Cybernetics and Systems Analysis, Vol. 51,
No .
6, November, 2015
ASYMPTOTIC BEHAVIOR OF A MODIFIED
STOCHASTIC OPTIMIZATION PROCEDURE
IN AN AVERAGING SCHEME
P. P. Gorun
1
and Y. M. Chabanyuk
2
UDC 519.21
Abstract. The asymptotic behavior of a modified discrete stochastic optimization procedure (SOP) in
a Markov environment in an averaging scheme is investigated. Additional SOP optimization parameters
are introduced and are used to investigate the behavior of fluctuations on increasing time intervals. The
normalization-dependent form of the limit generator is established, and a modified discrete SOP is shown
to be asymptotically normal w hen the introduced parameters assume certai n values.
Keywords: Markov process, stochastic optimization, asymptotic behavior, averaging scheme.
The investigation of the behavior of fluctuations of the classical stochastic optimization procedure (SOP) determines
an estimate for the rate of its convergence to the extreme point of an averaged evolutionary system. This problem arises in
using the algorithm of phase averaging of random evolutions [1] that is based on the closeness of the initial and averaged
evolutionary systems [2]. In particular, the behavior of fluctuations of a diffusive evolutionary system with Markov jumps
(a stochastic approximation procedure) is investigated in [3] where the rate function has a singularly excited addend with
a small series parameter.
The asymptotic behavior of a stochastic optimization procedure was investigated by the moments method described
in more detail in [4, 5], and, for more general cases, other limit distributions are obtained [6–8].
In [9], some randomized algorithms of stochastic optimization are considered under almost random disturbances
when trial perturbations are applied to the input. Two of them are randomized versions of the Kiefer–Wolfowitz procedure
(each of them requires two computations of an unknown function at every iteration step). The mentioned paper includes
a brief review of works of H. Kushner and D. Clark, B. T. Polyak and A. B. Tsybakov, J. Spall, H. F. Chen, and others. Each
author considered the mentioned procedures that supplemented one another under different conditions.
Since the asymptotic analysis of the classical SOP was made by the authors of this article in [10], this article makes
the asymptotic analysis of the modified procedure described in [9, procedure (2)] taking into account the aforesaid and with
a view to comparing the asymptotic behavior of the classical and modified procedures.
For simplicity, the one-dimensional case of regression functions is considered, but the obtained results are extended
by analogy to the multidimensional case.
The pseudo-gradient and also the SOP for a regression function
Cux C(; ) ( )Î
3
Ñ
,
xXÎ
, are considered in the
following modified representation:
Ñ=
+-
==
b
Cux
Cu bx Cux
b
uut bbt(; )
(;)(;)
,(),()
. (1)
The jump SOP in a scheme of series in a Markov environment is defined by the relationship (let
aCux
n
n
bnn
eee
=
-
å
Ñ=
0
1
0(; )
) [11]
956
1060-0396/15/5106-0956
©
2015 Springer Science+Business Media New York
1
2
Lviv State University of Life
November–December, 2015, pp. 137–146. Original article submitted June 8, 2015.
DOI 10.1007/s10559-015-9788-8
ut u a Cu x u u t
n
n
t
bnn
() ( ; ), () ,
(/ )
/
=+ Ñ = ³
=
-
å
e
ne
ee e
g
0
1
1
00
, (2)
where
g
is the time normalization coefficient,
a
n
e
is some normalizing sequence, and
n()t
is the counter of jumps before
a moment
t
.
The convergence of jump SOP (2) under the conditions
ut u t
e
ee() , ,
*
®®¥£
0
(3)
of the theorem on the sufficient conditions of its convergence (see [12]) means that the following equality holds:
¢
=Cu()
*
0
,
(4)
where
u
*
(without loss of generality, we consider that
u
*
= 0
) is an equilibrium point of the system
du t
dt
C u t C u q dx C u x
X
()
(()), () ( ) (; )=¢ =
ò
r
. (5)
Equalities (4) and (5) yield the balance condition
PCx¢=(; )00
, (6)
where
P
is the projector obtained by the stationary distribution of an embedded Markov chain
xn
n
, ³ 0
, i.e.,
Pjrj() ( )()xdxx
X
=
ò
. In this case, conditions (3) mean that it makes sense to study fluctuations of the jump SOP with
the following normalization:
u
e
e
u
e
g
ee
g
e
() (), () ()t
t
ut ut
t
t==
. (7)
In SOP (2), the following embeddings take place:
uu xx aa n
nnnnnnnn
eeeeeee g
tt ttte==== ³(), (), (), / ,
/1
0
,
where
t
n
are moments of Markov renewal.
Let us consider the functions
a
a
n
ab
b
n
b
nn
=>=>
ab
,, ,00
,
where
a
and
b
satisfy the following convergence conditions for SOP (2) (see [12]):
(1)
¢¢
£-CuVu kVu() () ()
0
;
(2) the following estimates take place:
max | ( ( ; )) ( ) | ( ( ))
xX
b
Cu x V u k Vu
Î
Ñ
¢¢
£+
2
1
1
;
max | ( ; )[ ( ) ]
~
(; ) ()| ( (
xX
bb
Cu x qxR I Cu xV u k Vu
Î
Ñ-Ñ
¢¢
£+
02
1))
,
max | ( ( ; )) [ ( ) ]
~
(; ) ()| (
xX
bb
Cux qxR I Cu xV u k
Î
Ñ-Ñ
¢¢¢
£
2
03
1+Vu())
,
where
~
(; ) ()(;) ()Cu x qxCu x Cu=-
;
(3) the restrictive condition on the growth of the Lyapunov function and the Lipschitz condition for
Ñ
b
Cu()
are,
respectively, as follows:
|| ( )|| ( ( ))
¢
£+Vu k Vu
4
1
;
max || ( ) ( ) ||
xR
bn
d
Cu C u k b
Î
Ñ-
¢
£
5
;
ki
i
>=005,,
;
957
(4) the sequences
a
n
and
b
n
are nonincreasing, positive, and satisfy the conditions
a
n
n=
¥
å
=¥
0
,
a
b
n
n
n
2
2
0=
¥
å
<¥
,
ab
n
n
n
=
¥
å
<¥
0
,
where
Vu()
is the Lyapunov function of averaged system (5).
We also consider the following two-component Markov process (MP):
ue
ee g
(), (/ ),
/
tx xt t
t
=³
1
0
.
(8)
LEMMA 1 [13]. The generator of MP (8) is represented in terms of test functions
ju(;) ( )×ÎC
1
Ñ
in the form:
LC
tt
xQx xx
eg ge
ju e ju e ju(; ) (; ) ()(; )
//
=+
--11
,
(9)
where
CP
tb
xxqx taC
t
xy
eagga
g
ju ju e
e
u()(; ) () ( ; ;
/
=+ Ñ
æ
è
ç
ö
ø
÷
--1
)(;) (;)
/
-
é
ë
ê
ù
û
ú
+¢ju e
g
uj u
g
y
t
x
1
, (10)
Pjj(; ) ( , ) (; )×= ×
ò
yPydz z
X
.
Proof. The proof of the lemma is based on the definition of the MP generator in terms of conditional expectation (for
example, [14, Chapters 3 and 5]).
o
For convenience, we introduce the auxiliary indicator function
Iy
y
yy
()
,;
,,(,].
=
<
=Î-¥
ì
í
î
00
10 0
Since a generalized case of the asymptotic behavior of SOP (2) is considered, we will first obtain some auxiliary
information and then fix values of the coefficients
ab,,
and
g
and pass to the formulation of a lemma and corresponding
theorems on the form of representation of the generator in terms of test functions
ju(;)×
.
Using representation (1), we expand the function
Ñ
æ
è
ç
ö
ø
÷
b
C
t
x
e
u
g
;
in the neighborhood of its point of extremum
u
*
as
follows:
Ñ
æ
è
ç
ö
ø
÷
=¢ + +
æ
è
ç
ç
ö
ø
÷
÷
¢¢
b
C
t
xC x
tt
bC
e
u
e
u
e
gg
b/g
b
;(;) (0
2
0; )x
+++
æ
è
ç
ç
ö
ø
÷
÷
¢¢¢
+
+
1
2
23
0
2
2
2
12
2
2
e
u
e
u
e
g
bg
gb
b/g
b
tt
b
t
bC
/
( ;)xo
t
+
æ
è
ç
ç
ö
ø
÷
÷
+
+
e
u
b/g
gb
1
, (11)
where
uu= ()t
and
b = const
.
For convenience and conciseness, we put
¢
=
¢
CC x(; )0
,
¢¢
=
¢¢
CC x(; )0
,
¢¢¢
=
¢¢¢
CC x(; )0
,
¢
=
¢
jju(; )x
,and
¢¢
=
¢¢
jju(; )x
. Substituting expansion (11) in representation (10), we obtain
eju
g
uj
ge-
=¢
1/
() (; )C
t
xx
t
+
¢
++
æ
è
ç
ç
ö
ø
÷
÷
¢¢
+
-
-
-
ee
u
ee
u
a
g
ag g
b/g
bg
1
1
2
2
2
2
1
2
t
aC
tt
bC
t
++
æ
è
ç
ç
ö
ø
÷
÷
¢¢¢
é
ë
ê
ê
ù
û
ú
ú
¢
+
+
e
u
e
j
b/g
gb
b/g
b
12
2
2
0
23t
b
t
bC Q
958
+
é
ë
ê
ê
ê
¢+ +
æ
è
ç
ç
ö
ø
÷
-
-
-
ee
u
e
a
g
ag g
b/g
b
21
2
22
2
2
2
2
2t
a
C
tt
b()
÷
¢¢¢
CC
++ +
æ
è
ç
ç
ö
ø
÷
÷
¢ ¢¢¢
+
+
+
e
u
e
u
ee
g
bg
ga
b/g
b
2
2
2
12
2
2
23tt
b
t
bCC
/ 2
2
2
12
2
22
4tt
b
t
bC
g
b/g
gb
b/g
b
u
e
u
e
++
æ
è
ç
ç
ö
ø
÷
÷
¢¢
+
+
()
++ + +
+
+
+
+
e
u
e
u
e
u
e
g
b/g
gb
b/g
gb
b/g3
3
3
2
2
2
21
2
2
3
7
12
6tt
b
t
b
t
3
3
b
bCC
æ
è
ç
ç
ö
ø
÷
÷
¢¢ ¢¢¢
++ + +
+
+
+
+
e
u
e
u
e
u
e
g
b/g
gb
b/g
gb
4
4
4
3
3
3
22
2
22
11
12
1
3
tt
b
t
b
()
31
3
3
4
4
42
0
1
9
b/g
gb
b/g
b
u
e
+
+
+
æ
è
ç
ç
ö
ø
÷
÷
¢¢¢
ù
û
ú
ú
¢¢
t
b
t
bC Q() j
+
æ
è
ç
ç
ö
ø
÷
÷
--
-
o
t
e
ag
ag
()/31 3
33
.
(12)
Proceeding from the analysis of representation (12) and the convergence conditions for SOP (2) and taking into
account the form of pseudo-gradient (1), it is expedient to study the asymptotic behavior of the SOP being considered with
the following relationships between the coefficients
ab,,
and
g
:
abg=<12,/
; (13)
abg==12,/
; (14)
abgg=Î12,(/;)
. (15)
When
a <1
, this procedure will not be asymptotically normal (see investigations in [10]).
Case 1. Let us consider relationship (13) when
a =1
and
bg< /2
.
LEMMA 2. Under condition (13), generator (9) has the following asymptotic representation in terms of test functions
ju(;) ( )×ÎC
3
Ñ
:
L
t
xQ
t
Qx
t
Qx
eg
g
bg
gb
ju e
ee
(; ) () ()
/
/
=+ +
é
ë
ê
-
-
-
-
-+
1
1
1
1
1
1
2
+++
-
-+
-
-
e
ju
e
q
bg
gb
g
g
e
21
12
34
23
33
1
//
() ()(; ) (
t
Qx
t
Qx x
t
t
xx)(;)
ù
û
ú
ju
,
(16)
where
Qx x aC xQ x
10
0()(; ) (; ) (; )ju j u=
¢¢
; (17)
Qx x
ab
CxQ x
20
2
0()(; ) (; ) (; )ju j u=
¢¢ ¢
; (18)
Qx x
ab
CxQ x
3
2
0
6
0()(; ) (; ) (; )ju j u=
¢¢¢ ¢
; (19)
Qx x bx x I
a
CxQ
4
2
2
1
22
0()(;) () (;) ( (;))ju u j u g=
¢
+-
æ
è
ç
ö
ø
÷
¢
0
¢¢
ju(; )x
; (20)
bx aC xQ() (; )=
¢¢
+0
0
g
, (21)
where the remainder term
qju
e
t
xx()(; )
is bounded.
Proof. Putting
a =1
and
bg< /2
in formula (12), we obtain representation (16) in terms of designations (17)–(21).
The boundedness of the remainder term follows from formula (12) and the boundedness of the first, second, and third
derivatives of the regression function.
o
959
The singular perturbation problem (SPP) for operator (16) is realized in terms of perturbed functions of the form
ju ju
e
ju
e
j
e
g
g
bg
gb
(; ) ( ) (; ) (
/()/
x
t
x
t
=+ +
-
-
+-
-+
11
1
2
11
1
3
u;)x
++
+-
-+
e
ju
e
ju
bg
gb
g()/ /
(; ) (; )
12 1
12
4
1
5
t
x
t
x
. (22)
LEMMA 3. The solution of the SPP for operator (16) in terms of test functions (22) such that
ju(;) ( )×ÎC
3
Ñ
is of
the form
LL
tt
x
t
x
ee e
ju ju q ju(; ) ( ) ()( )=+
1
, (23)
where the operator
L
acts by the rule
Lxk xI
a
xju uj u g
r
ju(; ) (; ) (; )=- ¢ + -
æ
è
ç
ö
ø
÷
¢¢
1
22
22
, (24)
where
kad=-g
; (25)
dqdxC x
X
=-
¢¢
ò
r() (;)0
; (26)
rp r
2
0
200 0=
¢¢
-
¢
òò
XX
dx q x C x R q x C x q dx C()()(;) ()(;) ()((;x))
2
;
(27)
in this case, the remainder term
qju
e
t
x()()
is such that
|| ( ) ( )||qju
e
t
x ® 0
,
e ® 0
.
Proof. Since the remainder term in representation (16) does not exert influence on the solution of the SPP [2, Sec. 3.1], [1],
we will consider the solution of the SPP only for the following operator truncated with respect to operator (16):
L
t
xQ
t
Qx
t
Qx
0
1
1
1
1
1
1
2
eg
g
bg
gb
ju e
ee e
(; ) () ()
/
/
=+ + +
-
-
-
-
-+
21
12
34
1
bg
gb
ju
/
() ()(; ).
-
-+
+
t
Qx
t
Qx x
The operator
L
t0
e
has the following representation in terms of functions (22):
L
t
xQ
t
QxQx
0
1
1
1
21
ee g
g
ju e ju
e
ju ju(; ) ( ) [ (; ) ()( )]
/
=+ +
-
-
-
+++
-
-+
-
-+
e
ju ju
e
bg
gb
bg
gb
//
[(;) ()()] [
1
1
32
21
12
t
QxQx
t
Qju ju
43
(; ) ()( )]xQx+
+++-
æ
è
ç
ö
ø
÷
é
ë
ê
ù
û
11
2
54 12
t
QxQx I Qx xju ju g ju(; ) ()() () (; )
ú
+
+-
-+
e
qju
bg
gb
e
()/
() ()
12
22
t
x
t
, (28)
where
qju
e
t
x()()
is bounded.
The solvability condition (see [2, Sec. 3.1]) for operator (28) is of the form
QxQxju ju
21
0(; ) () ( )+=
.
960
If balance condition (6) is taken into account, then, using the operator
R
0
, we find from the last equality that
ju ju ju
201 00
0(; ) () ( ) (; ) ( )xQx aCxQ==¢¢RR
. (29)
From the condition
QxQxju ju
32
0(; ) ()( )+=
, we find
ju ju ju
302 0 0
2
0(; ) ()( ) (; ) ( )xQx
ab
CxQ==
¢¢ ¢
RR
. (30)
From the condition
QxQxju ju
43
0(; ) ()( )+=
, we similarly find
ju ju ju
403
2
00
6
0(; ) () ( ) (; ) ( )xQx
ab
CxQ==
¢¢¢ ¢
RR
.
From solvability condition (28), we have
QxQx I Qx xju ju g ju ju
54 12
1
2
(; ) ()( ) () (; ) ( )++-
æ
è
ç
ö
ø
÷
= L
,
where the operator
L
is defined by the relationships
LLPPPju ju() () ()= x
,
L()() ()() () (; )xQxIQxxju ju g j u=+-
æ
è
ç
ö
ø
÷
412
1
2
. (31)
We now compute the right side of relationship (31). To this end, we use obtained representations (29) and (30),
LPju u r g j u() ( ) (; ) ()=
¢¢
+
æ
è
ç
ç
ö
ø
÷
÷
¢
ò
aq dx C x
X
0
+-
æ
è
ç
ö
ø
÷
¢¢
-
òò
I a dx C x R q x C x q d
XX
gp r
1
2
00
1
2
2
0
()(;) ()(;) (xC x)( ( ; )) ( )
¢
æ
è
ç
ç
ö
ø
÷
÷
¢¢
0
2
ju
or, using designations (24)–(27), obtain solution (23).
The computation of the remainder term
q
e
t
x()
is described in [15].
o
The asymptotic normality of normalized SOP (7) is established under the additional conditions
A1:
d > 0
; A2:
k > 0
; A3:
r
2
0>
,
where
dk,,
and
r
2
are represented by formulas (25)–(27).
Condition A1 provides the fulfillment of condition A2, which implies the ergodicity of the limit process
z()t
,
t ³ 0
,
and condition A3 provides the diffuseness of the process
z()t
,
t ³ 0
.
THEOREM 1 [10]. Let convergence conditions (1)–(4) of SOP (2) and also additional conditions A1–A3 be
fulfilled. Then the weak convergence of the processes
uze
e
() (),ttÞ®0
,
takes place on each finite interval
0
0
<££ttT
. The limit diffusion process
z(),tt³ 0
, is a process of
Ornstein–Uhlenbeck type [16, Vol. 2, Ch. III, Sec. 8e and Ch. X, Sec. 4b] and is defined by the generator
Lju uj u g
r
ju() () ()=- ¢ + -
æ
è
ç
ö
ø
÷
¢¢
kI
a
1
22
22
in terms of designations (25)–(27).
Proof. On the whole, the process of the proof is similar to that of the proof of the theorem on the convergence of SOP
(2) (see [12]). Therefore, using Lemmas 2 and 3, convergence conditions (1)–(4) of jump SOP (2), and also additional
conditions A.1–A.3, we obtain the statement of the theorem.
o
961
Remark 1. The Ornstein–Uhlenbeck limit process [17, Ch. 18, Sec. 4] defined by the generator
L
under the
conditions of Theorem 1 is ergodic with the stationary normal distribution
N = (, )0
0
2
s
, where the dispersion is computed by
the formula
s
0
2
=
Iakgr-
æ
è
ç
ö
ø
÷
1
2
2
22
/
.
Case 2. Let us consider relationship (14) when
a =1
and
bg= /2
.
LEMMA 4. Under condition (14), generator (9) has the following asymptotic representation in terms of test functions
ju(;) ( )×ÎC
3
Ñ
:
L
t
xQ
t
Qx
t
Qx
eg
gg
ju e
ee
(; ) () ()
/
/
/
=+ +
é
ë
ê
-
-
-
-
-
1
1
1
1
12
32
2
++
ù
û
ú
-
-
1
3
23
33
t
Qx x
t
xx
t
()(; ) () (; )
/
ju
e
qju
g
g
e
,
(32)
where
Qx x aC xQ x
10
0()(; ) (; ) (; )ju j u=
¢
¢
; (33)
Qx x
ab
CxQ x
20
2
0()(; ) (; ) (; )ju j u=
¢¢ ¢
; (34)
Qx x bx mx x I
a
C
3
2
1
22
()(; ) ( () ()) (; ) ( (ju u j u g=+
¢
+-
æ
è
ç
ö
ø
÷
¢
0
2
0
;)) (;)xQ x
¢¢
ju
; (35)
mx
ab
CxQ() (; )=
¢¢¢
2
0
6
0
; (36)
bx aC xQ() (; )=
¢¢
+0
0
g
; (37)
Qxqx x
0
ju ju(; ) () (; )= P
;
in this case, the remainder term
qju
e
t
xx() (; )
is bounded.
Proof. Putting
a =1
and
bg= /2
in representation (12), we obtain representation (32) in terms of designations
(33)–(37). The boundedness of the remainder term follows from representation (12) and the boundedness of the first, second,
and third derivatives of the regression function. o
Since this case is constructively similar to the previous one, we omit repeated computations and formulate the lemma
on the form of the limit generator.
LEMMA 5. The solution of the SPP for operator (32) in terms of test functions
ju(;) ( )×ÎC
3
Ñ
is of the form
LL
tt
x
t
x
ee e
ju ju q ju(; ) ( ) ()( )=+
1
,
where the operator
L
acts by the rule
Lx km xI
a
xju u j u g
r
ju(; ) ( ) (; ) (; )=- -
¢
+-
æ
è
ç
ö
ø
÷
¢¢
1
22
22
,
where
kad=-g
; (38)
dqdxC x
X
=-
¢¢
ò
r() (;)0
; (39)
m
ab
dx C x
X
=
¢¢¢
ò
2
6
0r() (;)
;
962
rp r
2
0
200 0=¢¢-¢
òò
XX
dx q x C x R q x C x q dx C()()(;) ()(;) ()((;x))
2
;
(40)
in this case, the remainder term
qju
e
t
x()()
is such that
|| ( ) ( )||qju
e
t
x ® 0
,
e ® 0
.
Proof. The proof of Lemma 5 is similar to the proof of Lemma 3.
o
The asymptotic normality of normalized SOP (7) is established under additional conditions A1–A3, where
dk,,
and
r
2
are represented by formulas (38)–(40).
Remark 2. In this case, Theorem 1 in terms of designations (38)–(40) takes place.
Remark 3. The Ornstein–Uhlenbeck limit process [17, Ch. 18, Sec. 4] defined by the generator
L
under the
conditions of the theorem is ergodic with the stationary normal distribution
NM= (, )s
0
2
, where dispersion is computed by
the formula
sg r
0
222
1
2
2=-
æ
è
ç
ö
ø
÷
Iak/
and the expectation is computed by the formula
Mmk= /
.
Case 3. Let us consider relationship (15) when
a =1
and
bg gÎ(/; )2
.
LEMMA 6. Under condition (15), generator (9) has the following asymptotic representation in terms of test functions
ju(;) ( )×ÎC
3
Ñ
:
L
t
xQ
t
Qx
t
Qx
eg
g
bg
gb
ju e
ee
(; ) () ()
/
/
=+ +
é
ë
ê
-
-
-
-
-+
1
1
1
1
1
1
2
++
ù
û
ú
-
-
1
3
23
33
t
Qx x
t
xx
t
()(; ) () (; )
/
ju
e
qju
g
g
e
, (41)
where
Qx x aC xQ x
10
0()(; ) (; ) (; )ju j u=¢ ¢
; (42)
Qx x
ab
CxQ x
20
2
0()(; ) (; ) (; )ju j u=
¢¢
¢
; (43)
Qx x bx x I
a
CxQ
3
2
2
1
22
0()(;) () (;) ( (;))ju u j u g=¢+-
æ
è
ç
ö
ø
÷
¢
0
¢¢
ju(; )x
; (44)
bx aC xQ() (; )=
¢¢
+0
0
g
; (45)
in this case, the remainder term
qju
e
t
xx()(; )
is bounded.
Proof. Putting
a =1
and
bg gÎ(/; )2
in formula (12), we obtain representation (41) in terms of designations
(42)–(45). The boundedness of the remainder term follows from representation (12) and the boundedness of the first, second,
and third derivatives of the regression function.
o
Eliminating repeated calculations, note that Lemma 3 on the form of a limit generator takes place.
The asymptotic normality of normalized SOP (7) is established under additional conditions A1–A2, where
dk,,
and
r
2
are represented by formulas (25)–(27).
Remark 4. In this case, Theorem 1 takes place.
Remark 5. Under the conditions of the theorem, the Ornstein–Uhlenbeck limit process [17, Ch. 18, Sec. 4] defined
by the generator
L
is ergodic with the stationary normal distribution
N = (, )0
0
2
s
, where dispersion is computed by the
formula
sg r
0
222
1
2
2=-
æ
è
ç
ö
ø
÷
Iak/
.
Thus, the asymptotic behavior of the modified stochastic optimization procedure is investigated in the
one-dimensional case in a scheme of series in a Markov environment. The conditions under which it is asymptotically
normal are established. The obtained results are extended by analogy to the multidimensional case.
In contrast to the classical procedure, to find the asymptotic behavior of a SOP in the case of some relationships
between the normalizing coefficients
ab,,
and
g
, it suffices that the regression function
Cux(; )
be twice continuously
differentiable,
Ñux C X(; ) ( ; )Î
3
Ñ
. In particular, when
a =1
and
bg< /2
(for the classical procedure) and when
a =1
,
963
bg< /2
, and
bg gÎ(/; )2
(for the modified procedure), the limit generators coincide, which testifies to the equivalence of
these procedures with respect to their asymptotic behavior.
However, despite their simila rit y, the use of the modified procedure in real-time systems is more expedient. The reason
is that, for the classical procedure, the condition of independence of external disturbances on a perturbation
bt()
is rather
limited since, at every iteration step, the vector
b
n-1
is used twice. In the case of the modified procedure, the simultaneous
occurrence of the perturbation
b
n-1
and external disturbances makes it possible to count on their independence (see [9]).
Thus, the obtained results expand the understanding and possibilities of investigating fluctuations of evolutionary
systems in the neighborhood of the extremum point even in the case of a nonlinear dependence of their regression function
on external disturbances. This, in turn, makes it possible to deepen the analysis of fluctuations of a SOP in investigating
conditions of optimization of stochastic systems.
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