Nonlinear Stochastic Optimization by the Monte-Carlo Method

INFORMATICA, 2000, Vol. 11, No. 4, 455–468 455
© 2000 Institute of Mathematics and Informatics, Vilnius

Nonlinear Stochastic Optimization by the
Monte-Carlo Method

Leonidas SAKALAUSKAS
Institute of Mathematics and Informatics
Akademijos 4, 2600 Vilnius, Lithuania
e-mail: sakal@ktl.mii.lt

Received: May 2000

Abstract. Methods for solving stochastic optimization problems by Monte-Carlo simulation are
considered. The stoping and accuracy of the solutions is treated in a statistical manner, testing
the hypothesis of optimality according to statistical criteria. A rule for adjusting the Monte-Carlo
sample size is introduced to ensure the convergence and to find the solution of the stochastic opti-
mization problem from acceptable volume of Monte-Carlo trials. The examples of application of
the developed method to importance sampling and the Weber location problem are also considered.
Key words: Monte-Carlo method, stochastic optimization, statistical decisions.

1. Introduction

We consider the stochastic optimization problem:

F (x) = Ef (x, ω) → min ,
n
(1)
x∈R

where ω ∈ Ω is an elementary event in a probability space (Ω, Σ, Px ), the function
f : Rn × Ω → R satisfies certain conditions on integrability and differentiability, the
measure Px is absolutely continuos and parametrized with respect to x, i.e., it can be
defined by the density function p: Rn × Ω → R+ , and E is the symbol of mathematical
expectation.
The optimization of the objective function expressed as an expectation in (1) occurs
in many applied problems of engineering, statistics, finances, business management, etc.
Stochastic procedures for solving problems of this kind are often considered and two
ways are used to ensure the convergence of developed methods. The first leads to class
of methods of stochastic approximation. The convergence in stochastic approximation
is ensured by regulating certain step-length multipliers in a scheme of stochastic gradi-
ent search (see, Robins and Monro, 1951; Kiefer and Wolfowitz, 1952; Ermolyev, 1976;
Polyak, 1983; Michalevitch et al., 1986; Ermolyev and Wets, 1988; Uriasyev, 1991; etc.).
The following obstacles are often mentioned in the implementation of stochastic approx-
imation:

456 L. Sakalauskas

– it is not so clear how to stop the process of stochastic approximation;
– the methods of stochastic approximation converge rather slowly.
The second way to ensure the convergence in stochastic optimization is related to the
application of methods of a relative stochastic gradient error. The theoretical scheme of
such methods requires that the variance of the stochastic gradient be varied in the op-
timization procedure so that to remain proportional to the square of the gradient norm
(see Polyak, 1983). This approach offers an opportunity to develop implementable algo-
rithms of stochastic optimization. We consider them here using a finite series of Monte-
Carlo estimators for algorithm construction (see also Sakalauskas, 1992; Sakalauskas and
Steishunas, 1993; Sakalauskas, 1997).

2. Monte-Carlo Estimators for Stochastic Optimization

First, introduce a system of Monte-Carlo estimators, which are applied for stochastic op-
timization. Solving the problems of kind (1), suppose it is possible to get finite sequences
of realizations (trials) of ω at any point x and after that to compute the values of functions
f and p for these realizations. Then it is not difficult to find the Monte-Carlo estimators
corresponding to the expectation in (1). Thus, assume that the Monte-Carlo sample of a
certain size N could be obtained for any x ∈ D ⊂ Rn :

Y = (y 1 , y 2 , . . . , y N ), (2)

where y i are independent random variables, identically distributed with the density
p(x, · ): Ω → R+ , and the sampling estimators introduced:

N
1
F (x) = f (x, y j ), (3)
N j=1
N 2
1
2
D (x) = f (x, y j ) − F (x) . (4)
N −1 i=1

Further note that a technique of stochastic differentiation is developed on the basis
of smoothing operators, which permits estimation of the objective function and its gra-
dient using the same Monte-Carlo sample (2) without essential additional computations
(see Judin, 1965; Katkovnik, 1976; Rubinstein, 1983; Shapiro, 1986; etc.). Thus, the as-
sumption could be made that the Monte-Carlo estimator of the objective function gradient
could be introduced as well:

N
1
G(x) = G(x, y j ), x ∈ D ⊂ Rn , (5)
N j=1

Nonlinear Stochastic Optimization by the Monte-Carlo Method 457

where G: Rn × Ω → Rn is a stochastic gradient, i.e., such a random vector that
EG(x, w) = F (x) (see, Ermolyev, 1976; Polyak, 1983). The sampling covariance
matrix will be of use further:

N
1
A(x) = G(x, y j ) − G(x) G(x, y j ) − G(x) . (6)
N j=1

Now we start developing the stochastic optimization procedure. Let some initial point
x0 ∈ D ⊂ Rn be given, random sample (2) of a certain initial size N 0 be generated at
this point, and Monte-Carlo estimates (3), (4), (5), (6) be computed. Now, the iterative
stochastic procedure of gradient search could be introduced:

xt+1 = xt − ρ· G(xt ), (7)

where ρ > 0 is a certain step-length multiplier.
Consider the choice of size of random sample (2) when this procedure is iterated.
Sometimes this sample size is taken to be fixed in all the iterations of the optimization
process and choosen sufficiently large to ensure the required accuracy of estimates in all
the iterations (see, Antreich and Koblitz, 1982; Belyakov et al., 1985; Jun Shao, 1989;
etc.). Very often this guaranteeing size is about 1000–1500 trials or more, and, if the
number of optimization steps is large, solving the stochastic optimization problem can
require substantial computation (Jun Shao, 1989). On the other hand, it is well known that
the fixed sample size, although very large, is sufficient only to ensure the convergence to
some neighbourhood of the optimal point (see, e.g., Polyak, 1983; Sakalauskas, 1997).
There is no great necessity to compute estimators with a high accuracy on starting the
optimization, because then it suffices only to evaluate approximately the direction leading
to the optimum. Therefore, one can obtain a not so large samples at the beginning of the
optimum search and later on increase the size of samples so that to obtain the estimate
of the objective function with a desired accuracy only at the time of decision making on
finding the solution of the optimization problem. We pursue this purpose by choosing the
sample size at every next iteration inversely proportional to the square of the gradient
estimator from the current iteration. The following theorems justifies such an approach.

Theorem 1. Let the function F : Rn → R, expressed as expectation (1), be bounded:
F (x) F + > −∞, ∀x ∈ Rn , and differentiable, such that the gradient of this function
sattisfies the Lipshitz condition with the constant L > 0, ∀x ∈ Rn .
Assume that for any x ∈ Rn and any number N 1 one can to obtain sample (2) of
independent vectors identically distributed with the density p(x, .) and compute estimates
(3), (5) and (6) such, that the variance of the stochastic gradient norm in (6) is uniformly
bounded: E |G(x, w) − F (x)|2 < K, ∀x ∈ Rn .

458 L. Sakalauskas

Let the initial point x0 ∈ Rn and the initial sample size N 0 be given and the ran-
∞
dom sequences {xt }t=0 be defined according to (7), where the sample size is iteratively
changed according to the rule:

C
N t+1 2 + 1, (8)
G(xt )

C > 0 is a certain constant, [.] means the integer part of the number. Then
2
lim F (xt ) = 0(mod(P )), (9)
t→∞

1
if 0 < ρ L, C 4K.

Theorem 2. Let the conditions of the Theorem 1 be valid. If in addition, the function
F (x) is twice differentiable and 2
F (x) l > 0, ∀x ∈ Rn , then the estimate

2 ρ· K
E xt − x+ +
Nt
t
2 ρ· K K· L2
x0 − x+ + 1−ρ l− , t = 0, 1, 2, . . . . (10)
N0 C
2
1 3
holds if 0 < ρ min L , 4(1+l) ,C K max[4, L ], and where x+ is the stationary
l
point.

The proof, given in the Appendix, is grounded by the standard martingale methods
(see Polyak, 1983; Sakalauskas, 1997; etc.).
The step length ρ could be determinated experimentally or using the method of a
simple iteration (see, e.g., Kantorovitch and Akilov, 1958). The example of the later is
given in the examples. The choice of constant C or of the best metrics for computing the
stochastic gradient norm in (8) requires a separate study. I propose such a version of (8)
for regulating the sample size:

n· F ish(γ, n, N t − n)
N t+1 = min max +n, Nmin , Nmax , (11)
ρ(G(xt ) (A(xt ))−1 (G(xt )

where F ish(γ, n, N t − n) is the γ-quantile of the Fisher distribution with (n, N t − n)
degrees of freedom. We introduce minimal and maximal values Nmin (usually ∼20–50)
and Nmax (usually ∼1000–2000) to avoid great fluctuations of sample size in iterations.
Note that Nmax also may be chosen from the conditions on the permissible confidence
interval of estimates of the objective function (see the next section). The choice C =
n· F ish(γ, n, N t − n) > χ2 (n), (χ2 (n) is the γ-quantile of the χ2 distribution with
γ γ
n degrees of freedom) and estimation of the gradient norm in a metric induced by the
sampling covariance matrix (6), is convenient for interpretation, because in such a case, a


random error of the stochastic gradient does not exceed the gradient norm approximately
with probability 1 − γ. The rule (11) implies rule (8) and, in its turn, the convergence by
virtue of the moment theorem for multidimensional Hotelling T 2 -statistics (see Bentkus
and Gotze, 1999).

3. Stochastic Optimization and Statistical Testing of the Optimality Hypothesis

A possible decision on optimal solution finding should be examined at each iteration of
the optimization process. If we assume all the stationary points of the function F (x) to
belong to a certain bounded ball, then, as follows from the theorem proved, the proposed
procedure guarantees the global convergence to some stationary point. Since we know
only the Monte-Carlo estimates of the objective function and that of its gradient, we can
test only the statistical optimality hypothesis. Since the stochastic error of these estimates
in essence depends on the Monte-Carlo samples size, a possible optimal decision could
be made, if, first, there is no reason to reject the hypothesis of equality of the gradient to
zero, and, second, the sample size is sufficient to estimate the objective function with the
desired accuracy.
Note that the distribution of sampling averages (3) and (5) can be approximated by
the one- and multidimensional Gaussian law (see, e.g., Bhattacharya and Ranga Rao,
1976; Box and Wilson, 1962; Gotze and Bentkus, 1999). Therefore it is convenient to test
the validity of the stationarity condition by means of the well-known multidimensional
Hotelling T 2 -statistics (see, e.g., Krishnaiah and Lee, 1980; etc.). Hence, the optimality
hypothesis could be accepted for some point xt with significance 1 − µ, if the following
condition is sattisfied:
−1
(N t − n) G(xt ) A(xt ) G(xt ) /n F ish(µ, n, N t − n). (12)

Next, we can use the asymptotic normality again and decide that the objective function
is estimated with a permissible accuracy ε, if its confidence bound does not exceed this
value:
√
ηβ D(xt )/ N t ε, (13)

where ηβ is the β-quantile of the standard normal distribution. Thus, the procedure (7) is
iterated adjusting the sample size according to (8) and testing conditions (12) and (13) at
each iteration. If the later conditions are met at some iteration, then there are no reasons
to reject the hypothesis about the optimum finding. Therefore, there is a basis to stop the
optimization and make a decision about the optimum finding with a permissible accuracy.
If at least one condition out of (12), (13) is unsatisfied, then the next sample is generated
and the optimization is continued. As follows from the previuos section, the optimization
should stop after generating a finite number of Monte-Carlo samples.
Finally note that, since the statistical testing of the optimality hypothesis is grounded
by the convergence of distribution of sampling estimates to the Gaussian law, additional

460 L. Sakalauskas

standard points could be introduced, considering the rate of convergence to the normal
law and following from the Berry-Eseen or large deviations theorems.

4. Application to Importance Sampling

Let us consider the aplication of the developed approach in the estimation of quantiles
of the Gaussian distribution by importance sampling as an example. Let us apply the
measure change procedure (Asmussen and Rubinstein, 1995):

∞ ∞
1 − t2
2 1 (t−a)2 (t−a)2 t2
P (x) = √ e dt = √ e− 2 ·e 2 ·e− 2 dt
2π 2π
x x

∞ ∞
1 −at− a − t2
2 2 1 t2
= √ e 2 dt = √ g(a, t)e − 2 d t, (14)
2π 2π
x−a x−a

a2
where g(a, t) = e −at− 2 and ﬁnd the second moment:

∞ ∞
1 − t2
2 1 2
e −2at−a − t2
2
D (x, a) = √
2 2
g (a, t)e dt = √ dt
2π 2π
x−a x−a

∞ 2 ∞
1 a2 2 ea t2
= √ e −at+ 2 − t2
dt = √ e − 2 d t. (15)
2π 2π
x x+a

Then the ratio

D2 (x, a) − P 2 (x)
δ2 =
P (x) − P 2 (x)

could be used as a measure of variance change (see Fig. 1). The variance of the Monte-
Carlo estimator of the integral obtained can be reduced by ﬁtting the parameter a.
Let us differentiate with respect to a the last by one expression of D(x, a) in (15).
The mimimum condition can be obtained after noncomplex manipulations by equating to
zero the derivative and changing the:

∞ 2
te −2at−a − t2
2
√1 dt
2π
x−a
a= ∞ . (16)
2
e −2at−a − 2
2 t
√1 dt
2π
x−a


Fig. 1. Dependency of δ2 on the parameter of measure change a (x = 3).

To demonstrate an approach to rationally choose the optimization step length, using
the method of a simple iteration, consider the iterative procedure

Nt i
i=1 y H(y − x + a )g(a , y )
i t t i
at+1 = Nt
,
i=1 H(y − x + a )g(a , y )
i t t i

where y i are standard Gaussian variables, H(t) = 1, if t > 0, and H(t) = 0 in the
opposite case. This can be used to solve (16) as a separate case of (7) when

1
ρt+1 = ,
Pt
Nt
1
Pt = H(y i − x + at )g(at , y i ). (17)
Nt i=1

This procces is iterated starting from a certain initial sample, changing the Monte-Carlo
sample size according to (8) and stopping when estimate (17) is obtained with a permis-
sible accuracy according to (13) and the hypothesis on the validity of condition (18) is
not rejected according to criterion (12). Then the decision could be made: P (x) ≈ P t .
The results of application of the procedure developed are given in Tables 1 and 2 for the
cases x = 3 and x = 5, and initial data: a0 = x, N 0 = 1000, ε = 1%. We see that 4 and
3 series respectively were used to ﬁnd the parameter for the optimal measure change. It
is of interest to note that the optimization for x = 3 has been required 107 times fewer
Monte-Carlo trials than with the number necessary to estimate probability (14) by the
standard Monte-Carlo procedure and 497514 times fewer for x = 5, respectively. The
Monte-Carlo estimators obtained can be be compared with those computed analythically.

462 L. Sakalauskas

Table 1
x=3

Proposed N t Generated N t Pt
t At according to (8) according to (8) ε(%) P (x) = 1.34990 × 10−3
and (13)

1 3.000 1000 1000 10.692 1.52525 × 10−3
2 3.154 9126 9126 3.770 1.34889 × 10−3
3 3.151 35.8 × 10−6 128327 1.000 1.34372 × 10−3
4 3.156 64.8 × 10−6 127555 1.000 1.35415 × 10−3

Table 2
x=5

Proposed N t Generated N t Pt
t at according to (8) according to (8) ε(%) P (x) = 2.86650 × 10−7
and (13)

1 5.000 1000 1000 16.377 2.48182 × 10−7
2 5.092 51219 51219 2.059 2.87169 × 10−7
3 5.097 46.5 × 10−6 217154 1.000 2.87010 × 10−7

5. Stochastic Weber Problem

As the next example, we consider the classical location problem, known as the Weber
problem (Ermolyev and Wets, 1986; Uriasyev, 1990). The problem is as follows: ﬁnd a
point x on a plane minimizing the sum of weighted distances up to the given random
points, distributed normally N (η k , dk ):

K
F (x) = βk |x − w| n(w, η k , dk )d w → min,
k=1 R2

where βk = 1, . . . , K, (values of β k , η k , dk are given in (Uriasyev, 1990).
We apply the method of simple iteration again. It is easy to see that:

K
∂F x−w
= βk n(w, η k , dk )d w.
∂x |x − w|
k=1 R2

Equating this gradient to zero the equation of a “ﬁxed point” for the optimal solution can
be derived:
K K
wn(w, η k , dk ) n(w, η k , dk )
x+ = βk dw βk d w.
|x+ − w| |x+ − w|
k=1 R2 k=1 R2


Table 3

iterations number of trials Nt
t

min 5 1226
mean 10.45 3770
max 20 10640

Now the optimization procedure can be constructed by iteratively using this equation
starting from a certain initial approximation x0 :

K K
wn(w, η k , dk ) n(w, η k , dk )
xt+1 = βk dw βk d w. (18)
|xt − w| |xt − w|
k=1 R2 k=1 R2

Substituting this expression into (7), we obtain that the corresponding step length as fol-
lows:

K
n(w, η k , dk )
ρt+1 = 1 βk d w.
|xt − w|
k=1 R2

This problem was solved 400 times by means of algorithm (18) evaluating the in-
tegrals by the Monte-Carlo method, adjusting the sample size according to rule (8), and
stopping according to (12), (13). The initial data were as follows: x = (54, 30), N 0 = 10,
β = 0.95, µ = 0.1, γ = 0.05. The optimum was found in all realizations with a permis-
sible accuracy. The amount of computational resources, needed to solve the problem, is
presented in Table 3.

6. Discussion and Conclusions

An iterative method has been developed to solve the stochastic optimization problems by
a finite sequence of Monte-Carlo samples. This method is grounded by the stopping pro-
cedure (12), (13), and the rule for iterative regulation of size of Monte-Carlo samples (8).
The proposed stopping procedure allows us to test the optimality hypothesis and to eval-
uate the confidence intervals of the objective function in a statistical way. The regulation
of sample size, when this size is taken inversely proportional to the square of the norm
of the gradient of the Monte-Carlo estimate, allows us to solve stochastic optimization
problems rationally from the computational viewpoint and guarantees the convergence
almost surely at a linear rate. The examples confirm theoretical conclusions and show
that the procedures developed permit solution of stochastic optimization problems with a
sufficient permissible accuracy using an acceptable volume of computations.

464 L. Sakalauskas

Appendix

Lemma 1. Let the conditions of Theorem 1 be valid. Then the inequality

2 ρ·L ρ·K
EF x − ρ · G(x) F (x) − ρ · E G(x) 1− + , (1A)
2 N

holds.

Proof. We have from the Lagrange formula (Diedonne, 1960) that

1

F x − ρG(x) = F (x) − ρ G(x) F x − τ · ρ · G(x) d τ
0
2 2
= F (x) − ρ G(x) + ρ G(x) − F (x)

+ρ ( F (x)) G(x) − F (x)
1
−ρ G(x) F x − τ · ρ · G(x) − F (x) d τ. (2A)
0

Formula (1A) follows if we take the expectation of both sides of (2A), apply the
Lipshitz condition and the estimate

2 K
E G(x) − F (x) , ∀x ∈ Rn , ∀N 1, (3A)
N

following, in its turn, from the independence of trials in the Monte-Carlo estimator (5).
The lemma is proved.

Proof of Theorem 1
∞ ∞
Let { t }t=0 be a stream of σ-algebras generated by the sequence {xt }t=0 and let us
introduce a random sequence

ρ·K
Vt = F (xt ) + . (4A)
Nt
1
Assume 0 < ρ L. Then by virtue of (1A) and (3A) we have that

ρ 2 1
E (Vt+1 | t−1 ) Vt − · E G(xt ) | t−1 +ρ · K · E | t−1
2 N t+1
ρ 2K 2
Vt − 1− E G(xt ) | t−1 , t = 1, 2, . . . . (5A)
2 C


It follows that Vt is a semimartingale when 2K < 1. Let C 4K. If now we take un-
C
conditional expectations of both sides of inequality (5A), then, after certain noncomplex
manipulations we get that
t
ρ 2
EF (x0 ) − EF (xt+1 ) + 2 · ρ · K > E G(xk ) .
4
k=0

The left side of this inequality is bounded because F (xt+1 ) F + > −∞, and there-
2 2
fore lim E G(xt ) = 0. In such a case, lim | F (xt )| = 0 (mod(P )), because
t→∞ t→∞
E | F (xt )|2 < E |G(xt )|2 . The proof of the theorem 1 is completed.

Proof of Theorem 2
Let us introduce the Lyapunov function:

2 ρ·K
W (x, N ) = x − x+ + .
N
We have by virtue of the Lagrange formula and (3A) that
2
E xt − x+ | t

2
=E xt − x+ − ρ · ( F (xt ) − F (x+ )) − ρ · (G(xt ) − F (xt )) | t

2 1
(1 − ρ · l)2 E xt − x+ | t−1 +ρ2·K ·E | t−1 . (6A)
Nt

Next, due to (8), the triangle inequality, and (3A) we get
2
1 E | F (xt ) − F (x+ )| K 1
E ·E +
N t+1 C C Nt
L2 2 K 1
·E xt − x+ + ·E . (7A)
C C Nt

Note, we can change the conditional expectations in (6A) by unconditional. Thus, by
virtue of (6A), (7A) we obtain:

E W (xt+1 , N t+1 )
ρKL2 2 K 1
1−2ρl+ρ2l2 + E xt −x+ +ρK ρ+ E √
C C Nt
K · L2
(1 − ρ · l − E W (xt , N t ) (8A)
C
2
1 3
if 0 < ρ min L , 4·(1+l) ,C K max[4, L ]. The proof of the theorem 2 is com-
l
pleted.

466 L. Sakalauskas

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L. Sakalauskas has graduated from the Kaunas Polytechnic Institute (1970), received
the PhD degree from this Institute (1974), Associated Professor (1987), Member of the
New- York Academy of Sciences (1997), presently is a Head of the Statistical Modelling
Group at the Institute of Mathematics and Informatics and an Associated Professor of the
Transport Engineering Department of the Vilnius Gediminas Technical University.

468 L. Sakalauskas

Netiesinis stochastinis optimizavimas Monte-Karlo metodu
Leonidas Sakalauskas
Nagrin˙ jami Monte-Karlo tipo proced¯ ru taikymai stochastinio optimizavimo uždaviniams spresti.
e u
Metodo stabdymo taisykl˙ s ir sprendinio tikslumas yra nustatomi statistiniu b¯ du, tikrinant
e u
statistine optimalumo hipoteze. Ivesta Monte-Karlo imˇ iu dydžio parinkimo taisykl˙ , užtikrinanti
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konvergavima bei stochastio optimizavimo problemos sprendima po baigtinio, praktiniu poži¯ riuu
dažnai priimtino, Monte-Karlo bandymu skaiˇ iaus. Pasi¯ lytas metodas pritaikytas sprendžiant
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reikšmingu imˇ iu ir stochastine Veberio problema.
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Nonlinear Stochastic Optimization by the Monte-Carlo Method

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Nonlinear Stochastic Optimization by the Monte-Carlo Method