Nonlinear stochastic programming by Monte-Carlo estimators

European Journal of Operational Research 137 (2002) 558–573
www.elsevier.com/locate/dsw

Stochastics and Statistics

Nonlinear stochastic programming by Monte-Carlo estimators
Leonidas L. Sakalauskas *

Department of Statistical Modelling, Institute of Mathematics and Informatics, Akademijos 4, Vilnius 2600, Lithuania
Received 18 January 2000; accepted 26 February 2001

Abstract

Methods for solving stochastic programming (SP) problems by a finite series of Monte-Carlo samples are considered.
The accuracy of solution is treated in a statistical manner, testing the hypothesis of optimality according to statistical
criteria. The rule for adjusting the Monte-Carlo sample size is introduced to ensure the convergence and to find the
solution of the SP problem using a reasonable number of Monte-Carlo trials. Issues of implementation of the developed
approach in decision making and other applicable fields are considered too. Ó 2002 Elsevier Science B.V. All rights
reserved.

Keywords: Monte-Carlo method; Stochastic programming; Kuhn–Tucker conditions; Statistical hypothesis

1. Introduction

We consider the standard stochastic programming (SP) problem with a single inequality-constraint

F0 ðxÞ ¼ Ef0 ð x; xÞ ! min;
F1 ðxÞ ¼ Ef1 ð x; xÞ 6 0; ð1Þ
n
x2R ;

where functions fi : Rn Â X ! R, i ¼ 0; 1, satisfy certain conditions on continuity and differentiability,
x 2 X is an elementary event in some probability space ðX; R; Px Þ, E is the symbol of mathematical ex-
pectation. Assume that we have a possibility to get finite sequences of realizations of x and that values of
functions f0 ; f1 are available at any point x 2 Rn for any realization of x 2 X.
The constraint optimization with mean-valued objective and constraint functions occurs in many ap-
plied problems of engineering, statistics, finances, business management, etc. Stochastic procedures for
solving problems of this kind are often considered and several concepts are proposed to ensure and improve

*
Tel.: +370-2-729312; fax: +370-2-729209.
E-mail address: sakal@ktl.mii.lt (L.L. Sakalauskas).

0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 7 - 2 2 1 7 ( 0 1 ) 0 0 1 0 9 - 6

L.L. Sakalauskas / European Journal of Operational Research 137 (2002) 558–573 559

the convergence behavior of developed methods (Ermolyev and Wets, 1988; Shapiro, 1989; Kall and
Wallace, 1994; Prekopa, 1995). The concept of stochastic approximation is quite well studied, when the
convergence is ensured by varying certain step-length multipliers in a scheme of stochastic gradient search
(see, e.g., Ermolyev, 1976; Mikhalevitch et al., 1987; Ermolyev and Wets, 1988; Uriasyev, 1990; Kall and
Wallace, 1994; Dippon, 1998). However, the following obstacles are often mentioned in the implementation
of stochastic approximation:
• It is not so clear when to stop the process of stochastic approximation.
• The methods of stochastic approximation converge rather slowly.
Improvement of convergence behavior using more accurate estimators is a well-known concept,
which has already found an application in a semi-stochastic approximation (Ermolyev and Wets, 1988;
Uriasyev, 1990; Kall and Wallace, 1994). However, the obstacles mentioned above occur here too. In
this paper, we attempt to develop this concept starting from a theoretical scheme of methods with a
relative stochastic error (see Polyak, 1983). The main requirement of this scheme is that the variance of
the stochastic gradient be varied during the optimization procedure so that it would remain propor-
tional to the square of the gradient norm. We show that such an approach offers an opportunity to
develop implementable algorithms of stochastic programming, using a finite series of Monte-Carlo
estimators for the construction of gradient-type methods, where the accuracy of estimators is adjusted
in a special way.
We briefly survey stochastic differentiation techniques and also introduce a set of Monte-Carlo esti-
mators for SP. The accuracy of the solution is treated in a statistical manner, testing the hypothesis of
optimality according to statistical criteria. Although we pay special attention to the case of the Gaussian
measure Px because of its importance in applications, the technique developed could be successfully ex-
tended to other distributions as well as to other cases of SP, that differ from those considered here. The
developed methods are studied numerically in solving test examples and problems in practice (see also
Sakalauskas, 1992, 1997, 2000; Sakalauskas and Steishunas, 1993).

2. Stochastic differentiation and Monte-Carlo estimators

The gradient search is the most often used way of constructing methods for numerical optimization.
Since mathematical expectations in (1) are computed explicitly only in rare cases, all the more it is com-
plicated to analytically compute the gradients of functions, containing this expression. The Monte-Carlo
method is a universal and convenient tool of estimating these expectations and it could be applied to es-
timate derivatives too. The procedures of gradient evaluation are often constructed expressing a gradient as
an expectation and then evaluating this expectation by means of statistical simulation (see Rubinstein,
1983; Kall and Wallace, 1994; Prekopa, 1995; Ermolyev and Norkin, 1995).
Let us introduce a set of Monte-Carlo estimators needed for construction of a stochastic optimization
procedure. Assume without loss of simplicity the measure Px to be absolutely continuous, i.e. it can be
defined by the density function p : Rn Â X ! Rþ . Assume, also, that values of p are available at any point
x 2 Rn for any realization of x 2 X. First, let us consider the expectation
Z
F ðxÞ ¼ Ef ðx; xÞ f ðx; yÞ Á pðx; yÞdy; ð2Þ
Rn

where the function f and the density function p are differentiable with respect to x in the entire space Rn . Let
us denote the support of measure Px as

SðxÞ ¼ fy j pðx; yÞ 0g; x 2 Rn :

560 L.L. Sakalauskas / European Journal of Operational Research 137 (2002) 558–573

Then it is not diﬃcult to see that the vector-column of the gradient of this function could be expressed as
rF ðxÞ ¼ Eðrx f ðx; xÞ þ f ðx; xÞ Á rx ln pðx; xÞÞ ð3Þ

(we assume rx ln pðx; yÞ ¼ 0, y 62 SðxÞ). We also see from the equation
Erx ln pðx; xÞ ¼ 0
R
(which was obtained by diﬀerentiating the equation X
pðx; yÞdy ¼ 1) that various expressions of the gra-
dient follow. For instance, the formula
rF ðxÞ ¼ Eð rx f ðx; xÞ þ ðf ðx; xÞ À f ðx; ExÞÞ Á rx ln pðx; xÞÞ ð4Þ

serves as an example of such an expression.
Thus we see that it is possible to express the expectation and its gradient through a linear operator from
the same probability space. Hence, operators (2)–(4) can be estimated by means of the same Monte-Carlo
sample. It depends on the task solved, which formula, (3) or (4), is better for use. For instance, expression (4)
can provide smaller variances of gradient components than (3), if the variances of x components are small.
Now, assume the Monte-Carlo sample to be given for some x 2 D Rn :

Y ¼ ðy 1 ; y 2 ; . . . ; y N Þ; ð5Þ

where y i are independent random vectors identically distributed with the density pðx; ÁÞ : X ! Rþ . We have
the estimates

~ 1 XN
Fi ðxÞ ¼ fi ðx; y j Þ; ð6Þ
N j¼1

1 X N
~
D2i ðxÞ ¼
F ðfi ðx; y j Þ À Fi ðxÞÞ2 ;
~ i ¼ 0; 1: ð7Þ
N À 1 i¼1
If we introduce the Lagrange function
Lðx; kÞ ¼ F0 ðxÞ þ k Á F1 ðxÞ; ð8Þ

which may be treated as an expectation of the stochastic Lagrange function
lðx; k; xÞ ¼ f0 ðx; xÞ þ k Á f1 ðx; xÞ;

then the estimate of the gradient of the Lagrange function

~ 1 X j
N
rx Lðx; kÞ ¼ G; ð9Þ
N j¼1

could be considered according to (3), as the average of identically distributed independent vectors
Gj ¼ rx lðx; k; y j Þ þ lðx; k; y j Þ Á rx ln pðx; y j Þ;

where EGj ¼ rLðx; kÞ, j ¼ 1; . . . ; N . The sampling covariance matrix

1 X j ~
N
~ 0
A¼ ðG À rx LÞ Á ðGj À rx LÞ ð10Þ
N j¼1

will be used later on too.


3. The rule for iterative adjustment of sample size

The Monte-Carlo sample size N should be chosen to compute the estimators introduced in the previous
section. Sometimes this sample size is taken fixed and sufficiently large to ensure the required accuracy of
estimates in all the iterations of the optimization process. Very often this size is about 1000–1500 trials or
still more, and, if the number of optimization steps is large, solution of an SP problem can require a great
amount of computations (Jun Shao, 1989). Besides, it is well known that the fixed sample size, although
very large, is sufficient only to ensure the convergence to some neighborhood of the optimal point (see, e.g.,
Polyak, 1983; Sakalauskas, 1997).
Note that there is no great necessity to estimate functions with a high accuracy on starting the opti-
mization, because then it suffices to evaluate only an approximate direction leading to the optimum.
Therefore the samples should not be taken large at the beginning of optimization, adjusting their sizes so as
to obtain the estimates of the objective and constraint functions with a desired accuracy only at the moment
of decision making on the solution finding. It is mentioned above that the concept of using more accurate
estimators has already found an application in semi-stochastic approximation methods. We develop here
this concept in an iterative scheme of methods with a relative stochastic error (see Polyak, 1983), choosing
at every next iteration the sample size inversely proportional to the square of norm of the gradient estimate
from the current iteration. It can be proved under some conditions that such a rule enables us to construct
the converging stochastic methods of SP.
Let xþ 2 Rn be now the solution of SP problem (1). By virtue of the Kuhn–Tucker theorem (see, e.g.,
Bertsekas, 1982) there exist values kþ P 0; rF0 ðxþ Þ; rF1 ðxþ Þ such that

rF0 ðxþ Þ þ kþ Á rF1 ðxþ Þ ¼ 0; kþ Á F1 ðxþ Þ ¼ 0: ð11Þ

In SP, the well-known gradient-type procedures of constrained optimization can be applied in iterative
search of the Kuhn–Tucker solution (Kall and Wallace, 1994; Ermolyev and Wets, 1988; Mikhalevitch et
al., 1987). We analyze the peculiarities of realization of such methods constructing a stochastic version of
the well-known Arrow–Hurvitz–Udzava procedure.

Theorem 1. Let the functions Fi : Rn ! R, i ¼ 0; 1, expressed as expectations (3), where x 2 X is an event
from the probability space ðX; R; Px Þ, Px is an absolutely continuous measure with the density function
p : Rn Â X ! Rþ , be convex and twice differentiable. Assume that all the eigenvalues of the matrix of second
derivatives r2 Lðx; kþ Þ, 8x 2 Rn , are uniformly bounded and belong to the interval ½m; MŠ, m 0, and, besides,
xx

m
jrF1 ðxÞ À rF1 ðxþ Þj 6 jrF1 ðxþ Þj 8x 2 Rn ; jrF1 ðxþ Þj 0;
M
where ðxþ ; kþ Þ is a point satisfying the Kuhn–Tucker conditions (11).
Let us state, in addition, that for any x 2 Rn and any number N P 1 there exists a possibility to obtain
estimates (7) and

~ 1 XN
rx Fi ðxÞ ¼ Gi ðx; y j Þ;
N j¼1

where fy j gN is a sample of independent vectors, identically distributed with the density pðx; ÁÞ : X ! Rþ ,
1
EGi ðx; xÞ ¼ rx Fi ðxÞ, and the conditions of uniform boundedness on variances are valid:
2 2
Eðf1 ðx; xÞ À F1 ðxÞÞ d; EjGi ðx; xÞ À rFi ðxÞj K;
8x 2 Rn ; i ¼ 0; 1:


Let now an initial point x0 2 Rn , the scalar k0 , and an initial sample size N 0 be given and the random se-
1
quence fxt ; kt gt¼0 be defined according to
~
xtþ1 ¼ xt À q Á rx Lðxt ; kt Þ;
ð12Þ
~
ktþ1 ¼ kt þ q Á a Á F1 ðxt Þ;

where estimates (6), (9) are used varying the sample size N t according to the rule
C
N tþ1 P ; ð13Þ
~ x Lðxt ; kt Þj2
q Á jr

q 0, a 0, C 0 are certain constants. Then, there exist positive values q, C such that
!
2
þ 2 jkt À kþ j 1
t
E jx À x j þ þ t B Á bt ; t ¼ 1; 2; . . . ;
a N

for certain values 0 b 1, B 0, as q q, C C .

The proof is given in Appendix A.

2
Remark. An interesting comment follows. As we can see, the error jxt À xþ j and the sample size N t have a
linear rate of changing, conditioned by the value 0 b 1, which depends mostly on the constant C in
expression (13) and conditionality of the Hessian of Lagrange function. It follows from the proof of the
theorem that N t =N tþ1 % b for large t. Then, by virtue of the formula of geometrical progression
X
t
Q
Ni % Nt Á ;
i¼0
1Àb

where Q is a certain constant.

Note now that the stochastic error of estimates depends on the sample size at the moment of the
stopping decision, say N t , and, in its turn, the accuracy of the solution. Thus, the ratio of the total number
Pt
of computations i¼0 N i with the sample size at the stopping moment N t can be considered as bounded by
a certain constant and not dependent on this accuracy. Thus, if we have a certain resource to compute one
value of the objective or constraint function with an admissible accuracy, then the optimization requires in
fact only several times more computations. This enables us to construct reasonable, from a computational
viewpoint, stochastic methods for SP. The procedure (12) and the rule (13) are developed further in
Section 4.
The choice of the best metrics for computing the norm in (13) or other parameters in stochastic gradient
procedures of the considered kind requires a separate study. The theorem proved is important in principle
and the proof could be extended without much difficulty to the case of unconstrained optimization as well
as that of optimization with several constraints (see also Sakalauskas, 1997).

4. The algorithm for SP by the finite series of Monte-Carlo samples

While implementing the method, developed in the previous section, we have to keep in mind that the
performance and stopping of computation of (12) and (13) could be done only on the base of finite series of


Monte-Carlo samples (5). In such a case, the application of methods of the theory of statistical decisions is
a constructive way of building an implementable method for SP.
A possible decision on finding the optimal solution should be examined at each iteration of the opti-
mization process. Namely, the decision on optimum finding could be made, if, first, there is no reason to
reject the hypothesis on the validity of Kuhn–Tucker conditions (11), and, second, the objective and
constraint functions are estimated with admissible accuracy. Since the distribution of introduced estimators
(5) and (9) can be approximated by the one- and multivariate Gaussian law, when the Monte-Carlo sample
size is sufficiently large (see, e.g., Bhattacharya and Rao, 1976; Box and Watson, 1962; Bentkus and Gotze,
1999), it is convenient to use the well-known methods of normal sample analysis for our purposes.
Thus, the validity of the first optimality condition in (11) could be tested by means of the well-known
multidimensional Hotelling T 2 -statistics (see, e.g., Krishnaiah and Lee, 1980). Namely, the optimality
hypothesis could be accepted for some point x with the significance l, if the following condition vanishes:

~ 0 ~
ðN À nÞ Á ðrx LÞ Á AÀ1 Á ðrx LÞ=n 6 Fishðl; n; N À nÞ; ð14Þ

where Fishðl; n; N t À nÞ is the l-quantile of the Fisher distribution with ðn; N t À nÞ degrees of freedom,
~ ~
rx L ¼ rx Lðx; kÞ is the estimate (9) of the Lagrange function gradient, and A is the normalizing matrix (10)
estimated at the point ðx; kÞ, N is the size of sample (5). In the opposite case this hypothesis is rejected.
Now we make two important remarks related to the statistical error of estimator (6) of the objective and
constraint functions. Since the real values of these functions remain unknown to us and we can operate only
with their estimators, produced by statistical simulation, first of all, the upper confidence bound of the
constraint function estimate has to be used for performance of (12) and for testing the second optimality
condition in (14), and, second, the sample size N has to be sufficiently large in order to estimate the
confidence intervals of the objective and constraint functions with an admissible accuracy. It is convenient
here to use the asymptotic normality and to approximate the respective confidence bounds by means of the
sampling variance (7).
Summarizing all such considerations, we propose the following algorithm for SP by finite Monte-Carlo
series. Assume some initial value k0 P 0, some initial point x0 2 Rn to be given, random sample (5) of a
certain initial size N 0 be generated at this point, and Monte-Carlo estimates (6), (7), (9), and (10) be
computed. Let us find a solution such that the statistical hypothesis on the validity of the first Kuhn–Tucker
condition (11) would not be rejected and the objective and constraint functions would be evaluated with
permissible confidence intervals.
Then, such a version of procedure (11) could be applied in stochastic solving of (1):

~
xtþ1 ¼ xt À q Á rx Lðxt ; kt Þ;
ð15Þ
~ ~
ktþ1 ¼ max½0; kt þ q Á a Á ðF1 ðxt Þ þ gb Á DF1 ðxt ÞÞŠ;

where q; a, are the normalizing multipliers, gb is the b-quantile of the standard normal distribution. Note
~ ~
that we introduce an estimate of the upper confidence bound F1 ðxt Þ þ gb Á DF1 ðxt Þ, taking into account the
statistical error of estimate (6) of the constraint function and using the normal approximation.
As mentioned above, we can choose the constant C and the metric to compute the norm while the rule
(13) is implemented. A possible way is to compute the estimate of the gradient norm in the metric, induced
by the sampling covariance matrix (10), and to set C ¼ n Á Fishðc; n; N t À nÞ. This is convenient for inter-
pretations, because, in such a case, the random error of the gradient does not exceed the gradient norm
approximately with probability 1 À c. Such a modification of (13) is as follows:

n Á Fishðc; n; N t À nÞ
N tþ1 P : ð16Þ
q Á ðrx Lt Þ0 Á ðAt ÞÀ1 Á ðrx Lt Þ
~ ~


In numerical realization it is recommended to introduce the minimal and maximal values Nmin (usually
$20–50) and Nmax to avoid great fluctuations of sample size in iterations.
Thus, the procedure (15) is iterated adjusting the sample size according to (13) or (16) and testing the
optimality conditions at each iteration. If:
(a) Criterion (14) does not contradict the hypothesis that the gradient of the Lagrange function is equal
to 0 (first condition in (11)).
(b) The constraint condition (second condition in (11)) vanishes with a given probability b:
~ ~
F1 ðxt Þ þ gb Á DF1 ðxt Þ 6 0:

(c) The estimated lengths of the confidence interval of the objective function and the constraint one do
not exceed the admissible accuracy ei :
pffiffiffiffi
2gbi Á DFi = N 6 ei ; i ¼ 0; 1;

where gbi is the bi -quantile of the standard normal distribution, there are no reasons to reject the hypothesis
about the optimum finding. Therefore, there is a basis to stop the optimization and to make a decision on
the optimum finding with an admissible accuracy. If one at least condition out of three is not satisfied, then
a next sample is generated and the optimization is continued. As follows from Section 3, the optimization
process should stop after generating a finite number of samples (5).
Also 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 standard points could be introduced,
considering the rate of convergence to the normal law and following from the Berry–Esseen or large de-
viation theorems.
An extension of the developed approach to the case of several inequalities-constraints is related, first,
with the consideration of joint confidence statements when the statistical error of constraint estimates is
taken into account. The simplest way is to consider a vectorial case of (12) or (15) and, separately for each
constraint, to compute the corresponding confidence bounds in (15) and condition (c). The case of
equalities-constraints is reduced to the one considered here, changing each equality by two inequalities.
Such remarks should be also taken into consideration applying our approach to other cases of SP as well as
to constraint optimization procedures, differing from that of Arrow–Hurwitz–Udzava.

5. Numerical study of the stochastic optimization algorithm

Let us discuss a numerical experiment to study the proposed algorithm. Since typical functions are often
in practice of a quadratic character with some nonlinear disturbance in a neighborhood of the optimal
point, we consider a test example of SP with functions of this kind

F ðxÞ Ef0 ðx þ xÞ ! min;
ð17Þ
P ðf1 ðx þ xÞ 6 0Þ À p P 0;

where
X
n X
n
f0 ðyÞ ¼ ðai yi2 þ bi Á ð1 À cosðci Á yi ÞÞÞ; f1 ¼ ðyi þ 0:5Þ;
i¼1 i¼1

yi ¼ xi þ xi ; xi are random and normally Nð0; d 2 Þ distributed, d ¼ 0:5, a ¼ ð8:00; 5:00; 4:30; 9:10;
1:50; 5:00; 4:00; 4:70; 8:40; 10:00Þ, b ¼ ð3:70; 1:00; 2:10; 0:50; 0:20; 4:00; 2:00; 2:10; 5:80; 5:00Þ, c ¼ ð0:45; 0:50;
0:10; 0:60; 0:35; 0:50; 0:25; 0:15; 0:40; 0:50Þ and n 6 10.


Such an example may be treated as a stochastic version of the deterministic task of mathematical
programming with the objective function f0 and the constraint function, f1 where the controlled variables
are measured under some Gaussian error with the variance d 2 . Note that our task is not convex.
The stopping criteria introduced above essentially depend on the rate of convergence of the distribution
of estimates (7) and (11) to the Gaussian law (see also Sakalauskas and Steishunas, 1993). We study the
distribution of statistic in criterion (14) by means of statistical simulation, solving the test task, when
p ¼ 0; n ¼ 2. The optimal point is known in this case: xþ ¼ 0 Let the gradient of considered functions be
evaluated numerically by the Monte-Carlo estimator following from (3). Thus, 400 Monte-Carlo samples of
size N ¼ ð50; 100; 200; 500; 1000Þ were generated and the T 2 -statistics in (15) were computed for each
sample. The hypothesis on the difference of empirical distribution of this statistic from the Fisher distri-
bution was tested according to the criteria x2 and X2 (see Bolshev and Smirnov, 1983). The value of the first
criterion for N ¼ 50 is x2 ¼ 0:2746 at the optimal point against the critical value 0.46 (p ¼ 0:05), and that
of the next one is X2 ¼ 1:616 against the critical value 2.49 (p ¼ 0:05). Besides, the hypothesis on the co-
incidence of empirical distribution of the considered statistics from the Fisher distribution was rejected at
the points differing from the optimal one according to the criteria x2 and X2 (if r ¼ jx À xþ j P 0:1). So the
distribution of multidimensional T 2 -statistics, in our case, can be approximated sufficiently well by the
Fisher distribution with even not large samples ðN ffi 50Þ.
Further, the dependencies of the stopping probability according to criterion (14) on the distance
r ¼ jx À xþ j to the optimal point were studied. These dependencies are presented in Fig. 1 (for confidence
l ¼ 0:95). The same dependencies are given in Fig. 2 for N ¼ 1000 and l ¼ ð0:90; 0:95; 0:99Þ. So we see that
by adjusting the sample size, we are able to test the optimality hypothesis in a statistical way and to evaluate
the objective and constraint functions with a desired accuracy.

Fig. 1. Stopping probability according to (14), l ¼ 0:95.

Fig. 2. Stopping probability according to (14), N ¼ 1000.


Now, let us consider the results, obtained for this example, by iterating procedure (15) 40 times and
changing the sample size according to the rule (16), where N 0 ¼ Nmin ¼ 50, and Nmax is chosen according to
the restrictions on the confidence bounds of the objective and constraint functions. The initial data were
chosen or varied: n ¼ ð2; 5; 10Þ, probability in (17) (i.e., p ¼ ð0:0; 0:3; 0:6; 0:9Þ), initial point (x0 ¼
ðx0 ; x0 ; . . . ; x0 Þ, x0 ¼ À1; 1 6 i 6 n), multipliers in (15) ða ¼ 0:1; q ¼ 20), probabilities in (b), (c) and (15)
1 2 n i
(b ¼ b0 ¼ b1 ¼ 0:95), confidence in (14) ðl ¼ 0:95Þ, probability in (16) ðc ¼ 0:95Þ, lengths of confidence
~
intervals in (c) (e0 ¼ 5% (from F ðxt Þ), e1 ¼ 0:05). The optimization was repeated 400 times. Conditions (a)–
(c) were satisfied one time for all paths of optimization. Thus, a decision could be made on the optimum
finding with an admissible accuracy for all paths (the sampling frequency of stopping after t iterations with
confidence intervals is presented in Fig. 3 ðp ¼ 0:6; n ¼ 2Þ). Minimal, average and maximal values of the
number of iterations and those of the total Monte-Carlo trials are presented in Table 1 that were needed to
solve the optimization task ðn ¼ 2Þ, depending on the probability p. The number of iterations and that of
total Monte-Carlo trials, needed for stopping, are presented in Table 2 depending on the dimension
n ðp ¼ 0; n P 2). The sampling estimates of the stopping point are ~stop ¼ ð0:006 Æ 0:053; À0:003 Æ 0:026Þ
x
for p ¼ 0; n ¼ 2 (compare with xopt ¼ ð0; 0Þ). These results illustrate that the algorithm proposed can be
successfully applied when the objective and constraint functions are convex and smooth only in a neigh-
borhood of the optimum.

Fig. 3. Frequency of stopping (with confidence interval).

Table 1
n¼2
P
P Number of iterations, t Total number of trials ð t Nt Þ
Min Mean Max Min Mean Max
0.0 6 11:5 Æ 0:2 19 1029 2842 Æ 90 7835
0.3 4 11:2 Æ 0:3 27 1209 4720 Æ 231 18 712
0.6 7 12:5 Æ 0:3 29 1370 4984 Æ 216 15 600
0.9 10 31:5 Æ 1:1 73 1360 13100 Æ 629 37 631

Table 2
p ¼ 0; n P 2
P
n Number of iterations, t Total number of trials ð t Nt Þ
Min Mean Max Min Mean Max
2 6 11:5 Æ 0:2 19 1029 2842 Æ 90 7835
5 6 12:2 Æ 0:2 21 1333 3696 Æ 104 9021
10 7 13:2 Æ 0:2 27 1405 3930 Æ 101 8668


~ ~ ~
The averaged dependencies of the objective function EF0t , the constraint EF1t , the sample size EN t , the
~ ~
Lagrange multiplier Ekt and the solution point Ext depending on the iteration number t are given in Figs. 4–
8 to illustrate the convergence and the behavior of the optimization process ðp ¼ 0:6; n ¼ 2Þ. Also, one
path of realization of the optimization process illustrates the stochastic character of this process in these
ﬁgures.
Thus, the theoretical analysis and numerical experiment show that the proposed approach enables us to
solve SP problems with a suﬃcient accuracy using a reasonable amount of computations (5–20 iterations

Fig. 4. Change of the objective function.

Fig. 5. Change of the constraint function.

Fig. 6. Change of the sample size.


Fig. 7. Change of the Lagrange multiplier.

Fig. 8. Change of the point of iteration: (Â) average, (Ã) path.

and 3000–10 000 total Monte-Carlo trials). If we keep in mind that the application of the Monte-Carlo
procedure usually requires 1000–2000 trials in statistical simulation and estimation of one value of the
function, the optimization by our approach can require only 3–5 times more computation.

6. Application to decision support in investment planning

The investment planning in many ﬁelds is related with the control of risk factors in the case of uncer-
tainty. It is often possible to describe this uncertainty by probabilistic models and thus to reduce the de-
cision making problem in ﬁnance planning to SP problems. Namely, the objective function could have a
meaning of average losses that should be optimized under the constraint on the probability of the desired
scenario of events.
Let us consider, as a typical example, the investment planning in the gas delivery (see Guldman, 1983;
Ermolyev and Wets, 1988). Omitting some details, the task is as follows. Choose x ¼ ðx1 ; . . . ; x12 Þ; z, to
minimize the expected costs of purchases, storage operations, and the increment of storage capacity,

X
12 X
12
F ðx; zÞ ¼ ct Á xt þ c0 Á E jxt À xt j þ c13 Á maxfxt g þ c14 Á z;
t
t¼1 t¼1

where xt is the amount of gas ordered from pipeline per month t ¼ 1; 2; . . . ; 12; z is the increment of the gas
storage, subject to the condition on probability of a reliable supply,


8

ðxt À xt Þ À a1 Á s¼1 ðxs À xs Þ 6 b1 þ a2 Á z

PtÀ1 =
Uðx; zÞ ¼ p À P ðAÞ p À P x

ÀðxtPt xt Þ À a3 Á s¼1 ðxs À xs Þ 6 b2 þ a4 Á z 6 0;
À

Nonlinear stochastic programming by Monte-Carlo estimators

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