Approximate stochastic systems of linear algebraic equations

U.S.S.R. Comput.Maths.Math.Phys.,Vol.29,No.6,pp.122-130,1989 Printed in Great Britain

0041-5553/89 $i0.00+0.00 0 1 9 9 1 Pergamon Press plc

APPROXIMATE STOCHASTIC SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS* A.I. ZHDANOV

The solution of ill-posed stochastic systems of linear algebraic equations with errors in the right-hand side and errors in some of the coefficients is considered. The existence and uniqueness conditions are derived using the statistical form of the generalized discrepancy principle. The effect of a priori information on the disturbances on the statistical properties of the solution estimates is investigated. Efficient computational algorithms are proposed and various techniques of selecting the discrepancy level according to the nature of the a priori information are examined.

IntPoduction The problem of solving ill-posed stochastic systems of linear algebraic equations by the regularized maximum-likelihood method was considered in /i/. The perturbations on the right-hand side and in the coefficient matrix were assumed to be stochastic with a normal distribution /i/. It was also noted /2/ that the solutions obtained by the regularized maximum-likelihood method for this problem with stochastic errors correspond to the solutions obtained by Tikhonov's regularized least-squares method /3, 4/ for the problem with deterministic errors. The regularization parameter was computed in /I/ from the maximum-likelihood equation. This equation is essentially a statistical form of the generalized discrepancy principle /5, 6/, which is also used in /3, 4/ for the problem with deterministic disturbances. The derivation of the discrepancy equation by the maximum-likelihood method /2/ relies on the asymptotic properties of the method and the normal distribution of the errors. In practice, however, regularization is most effective for a finite number of equations, and not asymptotically, and also with a non-normal or a priori unknown error distribution. This suggests that we should consider the statistical form of the generalized discrepancy principle for solving ill-posed stochastic systems of linear algebraic equations which is not related to the maximum-likelihood method. The best known statistical regularization methods /7-9/ do not use the discrepancy principle. However, a priori information on the statistical properties of the disturbances is usually insufficient for the application of the methods considered in /7-9/, and additional a priori information is required on the unknown solution. Moreover, the properties of the solution estimates obtained by these methods are usually only asymptotic. All this suggests that the discrepancy" principle should be restored to its leading role for solving stochastic regularization problems. In this paper, we derive the statistical form of the generalized discrepancy principle using a priori information on the first two moments of the disturbances in the right-hand side and in the coefficient matrix and also on the inconsistency measures of the exact system. We consider some applications of the statistical form of the generalized discrepancy principle when there is no a priori information on the variance of the disturbances. Many applied problems, for example, the problem of identifying the parameters of discrete transfer functions /9/, involve systems with inexactly defined right-hand side and a coefficient matrix which is partially defined with errors and partially without errors. The corresponding problem is sufficiently different from that considered in /I/ to justify an independent analysis. Moreover, the problem in this form is more general than that in

111. In applied problems, the exact system may be inconsistent. For deterministic disturbances, such problems were studied in detail in /5, i0/, whereas for stochastic disturbances consistency of the exact system is conventionally assumed, either explicitly or implicitly. Our paper focuses on the solution of all these problems. The treatment largely relies on the orthogonal projection method and on other results of /2/.

I. Statement of the problem. Consider the system of linear algebraic equations

Au= A where A is an (n)
~Zh.uychisI.Hat.~t.Fiz.,29,12,1776-1787,1989 122

123 We assume that the elements of the perturbation matrix A A and the perturbation vector A~ are random variables that satisfy the following two conditions (a.s. stands for almost surely).

Condition 2.

M(Ab,[~r,_,)=0 a.s.

ConditionL

D(Ab,19-,_,)=diag{~,' ..... o,~',at'} a.s.

Here Ab,=(Aa, ..... A~=, A])'; At, is the u-algebra, 5r,=a{Ab, .... ,Ab,}, induced by the random variables in braces, f=|, 2,...,n, and ~rQ=Z; a.a*=0 for all j=i, 2 .... ,m--s, 0 < ~'<~

for all

]=m--s+J,...,m

and

at'>0;

the superior index T

denotes the transpose.

Remark 1. In what follows, we will assume, without loss of generality, that a2=eo'= et'>O for all /=m-,+i .....m. Otherwise, if the variances a,,'.....a,~*, at' or their ratios are known a priori, we can always normalize the elements of the matrix AA and the vector AL as in /i/. This paper, unlike /i/, does not assume independence and normality of the elements of the sequence (Ab,)~>t of the perturbation vectors. Condition 1 implies that the sequence (&b,,Srv,)~, is a martingale difference /ii/. This condition is less restrictive than the condition of independence of a sequence of random variables and does not require identical distribution of the sequence elements, which is very important in many applications. The problem of solving an approximate system of linear algebraic equations with deterministic disturbances is formulated in /3, 4/. In the deterministic formulation of the problem /3, 4/, the class of systems that are accuracy-equivalent to some individual system is defined by quadratic norms. This metric is unsuitable for specifying the accuracy of stochastic systems and it must be replaced with some probabilistic metric. We naturally specify the accuracy of the approximate stochastic system by the quadratic norm averaged relative to some probabilistic measure (mean-square norm), i.e., the variance o2 of the errors in the matrix elements and in the right-hand side vector. The exacy system a.u=l. (i.l) may be deterministic of stochastic. unsolvable, i.e.,

An exact deterministic system (i.i), in general, may be

The quantity ~A is obviously a measure of the inconsistency of the .exact system (i.i). Then the normal pseudosolution ~' is defined as = arginf II~ll,

where U is the set of pseudosolutions of system (i.i). For a stochastic system (i.i), we assume that

M(hIA.) =a,u,

D (&lA.)=o0t~0,

/ = t , 2 . . . . . n,

(t.2)

where ~, l=i, 2,...,n, are the components of the vector ~" and at are the rows of the matrix A,~H.=, H.~ is the set of nXm rectangular real matrices. In this case, the normal pseudosolution of the stochastic system (i.i) is naturally defined as fi = arg inf ~ u ~,

where

U

is the set of pseudosolutions of the system

A.u=M(IJAJ. If we {~I} are stochastic stochastic

additionally assume that the collections of random variables {A,, I~, {AA}, and statistically independent of each other, then the approximate system with an exact system satisfying the conditions (1.2) may be formally regarded as an approximate system with the exact deterministic system

A.u=l~, where /~=M(~IA.), of'=D(/,IA.)=e,'+at', i=|, 2 ..... n, and ~A'=HA.u--I~[[=0. From the computational point of view, the problem of solving the approximate system with an exact stochastic system is obviously a special case of the problem of solving an approximate stochastic system with a deterministic exact system when ~a>0. Therefore, in what follows, we will consider this specific type of approximate stochastic szstem. We will thus assume that the approximate stochastic system is defined using the individual system (Z, [), the inconsistency measure ~,, a n d t h e variance o' characterizing the accuracy of the approximate system, and also properties 1 and 2 characterizing a certain statistical regularity of the disturbances. 2. Consistent estimation of normal pseudoso~utions. In this section, we assume that k experimental (perturbed) observations are available

124 for each row of the agumented matrix B,----(A~--].). Thus, the approximate matrix g is knXm and the vector [ is kn-dimensional, where k is the number of experimental observations of the elements of the original system of equations. Consider the problem of consistent estimation of the normal pseudosolutions ~ of approximate stochastic systems of equations (which in general are inconsistent, i.e., M,>0) as k ~ . Since we consider stochastically perturbed systems (~, [),, consistency of the estimates of normal pseudosolutions is understood in the sense of convergence in some probabilistic sense of s sequence of estimates to ~ as k - - ~ . In what follows, we consider strong consistency, i.e., convergence with probability i. We will calculate consistent estimates by a modification of the orthogonal-projection method (OPM), which in what follows is called the quasi-orthogonal projection method (QOPM). According to QOPM, the estimates of the normal pseudosolutions ~ are computed by solving the extremal problem

I(u),

inf

(2.t)

u ~ p , va

where

l(u)

k-'llff~U-foll'

u'Da+i

'

D ~ - d i a g { 0 , . . . , 0 , t , . . . , t), ~,n--s

A~ =

0l, m-$

l o = ( f ' , o . . . . . of, I

9

'IIAE,

t

is the identity matrix of order 8, and 0,,~-, is the w matrix of zeros. The main feature of the QOPM extremal problem (2.1) that distinguishes it from the OPM extremal problem considered in /i/ is that the function u'Du is not strictly positive definite and, as a consequence, the solution of problem (2.i) cannot be reduced to the solution of the extremal problem for the corresponding Rayleigh ratio. The solution of the extremal problem (2.1) relies on the following lemmas. E.

Le~rr~

2.

~,mln(Ba'rB~t!> O,

7%----- inf I (u)----I k-*lma* [(Ba'B")-*D']'

where B,

is the agumented matrix B~=(ga,--f0), D =diag{0,...,0, i,...,i}. m--s

Proof.

Define the a u x i l i a r y

s+t

function

v(v)=(B.'B~v, v)/(D.v, v), The proof of the equality

~.=~,

v ~ "+'.

where --'--k-*

inf

"v(v),

rE~m+l\{o)

is similar to the proof of Lemma 1 in /i/. Thus, in order to prove our lemma, we need to investigate ~. If X~,n(B;B~)>0, then the pencil of quadratic forms (B~'B~v, v) /13/, and therefore ~ is the least root of the equation

det(k-'B;B~-~D.)

--%(D.u, v)

=0.

is regular (2.2)

In this case, we clearly have X.=X=k-'I~[

(B~B~)-'D.].

If

X=,n(B~'B,)=0, then two cases are possible. In the first case, the matrix pencil is regular /13/, i.e., det(BJB,-~D.)~O. Then the minimum root of Eq..(2.2) is zero, and the corresponding characteristic vector y. has the property (D.u..v.)>0. Since (BdB~v., v.)=0, we have X~0. If the matrix pencil B~'B~--~D. is singular, i.e., det(B,'B~-~D.)--0, the minimum root of Eq. (2.2) is also zero, but (D.v.,v.)=0 and v(v.) is an indeterminate form. Partition the matrix B~B, into blocks,

BJB~--XD.

,

| Z,

B~ B. = IW-r

[ Z,

l

I z, I z, |

where

Zt is an

(sXs)-matrix and

Z, is an

[(m--s+i)X(m--s+i)]-:matrix.

125 By the singularity of the matrix pencil B~B~-LD., we have detZ,=0. Introduce the vector u'=(0,..r,0, p,)+~.+s, where p ~ ' \ { 0 ) is the eigenvector of the matrix Za corresponding to the zero eigenvalue, i.e., Z,p=O. Then v(v')=[Z,p, p)](p, p)=O. Therefos in this case also L = ~ = 0 . The lemma is proved.

CoroZZavy.

If s = m ,

then

~"=

D.=E.+t

and

0, k-%.,.[(B~'B.)-'I,

or

i f det(B~'B~)=O, i f det(B;B~)r

X.=k-'l,,~ (B,'B,).

The solution of the extremal problem (2.1) is not necessarily unique, and we therefore have to consider the uniqueness conditions.

Lenract 2. The extremal problem (2.1) has a unique solution if and only if this solution is given by

l.<~.,

and

u,= (X,'X~--kl.D)-'XJ/o and

l(uo)=X.,

where ~ . = II k-tl=~[(Jf~'l~)-tDl, 0,

Proof.

if if

J(u)

By the definition of the functional

grad l (u) Hence it follows that u minimizing

~mt,=k%..

X+=

we directly obtain

2k-t - - t (g;.~u-X;]',-kl(u)Du). u'D.u+

](u)

is the solution of the Euler equation

A,';[, -- YmtnDu = A ~ [ o, By Lemma I,

X=,,(I~'X~)>0, ~=,,(2[~'2[,)=0.

7mtn-----kinf l(u)~;~O.

(2.3)

v~ m

From the proof of Lemma i it also follows that

inf (z. A.~rA~.x) ~R"+'..tol (x, Dx) '

(x, B.XB.x) L. =~R,~.~,\infto> (x, D.x) "

Considering the variation over the set of (re+l)-dimensional vectors z with component zero, we obtain that l.<~.. Now, if A. <~., then

(m+l)-th

det (2[;/[~--k~.D)> 0 and there exists a unique solution of the Euler Eq.(2.3) and problem (2.1).

If :~.=~., then

det (I~'2;~'--kL.D) = 0 and the solution of problem (2.1) is not unique. The lemma is proved. We can now state a theorem which enables us to compute consistent estimates of the normal pseudosolutions ~ of systems (i.i).

Theorem I.

Assume that the sequence of perturbation vectors

matrix (~[, [) satisfies Conditions 1 and 2. known a priori, then the estimates

If

(Ab~)~=l

of the augmented

~,z=a~jz=a'>0, ]=m--s+1,..., rn,

and

a,= (.T~'X,-kk.D)+AT.'fo,

By Lemma i,

is (2.4)

where (.)+ is the Moore-Penrose pseudo-inverse, are strongly consistent estimates normal pseudosolutions ~, i.e., ~h-+~ a.s. as k-~oo.

Proof.

~A

of

the

k. is the minimum root of the equation

det ( k-t B~'B~-LD. ) =0. From Conditions 1 and 2 for the sequence (Ab,)~*=x it follows by the strong law of large numbers for uncorrelated random variables /14, p.146] that

lim [k-tB~B~] =/7.'B.+no'D.~--D.~ a . s . , where ~ =

(X~, --~,),

lo=(/:,

Since

o. . . . .

o)',

hi

126

X~'I~=AJA,+~x'D

a n d J:/0=A,V,,

then

and the least root of the characteristic equation

det(Q~--XD.)=9

has the form

Seeing that the eigenvalues are a continuous function of the matrix elements, we obtain lim X,=~mln a . s .

It thus follows that

lira [k-'h(a)l=t(u)=O a . s . , where

1~(u) = (A','A',-kX.D) a-A'Jt,, l(a) =(n~-X:,:D)--,4.'h. Thus, for each given 8 lira P{ sup Ilh(~,)-A(=)ll
CoPoZZaPy.

If

l.<~.,

the estimates obtained by solving the extremal problem (2.1)

are given by ~ = (X:X,--kZ.D) -'X:fo

(2.5)

and they are strongly consistent estimates of the normal pseudosolutions ~. The proof follows directly from Theorem 1 and Lemma 2. Despite the consistency of the asymptotic behaviour of the estimates ~i defined by (2.4), the effect of the parameter ~.>0 leads to a marked deterioration of the conditioning of the matrix III~--kl.E., because its conditioning measure is

Xm,n(IJI,)--kL and the estimates are computationally unstable for finite (especially small) k. It is accordingly relevant to consider the use of additional a priori information for the regularization of the consistent estimates obtained by solving the extremal problem (2.1).

3. Regulaz, i z a t ~ n of the estimate8 of the 8o~ut~on. Instead of the extremal problem (2.1), we will consider the extremal problem

inf F (u), u~ ~m

(3.t)

where

F(u)

k-'llWu-~iP llull:+t '

w --a Ir

Lermnu 3. The Proof. Since

+L

0

-, I ....

- Jl '

~ = ~0", 0 ..... 0f.

extremal problem (2.1) is equivalent to the extremal problem (3.1). ]~=I,'X,+kA.D,

]~'g=X~'fo=~'[

D:diag{l

and

. . . . . t, 0 . . . . .

we conclude that the solutions of the problems det (B~'R~--XD.) = 0 and

[Igli'=[If011'=llfll',where 0},

127

ire equivalent. Then Lemma 3 follows from Lemma i. The lemma is proved. Unlike problem (2.1), problem (3.1) exactly corresponds to OPM and therefore it can be :egularized using the results of /I/. We introduce the additional constraint IIaII~K in (3.1), which signifies regularization ~f an ill-posed problem, and consider the following dual problem: for a fixed functional ,alue F(a) = ~', it is required to minimize K. Thus, the regularized solution estimates Ire obtained by solving the problem

K=infllull on u~U~= {a : k-'ilr~a-ffil'-~ ' (t§

9 =0}.

We know /I0/ that this problem can be solved by the Lagrange multiplier method and this ;olution is obtained from the Euler equation (k]E,+~W) lith

~

ut=W~,

(3.3)

determined from the equality

k-qlWat-gil'-D'Hat[]~=D'.

(3.4)

Formulas (3.3) and (3.4) produce regularized estlmates of the normal pseudosolutions from inexact data (I, f), where the a priori information on ~ is defined by the parm e t e r D'. As noted in /I/, these formulas completely correspond to the formulas that letermine the solution by Tikhonov's regularized method of least squares /3, 4/. Let us consider the computational aspects of the simultaneous solution of Eqs.(3.3) and ~3.4) when ~' is a free parameter. From the dual problem it follows that the dependence K =K(~') established by formulas :7) and (8) should be one-to-one. However, the operator

Q-'(])=(k~Em+II~W)-, :s not continuous for all ~ ( - - ~ , ~). As a result the dependence K = K ( ~ 2) is not singletalued, and the previously developed schemes for determining the regularization parameter by :he discrepancy method /i0/ do not apply in this case. Thus, we need to determine the range )f the parameters ~ and D~ for which the dependence K = N(~*) defined by (3.3), (3.4) is }me-to-one and the regularized solution defined by these formulas therefore exists and is mique. Then from Lemmas 1 and 2 we obtain an analogue of Theorem 1 of /i/ for problem (3.3), :3.4).

Theorem 2. If D'<~., then the system of Eqs.(3.3), (3.4) has a unique solution if tnd only if ~.<~. and it is identical with the solution of the unconstrained extremal }roblems (2.1) and (3.1), i.e., it is uniquely defined by (2.5). Consider the case when ~G(--~., ~). As in /i/, represent the parameter ~ in the form ~=-X.+a,

a~(0, ~ ) .

;ubstituting this value into (3.3), (3.4), we obtain

I'WI17ur - (k~.,-a) u~,=I'W~,

u~= [W'W-(k~..-ct)E,.I-'W'~,

a>O, a>0,

Consider the f u n c t i o n r Theorem 3.

If

~.<~'
=k-'ll~'=~-~II'-~'(i+llu~li'). and IImTll>0,

then for all

.s continuous and strictly monotone increasing, and the equation Ition on (0, oo).

dE(0, o~) t h e f u n c t i o n ~(a)=0

r

has a unique sol-

Remark 2. If :)[~f=o. then

~=0. The proof of this theorem is completely analogous to the proof of Theorem 2 in /i/, )ecause the minimum root of Eq. (3.2) is kk. and l]~II=I]~'~l. As in /i/, application of fast-converging algorithms that find the root of the equation p(=)=0 requires an examination of the function ~(~)=k~(~-'). However, the result of /i/ ;hould be made more precise. We will show that for the application of rapidly converging algorithms, the parameter should satisfy the condition ~.<~'
Theorem 4. "ion

If X.<~'0, then for all ~(~) is continuous, decreasing, and downward convex.

Proof. Substituting

u~ from (3.3) into (3.4), we obtain

~ ( 0 , co)

the

func-

128

g'W[W'W- (kL-=) E.l-'~g+

(~X.-k~'-=)g'W[W'W- (kX.-=)E.l-'~g From this expression we obtain

a> (g) = l l ~ i ' - k t t ' - ~ , ~ G -, (g) ~,~g+

G(~)=(~-'-kl.)E,+I~'IV.

where

Direct differentiation of

O(~)

with respect to ~ gives

'~' (~)=2 (k~.--k~'--~-')~-'~WG-' (~)r~g
~(0, ~),

because

~t*>l..

Then

(~.) =2~.-'~WG-' (g) W"~4~-'(kT,.-k~'-~-') ~tYa-'(~) ~"g+

0"

~-'g'~a-' (~)(~'W-~'E.) a-' (~)w ' g -

because ~.<~'<~, and therefore I]7~I]7-k~tE,>O for all ~(0, ~). The theorem is proved. From Theorem4 it follows that when ~.<~'<~. the equation O(~)=0 can be solved by Newton's tangent method for ~, and this method converges for any initial approximation ~0>0. The approximation ~, to the required solution are calculated from the formulas

~,+,=[~,-o (~,)/o'(g,),

a>(f~) =llfU*-kt~'-f'm=o+ (kX.-,h~'-[~-9 llu~ll=, o" (g)=2 where

u~ and

(k~.-k~'-g-')u~'u;,

u~'=du~/d~ are the solutions of the following Euler equations:

[([~-'-k~..) E.+W'W] u~=W'g, [(f~-'--kL)E.+W'W] ut'=~-'ut. 4. Choosing the parameter ~' Note that Eq.(3.4) in general corresponds to the statistical form of the generalized discrepancy principle. Consider some possible techniques for choosing the parameter ~* from the feasible region which guarantee the existence of certain statistical properties of the regularized solution estimates defined by (3.3) and (3.4). One of the most undesirable effects associated with the ill-posed nature of the matrix Q(-~.) in calculations of consistent solution estimates from (2.5) is the strong increase in the average length of the estimate vector, i.e., the fact that Q(-l.) is ill-posed leads to M[]~kll'm]]uU'. Moreover, for the solution estimates obtained by solving the extremal problem (2.1) (without regularization), we cannot guarantee any finite-sample properties (especially for small k) other than the asymptotic properties for k - ~ . As shown in /15/, even with normally distributed disturbances A A and A] (although these conditions are highly restrictive in applications), the estimates obtained from (2.1) and also from some generalization of this criterion /15/ are merely asymptotically optimal and do not guarantee any properties for small k. It follows from the above that with an ill-posed matrix Q(--~.) it makes sense to ensure that the regularized solution estimates ~k have an unbiased Vector length, i.e., have the property M[]flA]l'=[]uI[' for all k.

Theorem 5.

If

X.~no'+~=
regularized solution estimates and

~i-~

as

k ~

l[g'[[]>0, then for

D*=~=*+na'

the

u~: defined by (3.3), (3.4) satisfy the conditions M[[u,l['=USl['

with probability i.

PrOOf. Consider the set U~={u : n [k-'ll Wu-~i'--Ix*iluli'=t**] } =

{u : M[k-'[lzTu-fU*-W.'D-=ttq }. By properties 1 and 2 of the elements of A A and

A], we obtain

k-'MIIfU'=U/~li'+na'=llA.~ll'+~'+na', k-'M[~'~] =A :A +(~,'+no')D,

k-'M[A'/J =A-~/o.

129 From these equalities we obtain that in[iluIl=U5 u the set U0 is an estimate of the set ~0, Where Since

on

u~,, U0=~ for M'=MA'+not for M'=M,'+no'.

Thus,

U,=U~

M[k-'i[Xu--fit'-~'u'D~] >0, then naturally its estimate

k-'ll~u-fU'-~'u'Du should also be negative definite, which is possible only if M'
CoroZZary. The results of Theorem 5 remain valid if o' is replaced by some unbiased ^ and strongly consistent estimate o'. As an estimate of o' we may use o~=kl./(k,-m) (if kn >m). Experimental computer results show that this estimate is sufficlently accurate for finite samples and its accuracy is not very sensitive to the conditioning of the matrix Q(-~.). We may then take ~'=kn~.]

(kn--m)+~A'.

If M A > 0 and |tA is a priori unknown, then we can find its robust estimate ~, by the method proposed in /16/. In this case, however, the results of Theorem 5 hold only approximately.

Rerr~rk

3.

Let

s=0

and therefore

/=A,.

In this case, Eqs.(3.3) and (3.4) take the

form

AJA~,+~=I=A.'[.

llA,~rfD'=~'.

(4t)

Here we may naturally take k = L If Mt=no'+M~m~ where

We know that p~l(n-m) is an unbiased and strongly consistent estimate of o' and, as shown in /17, p.203/, its accuracy is no t particularly sensitive to the conditioning of t h e 2 n matrix A{A,. We can s take ~ = n ~=/(-=), and if ~'
29:6-I

130 12. FEDOROV V.V., Regression analysis when there are errors in determining the predictor, Voprosy Kibernetiki, VINITI, 47, 69-75, Moscow, 1978. 13. GANTMAKHER F.R., Matrix Theory, Nauka, Moscow, 1966. 14. DOOB J., Stochastic Processes, IL, Moscow, 1956. 15. ISLAMOV I.M., Asymptotic regularization of the ill-posed identification problem, Zh. Vychisl. Matem. i Mat. Fiz., 28, 6, 815-824, 1988. 16. MOROZOV V.A., Methods of Regularizing Unstable Problems, Izd. MGU, Moscow, 1987. 17. DEMIDENKO E.Z., Linear and Non-linear Regression, Finansy i Statistika, Moscow, 1981.

Translated by Z.L.

U.S.S.R. Comput.Maths.Math.Phys.,Vol.29,No.6,pp.130-139~1989 Printed in Great Britain

0041-5553/89 $i0.00+0.00 O 1 9 9 1 Pergamon Press plr

DETERMINATION OF OPTIMAL CONTROLS IN A HIERARCHICALSYSTEM IN THE PRESENCE OF EXTERNAL FACTORS~ V.V. CHUMAKOV

The class of problems that determine the optimal controls in a hierarchical system in the presence of external factors is investigated. These are specific variational problems with coupled constraints on the set of all possible mappings of the control spaces of the system elements. These complicated variational problems are reduced to a number of maximin and extremal problems on the original control sets for two versions of the model, which assume different availability of initial information to the system elements and different optimality criteria. Some problems in the theory of hierarchical systems were considered in /i/. In particular, optimal control problems were formulated for the case when the upper level element in the hierarchy does not have full information on the efficiency criteria of the lower level elements, control problems with this kind of uncertainty were studied in /2-4/. It was noted in /i/ that optimal control problems should be examined for the case when this information is also missing on the subordinate level. Such problems are of considerable practical importance because a wide range of hierarchical systems are subjected to uncontrollable external factors. Some problems in the theory of hierarchical control systems in the presence of uncontrollable factors were formulated and solved in /5, 6/. In this paper, we consider a class of problems that develop these concepts in the following direction. We assume that, due to the action of external factors on the hierarchical system, the elements select the controls without exact information on the efficiency criteria of both the upper and lower levels in the hierarchy. The paper continues the studied of /7, 8/. The theory of active systems /9/, which is conceptually close to the information theory of hierarchical systems, has not dealt with these problems so far.

I. Statement of the problem. Let the reward functions of the centre and the subsystem be respectively M0(u, z, ~) and ~,(u, z, ~), where u~U is the centre control, z~X is the subsystem control, ~ A is a parameter that stands for uncontrollable factors. We assume that the sample ~ A is observed after the subsystem has selected a specific control z~X, and the centre sets a specific control u~U given z~X and ~ A . Let us state the basic assumptions that are needed to formalize the concept of o p t i m a l interaction in hierarchical systems (in what follows, the value set of a variable is specified only if it differs from the original definition set).

Assumption 1. The sets and functions specified above are known to the centre. In addition to information about the set A, the centre may also know the distribution function of the parameter ~ on the set A. Ass~dnptlon 2. The centre makes the first move and sets the rules for the exchange of information with the subsystem. The ability to set the rules for the exchange of information will be modelled by passing from the sets U and X to some extensions U and X of these sets and specifying the mapping n: UXXXA-~UXXXA. In addition to the mapping ,, the strategy sets U and ~ are also required to satisfy the following conditions (in what follows, we put x(~, ~, ~)=(xu(.), n,(-),

_~(:))):

~Zh.vych{s1.Mat.mat.Fiz.,29,12,1788-1799,1989