glushkov Institute of Cybernetics,
NAS of Ukraine, Kiev
Б.В. Норкiн
Статистична апроксимацiя багатокритерiальних задач стохастичного
програмування
Обгрунтовано апроксимацiйний пiдхiд до розв’язання задач багатокритерiальної стохас-
тичної оптимiзацiї. В якостi узагальненої моделi стохастичної системи, що оптимiзуєть-
ся, використовується векторна модель типу “вхiд–випадковий вихiд”. Випадковi виходи пе-
ретворюються в вектор детермiнованих показникiв ефективностi i ризику. Проблема поля-
гає в тому, щоб знайти тi входи, якi вiдповiдають Парето-оптимальним значенням вихiд-
них показникiв. Ця задача наближається послiдовнiстю задач детермiнованої багатокрите-
рiальної оптимiзацiї, де, наприклад, цiльова вектор-функцiя є вибiрковим середнiм набли-
женням вихiдної функцiї, а допустима множина є дискретним наближенням можливих
входiв. Наближенi оптимальнi розв’язки визначаються як слабо ефективнi (з деякою точ-
нiстю) за Парето. Аналiз збiжностi включає в себе обгрунтування збiжностi загальної
апроксимацiйної схеми i встановлення умов збiжностi з ймовiрнiстю одиниця при адекват-
ному регулюваннi параметрiв вибiрки.
40 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2015, №4
Б.В. Норкин
Статистическая аппроксимация многокритериальных задач
стохастического программирования
Обосновывается аппроксимационный подход к решению задач многокритериальной стохас-
тической оптимизации. В качестве обобщенной модели оптимизируемой стохастической
системы используется векторная модель типа “вход–случайный выход”. Случайные выходы
преобразуются в вектор детерминированных показателей эффективности и риска. Пробле-
ма состоит в том, чтобы найти те входы, которые соответствуют Парето-оптималь-
ным значениям выходных показателей. Эта задача приближается последовательностью
задач детерминированной многокритериальной оптимизации, где, например, целевая век-
тор-функция является выборочным средним приближением исходной функции, а допусти-
мое множество является дискретным приближением возможных входов. Приближенные
оптимальные решения определяются как слабо эффективеные (с некоторой точностью) по
Парето. Анализ сходимости включает в себя обоснование сходимости общей аппроксима-
ционной схемы и установление условий сходимости с вероятностью единица при адекват-
ном регулировании параметров выборки.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2015, №4 41
|
| id | nasplib_isofts_kiev_ua-123456789-96220 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-12-07T18:06:49Z |
| publishDate | 2015 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Norkin, B.V. 2016-03-12T15:32:23Z 2016-03-12T15:32:23Z 2015 Statistical approximation of multicriteria problems of stochastic programming / B.V. Norkin // Доповiдi Нацiональної академiї наук України. — 2015. — № 4. — С. 35-41. — Бібліогр.: 12 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/96220 519.6 The article validates an approximation technique for solving multiobjective stochastic optimization problems. As a generalized model of a stochastic system to be optimized, a vector “input–random output” system is considered. Random outputs are converted into a vector of
 deterministic performance/risk indicators. The problem is to find those inputs that correspond
 to Pareto-optimal values of output indicators. The problem is approximated by a sequence
 of deterministic multicriteria optimization problems, where, for example, the objective vector
 function is a sample average approximation of the original one, and the feasible set is a discrete sample approximation of the feasible inputs. Approximate optimal solutions are defined as weakly Pareto efficient ones within some vector tolerance. Convergence analysis includes
 establishing the convergence of the general approximation scheme and establishing the conditions of convergence with probability one under proper regulation of sampling parameters. Обгрунтовано апроксимацiйний пiдхiд до розв’язання задач багатокритерiальної стохастичної оптимiзацiї. В якостi узагальненої моделi стохастичної системи, що оптимiзується, використовується векторна модель типу “вхiд–випадковий вихiд”. Випадковi виходи перетворюються в вектор детермiнованих показникiв ефективностi i ризику. Проблема полягає в тому, щоб знайти тi входи, якi вiдповiдають Парето-оптимальним значенням вихiдних показникiв. Ця задача наближається послiдовнiстю задач детермiнованої багатокритерiальної оптимiзацiї, де, наприклад, цiльова вектор-функцiя є вибiрковим середнiм наближенням вихiдної функцiї, а допустима множина є дискретним наближенням можливих входiв. Наближенi оптимальнi розв’язки визначаються як слабо ефективнi (з деякою точнiстю) за Парето. Аналiз збiжностi включає в себе обгрунтування збiжностi загальної
 апроксимацiйної схеми i встановлення умов збiжностi з ймовiрнiстю одиниця при адекватному регулюваннi параметрiв вибiрки. Обосновывается аппроксимационный подход к решению задач многокритериальной стохастической оптимизации. В качестве обобщенной модели оптимизируемой стохастической системы используется векторная модель типа “вход–случайный выход”. Случайные выходы
 преобразуются в вектор детерминированных показателей эффективности и риска. Проблема состоит в том, чтобы найти те входы, которые соответствуют Парето-оптимальным значениям выходных показателей. Эта задача приближается последовательностью
 задач детерминированной многокритериальной оптимизации, где, например, целевая вектор-функция является выборочным средним приближением исходной функции, а допустимое множество является дискретным приближением возможных входов. Приближенные
 оптимальные решения определяются как слабо эффективеные (с некоторой точностью) по
 Парето. Анализ сходимости включает в себя обоснование сходимости общей аппроксимационной схемы и установление условий сходимости с вероятностью единица при адекватном регулировании параметров выборки. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Інформатика та кібернетика Statistical approximation of multicriteria problems of stochastic programming Статистична апроксимацiя багатокритерiальних задач стохастичного програмування Статистическая аппроксимация многокритериальных задач стохастического программирования Article published earlier |
| spellingShingle | Statistical approximation of multicriteria problems of stochastic programming Norkin, B.V. Інформатика та кібернетика |
| title | Statistical approximation of multicriteria problems of stochastic programming |
| title_alt | Статистична апроксимацiя багатокритерiальних задач стохастичного програмування Статистическая аппроксимация многокритериальных задач стохастического программирования |
| title_full | Statistical approximation of multicriteria problems of stochastic programming |
| title_fullStr | Statistical approximation of multicriteria problems of stochastic programming |
| title_full_unstemmed | Statistical approximation of multicriteria problems of stochastic programming |
| title_short | Statistical approximation of multicriteria problems of stochastic programming |
| title_sort | statistical approximation of multicriteria problems of stochastic programming |
| topic | Інформатика та кібернетика |
| topic_facet | Інформатика та кібернетика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/96220 |
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