On an extremum problem

ON AN

EXTREMUPI

I.

PR~~~~~

A. BAKHTIN (Voronezh)

(Rece iued

3 July

1962)

This paper deals with the problem of finding the smallest value, and the points where this value is reached, of the function L

N

on the polyhedron

bl

XIj =

??lj,

mj>O,

j=i,...,

N;

=ij

Z O*

(2)

This problem for h’ = 1, 2 has been considered in 111. In the present paper we indicate a method of reducing the cases IV> 1 to the case M - 1. We shall assume that none of the sets &xi,), &xi,> , . . . , {aiiv> sists entirely of the same units, since otherwise the problem 1s obviously reduced to one of the cases 1, 2, . . . , i?’ - 1.

1.

Characteristic numbers. Reduction of the general case to the case N > 1 by means of the characteristic numbers N = 1

1. we shall introduce some notation and a number of definitions. The points of the polyhedron (2) will be denoted by the matrices

*

Zh.

vych.

mat.,

4, No. 1. 120-135, 1964. 160

con-

161

The point (or the matrix)

(8.2) in which the function (I) reaches its smallestvalue over the polyhedron (2). we shall call the optimum set. Since sj > 0, j = 1, . . . . N, thereforein the optimum set (1.2) there are always positivenumbers

(some of the numbers i,, i,, . . . . iN may be identical). We put

As for the cases N = 1, 2 (see tll), it is shown that for all .g) + 0 (1.4) and for all XI!) = 0

k, j = 1, . . . . N will be The numbers h,, *.*, hF and Kkj = hk/hj, C811ed the ChW8CteriStiC numbers of the functions(l)-(2).

In order to simplifythe calcul8tionswe Sh811 use the notations K1

=

Kll

The

=

1,

K2

=

Kzl,

.a.,

KN

set (a. , a. , . . . . aiN)

=

KN~*

will be called active if at least one

of the number:lr~~T~*ii), . . . . x$’

of the optimum set is different

from 0. Otherwisewe shall call this set inactive. aij,

Be define analogouslythe activityor inactivityof each of the bases i = I, ..,, L, j = I, . . . . N.

.A. Bakhtin

162 The set

(a.

a.

‘1’

two numbers of this

Lz’ -*-

aiN) will

(jl,

j2#

j2,

. . . . j,)

then the set

2. Let in the set (1.2)

will

xi!;,

. . ., (ai

jk)-active.

. ..,

if at least

set are active.

If in the optimum set (1.2) j f jl,

he called poliactive

zrs:

, a.

1

x~~~~, ziij,

a2’

f Cl and _xig) = 0 for

.*.,

all

aXiN)will be called

f 0. Then from the relationships

follow that

In particular,

if

i, = i, and x!‘!

L1ll’

,%tJ, =

xi:;,

f 0, then

I In a< j, I I In t&j, I

-(1:.7) l

Since

Therefore from relationships

Thus,

we

(1.6),

(1.7)

it follows respectively,

that

have

‘Iheotem 1. In the functions bases aij, for which

3. Consider the function

(l)-(2)

there can be active only such

An

extrerua

163

problem

(I&?)

cp= i P;{, 44

where Pr = min {al, up,

. . 0, u

%$over the polyhedron

We have ‘Iheorem 2. The smallest value of function smallest value of functions (1.12)-(1.13). Proof.

Then

(l)-(2)

Let the set (ai , ai , . . . . aiN) be (jl, 1 2

1 In f&i,1

coincides jp,

. . .,

with the

jk)-active.

a$ . .a$ = hj,,

(1.14)

. . .. . .. .. ... . .. ....

,ha(j*l&.. l &hj&,. and for the remaining j,

j f jl,

j2,

. . ., jk (1.15)

Consequently,

for all j f jl,

j2,

. . . , jk

(iA6) Therefore

Let us assume now that the set (a. ‘1’ Then

ai , . . . , aiN) iS inactive. 2

184

I.A.

Bakhtin

and this means that in this case

Consequently l&@&Y, i=l

where x10’ = zg’ + x&

+

l

’ * + XNX$j > 0

and

$l4”’ =

ml +

x,m, +

a--

xN*mN.

+

Furthermore from relationships (1.14) and (1.15) aiN) is active then the set (n. a. ll’

22’

*-*’

(0)

+

and if it is inactive

it follows that if

atIP;’

=

h,,

q)

>

(1.17)

0,

then (1.18)

Ila PI I< h,. this means (see

[II,,that

= Be:‘”

IninqL

i-l

Thus

mincD=min+

2. The uniqueness of a system of characteristic

numbers

In this section we shall prove the uniqueness of the system of characteristic numbers h 1, of the functions

*

*

*,

h;

(l)-(Z).

Xl = i, x, +,

. .

.)

xN=-

hN

hr

problea

165

be an arbitrary

subsystem of the system of

An extream

l.ktK.

K.

11’

**‘t

J2'

characteristic

numbers K~

Kjk =

I,

K*,

. . . ,

K~ of

functions

(l)-(Z).

We construct with the help of this subsystem the set g of all indices i, i = 1, . . . , J!+ for each of which the number

is equal to 8) either to some a:& L * b) or to some

aUxj jj

. =

9

J

’

32’

‘..I

Ik”

while in this case for some

i, the equation &% =ap=ph ti, and p. must be active ‘in the functions (1.X2)-(1.13); ‘1 (r is the last unused number of the set jr, jg, c) or to some a3

must be satisfied

. . . . jk},

while for some i, the equation

must be satisfied, where s is one of the numbers j,, j,, . . ., jk used in the previous step, and p. must be active in the functions (I. X2)l2 (1.13). We mark in the set j2, . , . , j k all numbers yI, . . . , y to eadi of which according to the rule of constructing the set cf tiere corresponds at .least one number i CZ %. Definition,

The function

given over the polyhedron

(2-2) where wt” ) are the COOrdin8teS of the optimum Point of functions (l.lZ)(1.13). we shall call the’branching tree Of the S%CtiOn Key, K~*,...,K~~ with the trunk jl and the branches yl, shall call it a tree. It follows

from the definition

yz, . . . , yp, or more simply we

of the tree that the charfm%teriStic

166

I.A.

Bakhtin

number of the functions(2.1)-(2.2)is h. . Jl

&finition.

W@shall say that the tree s Of the sectionK. II’

with trunk jl grows with the additionof the sectionK the tree S' of the sectionK.

K.

..t,

Jk

J 1'

,

Kn , . . . . Kn

"1'

‘jk

**”

--*B

with

trzli

j:’

has new branchesnot containedin L% Other&e we shal! say that the tree s does not grow with the additionof the sectionK Kil * “1’

...’

Q

2. Theorem 3. Functions(l)-(2)have a unique system of characteriStiC numbers h,, . .., hN; K1 = 1, K2, m.., KM. Proof.

Since the existenceof a system of characteristic numbers is obvious, it is sufficientto establishthe uniquenessof th%s system. 'Ihisfact will be proved if we can show that the system K~, ..** KN is unique, since accordingto the system K*, . . . . K~ is constructeda unique function (1,12)-(1.13), the characteristic number of which coincideswith the characteristicnumber h, of functions(l), (21, and since by definition & = x.&

* . ., hN = St&.

(2.3)

K:I), . ..# K~I) be another Let us assume the opposite:let ~jl), system of characteristicnumbers for functions(l)-(2).Here we have two cases:

a) all numbers I+~), .... numbers K~, . . . . KN, i.e.

K$”

Ic, <"f',

are

respectivelynot smallerthan the

f . .*2$


@-G

b) among the numbers K:~), . . . . 1(1$11 there are and smallernumbers than the numbers K~, . . . . K~ Let

us

considercase (a). Let K1

= 1,

characteristic numbers of the system K~,

K. , . ..) J2 I
changed during the transitiont0 the System K:‘) We denote Kl,

Kj,,

. . . .

by S and Ka

Sr ‘)

and .:I),

reSpeCtiVeb

K. Ik KN, = I,

larger

be all the which remain un.:I’,

. . . .

KA”.

the trees with the trunk I, with the sections K(I), . . . . K!‘) , respectivelyand by 8 and

Dzi(') we denote t;eksets of num&s

over w&h

these trees are extended.

It is obvious that either n Q: %'"',or % C so). Let 3 c 3"'. Then, from this and from the definitionof the tree it will follow that this is possibleonly when one of the bases of the tree s is active, and the correspondingbasis of tree S"' is inactive,i.e. when

An

167

probler

cxtrenun

(hi” , . .., hi’) are the characteristic corresponding to the numbers ~:l), .il),

numbers of functions . . . . I$‘)).

(l)-(2),

9 C ‘8(l). We construct with the help of the

Let us assume now that

set 9 a new function g, using the rule of constructing the functions SC’) by means of the set 3(l). We denote by yB, . .., yp the branches_ of the tree S. It is easy to see that over the bases di = p,. of tree S is. extended a sum not less than ml + K III + . . . + eandover the

y,n+,:

y2 y2

corresponding bases d!’ ) = 9., of function S is extended a sum not ;:;tt;;I;ha ml + K~~+,, + ... Hence it follows that h, > h, + KYpmYp’ .

Thus, in case (a) always

hf’ >

(2.5)

h,.

Let us turn now to the relationships

Let us select among these numbers the smallest. Let for the sake of explicitness this be K/K:‘). Obviously, s2 < ,:I). Let

K2,

for which

K

be all the numbers of the system

Kn "2'

'**'

We denote by u and o(r) .... B(l)

Kk'),

obvious that either The relationship d(l) 1 tree

the trees of trunk 2, and

SeCtiOIIS

K2,

K,

respectively, and by 88 and n9 tie sets of numbers over which these trees are extended. It is

K,

and

K~,K~, . .., K~

9

K';;,

. . .) Kc l)

m(‘) o? @I or aIRi”C 402 . 902(l)o? %R is possible

only when one of the bases

of tree o(r) is active, and the corresponding 0 is inactive, i.e. when

Let

9R(‘) CI: a.

, 2

basis

We construct with the help of the set

di = di ‘) of

@

a new

function g using the rule of constructing the function u B means of the set S. We denote by S,, . . . , 6, the branches of tree o(’ ).

168

I.A.

Bakhtin

It is easy to see that over the bases d(i‘) of the tree u( ‘) is extended a sum not less than

and with respect to the corresponding bases d,.= extended a sum not greater than

d(l) i

of function 8 is

Consequently

Thus, in case (a) always

J$’ Q F’rominequalities

which contradicts Thus,

case

(2.5).

(2.6)

h,.

(2.6)

follows

the assumption ~$l) >

K?.

(a) is impossible.

The impossibility of case (b) is established in the same way. We remark only that in case (b) from among the relationships

we select

the smallest (let this be for example,

K2/Kj1)

)

and the

greatest (let this be for example, K,,,/K~‘)). Then, as in case (a) we construct trees with trunks 2 and N and it is shown that

which contradicts

the assumption

Thus, functions (l)-(2) numbers K2, KS, . . . , KN’

have a unique system of characteristic

An

crtrcrun

169

problem

3. Finding the characteristic numbers 1. Let

be sn arbitrary

or;

nmnber. We put pi1 = min (wl,

Positive

aGNland Fik

= aik when k = 2, . . . . fi! - 1, i = 1, . . . . L, where aij are the basts of functions (l)-(2). We construct

the function L

(3.2) on the polyhedron (3.1) 6-l

where Pl =

ml +

, XNmN;

j==l

pa = %, . . ., pN-1

=

mN+

Let (x’ij) be the optimum set of functions (3.1)-(3.2). We denote by 1’ the suunkation extended over all such indices i for which pi1 = air, I

i/XN i, for which Pfl = qN .

and by 1” the swnmation sign extended over all

We remark that from the definitions of X’ and 1” it follows will be extended also over some comnon indices. Lemma.If

K,;r

-c Kp

that they

then P’r;l

<

K&N Proof.

Let

h;

, . .

x;=$x;=&

.,h;_l;

, hN-1

(N-1 = -

hi

1

be the characteristic numbers of functions (3.1)-(3.2). ~$a;. We construct by means of the set (X~j) a new set &

putting Iij Fil

=

ai

1

= xi j when j and ii-

1

=

. . . . . . . .

2,

= 0, iiN =

. . . .

iN

/Ki,

IT -

A =lN

1; if

We assume hi =

Si, pi

= 1

f

xii, ai

,

1

ziN

=

0,

if

. 1

first we shall prove the first part of the lemma.,Let us assume the opposite: let for some positive K; < KN the suw Z’xil > ml. It is obvious that only one of the two following cases is possible: At

I.A.

170

Rokhtin

b) “; ’ ‘tr,.

a>h;<$,

Let us consider case (a) 11;< h,.

z

(> I),

Let

;+,

. . .

,

Ir

3 *fk

be all the numbers of the system

I,?

Xl -;-=

equal to uN/t$., and let

2

)...,

%

x1

%N

(3*4)

’

be the largest number of the series

KN/K~

(3.4).

We denote by S and s’ the trees corresponding respectively to the OPtimumset (1.2) and to the Point (3.3), with the trunk I? aud with the , sections KN, K. and tci;, K: Let y2, .*., yp be 12’ -’ Kjk J2’ -**’ Kjk’ the branches of the tree ~7, end 92 and 8’ be the sets of indices over which are extended respectively the trees I’? and l?‘. It is obvious that either % Q W, or 9 C 92’. The relationship m C 3’ is possible only in the case when one of the bases di of the tree 1’5is active and the corresponding basis #!c = tii of tree S’ is inactive, i.e. when

hiv> We assume that

!% C 3’.

hv.

We construct with the help of the set

and of the set (3.3) a function ATusing the rule of constructing function S by means of the set (3.3) and the set 8’. It is easy to see that over the bases ‘i not less than

. . . + X;,myp)<

since in view of the definition

’ = &ryc

mN

the

of tree S is extended a sum

and over the bases di = di is extended a sum not greater than

& (4~m’N+ xl, my,+

%

of set (3.3)

mi> ml

extrcaur

An

prob

171

lea

and this means that n$,\(m,,, Hence it follows that hi 21~~ Consequently, always I hrv

K,h

>

(3.5)

hN.

&I the other hand, since 11;
KI;

<

KN,.

The obtained inequality contradicts inequality number KN/KJ$ cannot be the largest one in the Set We mark in the series

(3.4)

therefore (3.5). (3.4).

hi =

~r;h;

<

Thus, the

all ratios (3.6)

greater than

KN/K~,

and

COnStI'UCt

from

K' =2'

K; l

**'

a group of all

numbers for each of which the corresponding ratios’ i3.6) We assume that all trees constructed we add t0 these groups the Set K,;, Kt constructed

for the sections

J2’

are equal.

for this group do not grow when I Then as before, having

***’

Kjk*

and Kr;, K: . . . . Ki, the Jk J2 JP’ trees S and s’ with the trunk A’ we can ascertain that this case is impossible. K~,,

K.

,

. . . ,

K.

Thus, among the groups constructed there are such for which some of , the trees grow when oh, K : is added. 12’

Kjk

-’

We mark in the indicated set of groups all those for which there are ’ is added. trees which grow when ~6 K: J2’

-

Kjk

If this leads to a marking of all groups then the Process terminates. Otherwise in the set of unmarked groups we mark all those for which there are trees which grow when some groups marked in the previous step are added. ‘Ihis process cannot continue indefinitely since the set i , is finite and consequently there are a finite number of K=2* -‘*’ Ka tgroups. We select after the completion of the process among all marked groups the group which has the largest ratios corresponding to it in the series (3.6). Let this be , , q3# # * * * t Qt 9

(3.7)

It is obvious that all trees of the unmarked groups, if they exist,

172

I.A.

will not grow when section Let us construct

(3.7)

Bakht in

is added.

for the sections

1~~

1’1’

. . . ,

Kr,

and 1G:

PI’

.“‘

“I’,

the trees CTand CT’aith the trunk fil. Let F,, . .: ,’ Ss be brauches of the tree a and % and %’ be sets of numbers over which the trees o and CT’ are extended. It is obvious that either

Z@ Q: @2’, or %RC a’.

The relationship

@

CZ 9)2 is possible only when one of the bases di of the tree u and the ~rresponding basis di = di of tree CT’is inactive, when active,

4, > hat* Let %RC %X’. Let us construct with the help of the set set (3.3) a function G using the rules for the ~onst~ction tion u’ by Beaus of the set (3.3) and %?.

is

i.e.

(3.8) S2 and the of the func-

Since over the bases $3; of the tree o is extended a sum not less than $ (xp, mp, * %p& I

+ - * * + %pa,h

and with respect to the bases Ji = dt of function ;i a sum not greater than

(3.9)

i.e.

An

Inequality

(3.9)

contradicts

extrearun

prob

inequality

tenr

1?3

(3.9).

Let us now investigate the case (b) hi > h,. We select from the system (3.4) all numbers which are equal to 1. Let these be (3.10) We assume that unity is the smallest number in the sequence (3.4). we aIXi K;, Kf , . . . . K; construct for the systems K~, K I , . -. , K[ the trees r and r’ with trunk 1. Let’these tre& be extendid respectively, over the sets of numbers 5? and f’ and c2, +. . , c V be the branches of tree r’. x’ Z $ and 8’ c $?. Obviously We shall consider two possibilities: relationship JCIQ: Jc is possible only when one of the bases $1: of tree r’ is active and the corresponding basis L!~= 12: of tree r is inactive, i.e. when hi < It,, w!lich is impossible. We shall assume that

3’ (= 3.

7e construct

with the help of the set

$’ and the optimal set (1.2) the function ?“, using the rule for the construction of function r by means of the set 3 and the set (1.2). Since uver the bases 2; of tree r’ is extended a sum not less than

4 + xc,mc, +

.-q+

Xc,mcvr

and with respect to the bases d.z = C$ of function ;i: a sum not greater than I ml + xc, MC,

-I-

’

** +

x&m,,Q 4 +

xc,mc,4

+++ + xcomca,

therefore

which is impossible. Thus, unity cannot be the smallest number of the series mark in the series (3.4) all numbers %k, 7, %k,

%k*

-;

%k,

,

f..)

%kP 7, %kP

which are less than unity, and from the system

(3.4).

We

(3.11)

174

I.A.

xk,,

Xk,r

Bakht

.

*

-,

in

(3.12)

xk+

we CornPosea group consisting of all numbers for which the membersof the series (3.11) are equal. At first we shall assume that all trees of the group constructed wiii not grow when the system K1, Kl , . . . , K l U 2 is added. Then by means of the trees r and r’,

as before,

it is shown that

h’ < A,, end this is impossible. We shall assume now that some of the trees of the groups considered grow when System K~, K l , . . . , K I is added. 2 U Te mark among these groups all those for which some of the trees will grow when K1, KI ‘ . ..* KE is added. If as a result of this all groups 2 are marked (3.12) then theUprocess terminates. Otherwise among the remaining groups we mark all those (if any) for which Some of the trees grow when one of the groups marked in the previous step is added. This process cannot continue indefinitely since the system (3.121 is finite and therefore there are a finite number of groups. We select after the completion of the process from among the marked groups the one with the Smallest ratio (3.11). Let this be

It is easy to see grow when the section

all trees of unmarked groups, if any, will not (3.13) is added.

that

We denote by h and If’ the trees with section K

rl’

K’

r2’ ***’

K’

K

‘II

K

r2’

l

**’

K

T” and

r” with trunk rl.

Let 9 and R’ be sets of numbers over which these trees are extended, and -r2, . . . . ‘a be branches of the tree A'. Ve shall consider two possi_bilities: 9’ Ct 9 and R’ c CL It iS obvious that relationship Q’ & R is possible only when one of the bases cl: pftree A’ is active, and the corresponding basis Ji = di of tree A is inactive, i.e. when

An

extreaum

Let us assume now that R’ C

prob

lcn

R. me construct

175

with the help of the

set R’ and the optimum set (1.2) a function x, using the rules construction of function A by means of the sets (1.2) and Q. It is easy to see that over the bases CI: of tree sum not less than

and over than

the bases

$ (%, mr, +

x7,

‘I’; = I?; 0:’ function

m,, + . . . +

A x (Xi,m,

-

A’ is extended

x is extended

xTa

m,=) =

+

xi,m,, +

e- - +

for

in this

xi0 my=),

case &, Q h,, .

Thus,

both when 9’

Q: I! and when 9’ C

h;, < Rence and from

K'

‘1

>

K

‘1

it

follows ,

r!

h,. that

h;++

=h, rI

rl i. e.

h; < and this

is impossible.

I,et us now prove

h,,

Consequently

the second part of

the lemma. Let

assume

Ql = 1, a,

a;= 1. a;

%N-1 XN

=-,...,upJ=

=+,

KY-1 X,V

Obviously

-

z

XN

. . .,cpJ==

1 XN

a

a sum not greater

rl

Consequently

the

.

Ki

>

K~

Fe

176

I.A.

!Ye construct

Bakhtin

the function

(3.14) where pij

= aij

when I. = 2, . . . . ,“I-

and

1

,

ii;1=

mill {QN,

a?Nh

over the polyhedron

(3.15) where

p, =

?nN

+

ON ml,

pa

=

tit,

. . .,

pN-1

=

mN-1 .

Ce shal,l show that for each optimal set (xfi) (3.2) the set (%i)), where XI) = xij when j = 2.

of

n !’

(3.14)-(3.15)

‘1

= X!

versely (Fi j),

I1

/K,&

for

optimal,

each optimal

where Fij

oDtima1 for tion

is also

= Fij

set

(xTj)

(Zij)

of

functions functions

(3.14)-(3.15)

when j = 2, . . . , P’ - 1 and Z.

the functions

of the sets

but for

functions (3. I). . . . 1”’ - 1 and

(3.1)-(3.2).

Indeed,

‘1

and conthe set

= Yi /o,i, is 1

according

to the defini-

and (Fij)

i = 1, . . . , L; i = Z , . . , , N; j=2,...,N-1, i=l

i=l

and

4:zil = -i-F1 i& =

d

(mN +

0;

ml) =

ml

+

xkmN.

i=1

This means that L N-I

min F = 2 J-J pi7 4x1 j==l

l

L N-I

-

= 2 fl pi:“j > i=l j=l

min 3

An extrenun

problem

177

min F = min p and thz sets (x:)) tions I: and F.

and (fij)

are optimal,

respectively,

JIence, and from u,k < cs?+ in view of the validity of the lemmait follows that

for the funcof the first

half

i. e.

The lemmais fully proved. 2. There follows

from the lennaa

Theorem 4. If l’xil K,$lp

then

K;

=

K,p

< ml, then

and finally

K~
if

if 1” xi1 <

I’xil>ml

K$$,,

then

and 1” xi1 > K~>KJ,I.

3. Theorem 5. If K; = 1, K;, . . . , pi 1 are the characteristic numbers of functions (3.1)-(3.2) and KI; = K~ then the smallest Value of functions (3.1)-(3.2) coincides with the smallest value of functions (l)-(2) and K;, K;, . . . . K; are the characteristic numbers of the functions (l)(2).

Proof. while

Let (xij)

be the optimal set of the functions

S’zk > ml,

(3.1)-(3.2),

Z’XZ> X;J~N,

(3.16)

where the signs 1’ and IWare extended respectively

over all such 4v numbers i = 1, . . . , L, for which ptr = ml and &I = a~ . We construct

a set

(3.17)

178

I.A.

where

I

X+j =

Bokhtin ._

when j = 2, . . .,N -f, Q=

%if

~31,Z& = 0 or t

pi, =

and ‘& = 0, =tlv = x&v for @t1 = ai.v s’wN< WI. For the l/x; remaining Qtl, i.e. & and &V Pfl = a{1 = &Vf. 9 if these exist is selected arbitrarily, but according to the conditions: 4); ‘izill ~trs i

2) $r + z&2{* = z?&;

0;

x8

L

W-1

i=l

For example, this can be done as follows.

If the sum of all determined

xii is equal to ml fin view of inequality

(3.15)

ml), then we set the indeterminate quantities

ril

it is not greater than = n and iin = zI1,k;i.

If however, this sum is less th-an nl, then we shall assume the indeterminate -ril = zil,

XiN = 0 until the sum ;I of all determinate zil

remains less than ml. Let during the addition of a certain zi 1 (the number xi 1 has not yet been determined) G1 + .z: 1 > ml which: in view 0 of (3.15)“will necessarily happen at some stage. Ye put

and all the remaining ones, if ‘any

It is easy to see that the point (3.17) belongs to the po~hed~n (Z), while the value of function (1) at point (3.17) is equal to the optimum Value of functions (3.1)-(3.2). Consequently min Q, < min I+*, On the other hand, if

(3.18)

(x~~)) is t!le optimL% set of functions

then min

Q

=

i

G ;i;’ = i

i==l j=i

Hence and from (3.18)

(p$?+XNx:%

i==l

it follows that min a == min F.

“ri’ pi:“))> ?==+a

min

F,

(l)-(2),

An txtremum

~rthe~ore, the set Therefore the n*mbers K; of the function (l)-(2).

is optimal for the functions (l)-(3). . . . , pi, are the characteristic numbers

(3.17) =

1,

179

problem

K;,

The theorem is proved.

4. Finding the optimai values 1. Theorems 4 and 5 enable us to use the following method for determining the smallest value of functions (l)-(2) and the points in which this value is reached. me construct from the different . ..* L, a series

numbers 1111aiN { //In at11, i = 1,

(here we assume O/O = 0~). If the series (4.1) consists only of one term al, fl < al < m, then = ai. Then, in view of Theorem 5, the smallest value of functions (l)-12) is equal to the smallest value of functions (3.1)-(3.22, where Ki = a1 and the set (3.17) will be oPtima1 for functions (l)-(3).

KN

Let the series (4.1) series ttv its middle

consist of more than one term. !Ve divide this

and 1; :

up,+l

<

* ’ ’

<

up

and we compare with the help of Theorem 4 the numbers up and aPl

= KN

then the smallest value of functions

the smallest value of functions

(3.1)-(3.2)

(3)17) will be optimal for the functions Ifap in se&i&


then the division

(l)-(2)

K~

If

wiil be equal to

with ki = up and the set (l)-(2).

1

into parts by the middle term is made

1;:

if however, ap > K~ then in section I;. 1 This process is continued until it either terminates at some stage or an interval is found containing K,,,, the ends of which consist of two neighbouring numbers of the series (4.1). In the latter

case a similar process is carried out for the numbers

I.A.

180

K,v_~. for

which we construct

Bakhtin

out of a11 different

numbers 1In aily_lj/

i = 1, . . . . L, a series

end so on. Let for all numbers K~, . . . , K~ the process described lead only to the determination of intervals consisting of these numbers. We select in each of these intervals in an arbitrary manner one number ~l;f, K,(rl,. . . , K; and we define the numbers

Ve mark among the numbers ail,

i = 1, . . . , L, all

those

for

which

ci = acl respectively, the sake of

(at least explicitness

one such number must necessarily this be aU, Q,.

exist).

Let for

. ., cr,

and the function

(4.2) be determined

over the polyhedron

a1

=

xii =

ml;

i=1

Then the smallest value of value of, functions (4.2)-(4.3) (l)-(Z)

is constructed

(4.2)-(4.3):

put for

according

N;

Stj >

0.

(4.3)

all

the remaining

terms Tij

xi!)

is equal to the smallest set (~$9 ‘) of functions

to the optimal

(0)

all

i=2,...,

functions (l)-(2) and the optimal

Xij

replace

mj,

i=n+1

of matrix =

set

(nii)

of

functions

(Tij)

-.

Xij,

by zeros.

We remark that a series of the form (4.1) can be divided not only by its middle term but by any numbers which divide the series into equal (with an accuracy up to one term) Parts. Finally

we remark that one illustration

of problem

(l)-(2)

is

An

extremum

problem

described in the book [21 (see pages 125-131). similar problem is given in 131.

181

An investigation

Translated

1.

Bakht in, 1. A., Krasnosel’ extremum of a function 400-409, 1963.

skii, M.A. and Levi& A.Yu., over a polyhedron. Zh. vych.

of

a

by G.R. Kiss

On finding the mat ., 3, r:o. 2.

2.

Yudin. D.B. and Gol’shtein, E.G., Problems and Methods of Linear Programming (Zadachi i metody lineynogo programmirovaniya). !~OSCOW, Fizmatgiz, 1961.

3.

problem. Marine. A. S. , A target-assignment America, 6, No. 3, 307-466. 1958.

J.

Operations

Res.

sot.

of