A method of solving bilinear minimax problems

U.S.S.R. Cornput. Maths. Math. Phys. Vol. 25, No. 5, pp. 6613, Printed in Great Britain

1985

0041~5553/85$10.00+0.00 0 1986 Pergamon Journals

Ltd.

AMETHODOFSOLVINGBILINEARMINIMAXPROBLEMS* B. D. DUISEBAYEV

An algorithm

for solving bilinear

I. Formulation of the problem moves [ 1 1, in which the first player

minimax

problems

and its properties. has the efficiency

using the branch

Consider criterion

and bound

a game between

method

two players

is proposed. with a fixed order of

f,@, Y) =rAY+Qv, and the second fi(Z, Y) =zBy+by. Here x and y are the strategies

of players

1 and 2 respectively,

X={sdi?IDzGd,

which are chosen

z>O} and Y=(y=E’IGy=g,

A, E, D and G are matrices, d and p are vectors of corresponding dimensions, communicates his strategy x to the second player, the best guaranteed result

from the bounded

polyhedral

sets

y>O), and int X#@. of the first player

If the first player equals

(1)

The solution of problem (l), i.e. obtaining w and the e-optimal strategy of the first player, since the function of the responses of the second player to the control of the first

is a rather

difficult

can be discontinuous and multi-extremal. An example of this kind is considered in Sect. 3. To obtain the algorithm for solving problem (1) we will consider three problems of linear programming depend on the vector of the parameters xeX. The unperturbed problem:

The lexicographic

the following

problem,

which

f2k y)-+max (y+=Y.

(2)

(12(2, y) , -f, (2, y) ) -+max 1y= Y;

(3)

problem:

quantity

is determined:

max -k

(~4Y)

yEY.W

and the point

y’~Y~(z),

for which

-

fl (3, y’) =

max - fl@,Y). lEY&)

The perturbed

problem:

f2(& Y)-cfl(5, y)+mas

ly=Y:

(4)

here E is a small fixed scalar parameter. K=(i. 2,..., ‘Y) is a set of numbers of the Suppose y’,..., yN are the extreme points of the set Y, and extreme points. The extreme point ys is called an optimal extreme point of the problem (i). i=?. 3, 4. if the vector Z’fEk exists, such that for the problem (i) lIxi. the point ys is a solution. The sets of numbers of all the optimal respectively. The sets of parameters for extreme points for problems (2), (3) and (4) are denoted by x0, XI, K which the extreme point y’, s=.K, is optimal in problems (2), (3) and (4) are denoted by S.:. .y(:,‘. S: we will determine the quantities (;“=Ny’. ri”=dy’, b’=by’, cr‘=o!y”. respectively. For each extreme point ys, s=K, *Zh. vychisl.

Mat. mat. Fix. 25, No. 9, pp. 1293-1303,

1985.

6

7

Let us consider problem (2) in more detail. The extreme points ys and y’ are called equivalent for problem (2) if hold. The set of numbers of the extreme points which are equivalent to the pointy’ the equations ^b’=&‘, b’=b’ for problem (2). For extreme pointy’ we will determine the sets are denoted by Lo(s) X,‘={x~kIx~‘+b’>xb;+b’,

rsK\L,(s)],

8,‘=(x~E*1xb^“+b’>xb’+b’,

It is easy to prove

the following

r=K\L,(s)}.

facts:

I ) X,‘=X,‘; 2)if

X0”+@

3) if

Xss#

It follows

Lemma

@

and

.z*E~<,

then

and

z* E X0’,

then

from the lemma presented

1. Suppose

X, X,,

Y, (I*) 3

with y’; i&t&)

Ya (I*) = below

, X,cE’

with y’. i-L,(r)

that we can choose

are convex

int X#

@

closed

and

the subset

It0 from the set KO, such that

sets, and

U

Xi3

X;

Ibi
then we can choose

a subset K from the set of indices Vi E K

, N) , such that

(1, 2..

int (X n Xi) # (21, and also

IJ (X fi Xi) = X. iEK

Proof.

It is obvious

that vn

u

l
Suppose

X={lGiGNIXCIXi;f-~);

then it is obvious

x,)=x.

that u (XflXJ=X. iEK

If V&K int (XilX,) ZQ. will show that in this case

then the theorem

is proved.

u

Suppose

the index

t=K

exists, such that

int (XllX,)==0.

We

(x n xi)=x.

K\iil

For this we will show that any point s”~XflX, Suppose xoexnx, and x’eXflXr, &K\(t). e=i~;ri,t,o(P, The set IllxjlGf}).,

(to+&) IlX, where B is a closed belongs to XflX,, otherwise

must also belong to one of the sets Xl-IX,, &K\(t). This means that

?I f? Xi)>% unit sphere

P(z”,X

ll Xi)=luE~~x,Il~O-xU~

I

with its centre at the origin of the coordinates

(i.e.

&={xd?*

u (X f? Xi)#X. iEK x’EintX and z’#x’, then the points Suppose simultaneously internal points both for x0-i-eB, int(XnX,)~int((x?-aB) flX)+o follows Let us proceed to problem (3). The extreme if the equations ?I*=%‘, b’=b’, 8’=h’, a’=~’ toys for problem (3) are denoted by L,(S). We will determine the sets

of the form .rO(l-A)+?& where O<~
X,‘=(x~~*~x^b’+b’~x~+b’, CZiL’+ff,

=L,

(s) \Ll (s) 1,

X,‘={x~E*Jx6’+b’>x~~b’,
,

r=La(s)\L(s)}.

reK\L,

(s) (

sc?+u

a

Lemma 2. Suppose X,+0, then &‘cX,‘cX,‘. Proof. We will prove the first inclusion. Suppose reaK\L,(s), hold. Therefore

it,‘+@ and

z E.%,~,

Ye (z’) = with =I&) Further,

s’d’+a’
rd,

(s) \L, (s),

Consequently, z’EX{‘. We shall prove the first inclusion.

Suppose

hold, otherwiseY is not a solution of problem solution of problem (2) when x = x*, i.e.

Therefore problem

all the inequalities

z~‘fa’~z6’~a’,

s’b’+b’>z‘~‘+b’,

i.e. the inequalities

yT.

i.e.

z’EX,‘.

s’b^‘+b” >z’k+b’,

All the inequalities

(2) when x = x*. Further,

r&(s)\L,(s),

all the extreme

points

must also hold, otherwiseyS

r=K\L,(s),

must ate a

y’, r~-L,(s),

is not a solution

of the

min !I (x*, y). yEYW)

Consequently, X,‘cX,‘. For the extreme points

which are optimal

in problem

w,*=

(3) we shall determine

tk f

max

the quantities

a’.

l&,%X

By virtue of Lemma that

Theorem

1, from the set of numbers

I. The guaranteed

of the extreme

Kt

points

we can choose

the subset

l?!

, such

result of the first game equals w = max w,‘. “EfG

Proof. We shall take the arbitrary point Z*EX and determine the number Since the extreme pointy’ is optimal in problem (2) when x =x*, then

F (2’) Q

x:2 +

a’ <

&I?,

such that

.r’~Xflr,’

wl’ < max w,‘. I&,

Therefore, w = sup F (5) < max wl’. SEX

Suppose

the number

t=R,

and the point

z~EXOX,’

I&[

are such that

^

max wr’ = w,’ = x”af + a’. SE:, where We will take the point z’=X,‘OS. The sequence s’=zO(l-E)+Ez’, when O
Let us proceed to problem Lemma 3. For any x and

Proof.

(4). We shall denote the set of its solutions e>O the following inequality holds:

F- 0,

when Y = x* by

Suppose

y1 E Ye(4 and k (2, Y') = and

y*~Y.(z) and f~(x, y”) =

,“y”m-x, e

h (x, Y).’

min h

VEY.(X)

is e-optimal

(2, 1/j,

YF(z’)

for the function

9

Since

y’EY,(Z),

then the inequality

E]~,(z, y’)-frk y’) l>f2(-t, y’) -f~k fl(5, Y’). We will determine the sets

ft(z,

Y’)-&f,(t,

Y’). Since

y’) >ft(z, y’)--E/,(z, then f2(z, y’) -f2(z,

y’mYo(x),

y’) holds. Hence we obtain y’) 20. Therefore, 5, (2, y’) >

~.‘=(1~klz(~‘-~8’)+b’-EU~~=(~~-~~)+b’--ea’, rd\L,(s),

z8’+a’
rd,(s)\L(s)),

X,‘=(x~E*Jz(~‘--EiL’)+b’--E~~>X(b^’--E(i~)+b’--Ea’, rd\Lo(s),

zb’+a’cx8’+a’,

FLo(S)\L,(S)

It is obvious that X,*=X.’ when e>O. Lemma 4. Suppose SE&; then e>O ^ exists, such that for any for all e7.v-c; the followtng equation holds:

).

the set

OCe
X.‘OX

is not empty,

and

Ye(z) = i=t’~h, Y’.

I

Proof. Suppose

z&,‘OX,

set of inequalities

i.e. for x the following z(&^b’)+(b’-b’) z(fi’-Z)

holds:

rdc\L,(a),

>O,

+(a’-a’)
IELO(S)\LI(S),

2=X. The vector hold when

s=X.‘IlX, E)O and

z(~-b^‘)-EX(ii~--(i’)f(b~-b’)

if

e
--~(a’-a’)>O,

(b’-bb’)

s@-@)+

min raf\La

5 (2 -

2) + (a‘--

at) ’

rdl\Lo(s)

z(ci*-a’)

+ (d-a’)

holds.

These inequalities

will

>o.

Thus,

zmX*‘flX when O-G-& The second statement directly follows from the definition of the set X.‘. Lemma 5. Suppose the set X-(sd?~Asfb, s>O} is bounded and int X+0, then the linear-programming problem cz-tmax IIEX is stable. Here A is a (mXk)-matrix, bEEm, c=E’. The linear-programming problem is stable if the optimum value of the functional depends continuously on the coefficients of the matrix A and the vectors b and c. For more detail see [3]. Proof. Since X is bounded, the problem (c+e)s+max]rmX, where eEE* is a vector all of whose components equal unity, has a solution. Thus, the set of double estimates Y=(ydP’lyA>c+e, y>O)f@. Therefore the vector y&P’, y>O, Yd>c exists; consequently [3, p. 2831, the problem ct+max]zmX is stable. For the extreme points which are optimal in problem (4) we will determine the quantities

wea= max &7,%x Theorem

2. Suppose

K.

is a set of numbers

of the optimum

IJ (X,“n JERE then the following

relations

By virtue of Lemma

points

of problem

(4), such that

we= max wa’; a&,

X)=X,

IEX

and w,‘=w. Suppose smK, belong to .z,. Therefore

and also

m>me.

lim tcE= w. e-10

3, u = sup F(I)

> sup

max

fi (r, y) > max wp’ = w,.

L&X i/EY,W

By virtue of Lemma

4, if

O
lim we > lim we‘. E-+a E-O By virtue of Lemma

extreme

hold: VE>O

Proof.

SEEK,.

za( T a’,

5 we have lim w:=w,l. e-+e

s&z,

then one of the elements

of the set

L,(s)

must

10

Consequently, limw.=w. c-to

Note. In the nonlinear separation

case the relations of Theorem 2 are established in [4 1. However of the set R. enables us further to propose an efficient numerical algorithm.

in linear problems

the

2. An algorithm for solving the problem using the branch and bound method. Theorem 2 justifies the following method of solving the initial problem: it is required to choose a sequence of positive numbers (E*) , such that lim ek =O, X-W and successively to determine the quantities result can be obtained using W.a . A lower estimate of the guaranteed of w.. The the single definition w, for fairly small E. We shall consider in more detail the practical calculation optimum extreme points of problem (4) can be constructed using an inspection algorithm [ 51, based on the concepts of parametric programming, or the method of subdividing the set of parameters into nonintersecting subsets [6]. We shall describe the scheme of the latter method as it applies to problem (4). For simplicity we will assume that the set I’ is not degenerate, i.e. a single permissible basis/of problem (4) corresponds to each extreme point ys. (Here the symbolJ, denotes the number of basis columns of matrix G.) In this case it is sufficient for the optimality of the X.’ is deterextreme point that its basis is also simultaneously a pseudobasis [7]. Therefore the area of optimality mined using the system of n-m inequalities

X.‘-(z~E’~A,(zC+c);20,

j=(i,

2,. . . , n)\l.),

c=b-en, are square matrices which where C=B-eA, A,izC+c)=t(CJ.G,,-‘G,-C,)+(c,.C,.-’G,--c,); C,., G,, cI, is a vector which consists of the consist of the columns C,, G,, jml., of the corresponding matrices; components C,, j”J., of the vector c. The scheme of the-method is as follows. For the beginning, the point S’EX is chosen and the optimum extreme 9’. ‘+ pointy’ of problem (4) when x =x* is obtained. The set X.’ is specified using some system of inequalities The set X\X.L is further divided into n-m non.intersecting sets of level 1: p’,‘>O, i--l, 2, . , n-m.

sI={z~x12P i+p’. ‘CO), S,-((2~XlzP’.‘fp’,‘<0, zP’.‘+p’. ‘10, i-2,

j=l,

2,.

. i-11,

3,. . . , n-m.

exactly the same procedure is carried out, i.e. the point .z’ES,, is For each non-empty set S,, i-1, 2,. . . , n-m, is divided into XcL, are found, and then S.\X,’ chosen, the optimum extreme pointy’ and its area of optimality n-m sets of level 2. The procedure is repeated for the sets of level 2, etc. By virtue of the finiteness of the set of X.’ of extreme points at some level 0 all the sets S,,, __..‘0 will be empty. At the same time the sets of optimality the extreme pointsy’ obtained cover X (see 161). We can choose a point from the set S,,. ., fV by solving the linearprogramming problem fJ+max

for the limitations ZD L_pl, 4, . . . @<-@.

.

j=l,

zP’, j+p’a ‘20,

. . . S-p’. k,

.

.

.

.

.

XPS j+p*. $0,

.

.

.

j=l,

2, .

.

2. .

.

. , i,-1, . .

, i,-l.

Suppose a?, 8’ is a solution of the given problem. If O’>O, then YE&,. .., ip, if &GO, Note that the number of limitations in problem (5) can be significantly reduced if we bear in mind that the limitation zP’* “+pg, “20 follows from SC,,__,,jtEO is not essential for all the sets S,, j,
, i,) = dir,

where

S,,, .._.I,,

is the closure

of the set

S,,. .... ,v

.

then S,,. .,.,a”=QJ. the following: it _,, ....,, ( t
max max jr (.G Y), ... , i P “EY

An algorithm

for calculating

Y, is described

in

[ 81

11

Table

1

I 2 3

The procedure estimate

Xi

Yi

SAY + ‘Y’

I

1.5+c (LO,

0)

*G

((40,

-os+r

i-u.5z l.SCP 1--_u.5E4=*

(0, CO)

=<

2.5-0.58 ___1+0.5E

0.5+0.5~

2.5-0.5~ -Gr 1 + 0.56

1)

for determining

Y, is rather

complex

I

1

and it is therefore

reasonable

in practice

to use a rougher

where a,’ =

mar

Ai is the j-th column of the matrix linear-programming problems.

(sAj + e,),

, iq

ssi.

A and ci is the j-th element

3. An example of the operationof in the following problem:

the algorithm.

f, (5, y) = (Z-0.5)

of the vector a. To determine

We shall illustrate

y,+ (0.52-0.5)

the operation

YZwe need to solve n+l

of the algorithm

for calculating

yr+y,,

fz(z, y)=(-z+1.5)y,+(s-2.5)~,, X={2~E’~0~2~3), The polyhedron Y has 3 extreme functions (zB+b)y’, i=i, 2,3,

F(x).

y,>o,

i=l,

2, 3).

points: y’=(l, 0, O), ~‘“(0, 1, O), ~“‘(0, 0, 1). Figure the upper curve of which is a graph of the function

and we also show the sets X,,i, i=l, 2, 3. Figure 2 shows graphs of the functions function

and also the function

Y=(y&‘Iy,+y,+ya=1,

The function

(sA+a)y’,

F(x)

i=l,

has discontinuities sup F(I) IZX

Fig. 1

2, 3,

1 shows graphs of the

the upper curve of which is a graph of the

at the points x = 1.5 and x = 2.5,

=_I,75

Fig. 2

Table 1 shows the boundaries of the sets X.8 and the values of the quantities w.‘, i=l, 2, 3, E=O, 0.1, 0.01. E=O.I and shall present the modus operandi of the algorithm without using the upper estimates (suppose the initial approximation x* = 1): w.4.18, c&=1.68 we have S,=(z~X~1.68Cz) (level 1). for z’=i, y’, W*‘-1.18, for z-=3, yS, w.~=I, w.=i.19, ~.=I.68 we have S,, ,=(z~X(1.68-~, ~K2.33) W-l 2). We

12

for z-=2, y’, S,, ,,,-(2~X11.68-+ stop: ~~1.67, We shall present approximation x* = for t-=2, I/*,

w.‘= 1.67, w.=1.67, x.=2.33 we have S,. ,, ,= {z~X11.68~2, s<2.3Q, r<1.68)=0 and zt2.33, z>2.33)=0 (level 3); x.=2.33. the modus operandi of the algorithm using the estimates r1 (suppose c=O.Ol and the initial 2): w.‘=1.74, x.=2.48, ~.=I.74 we have S,-(z~X1~<1.62) (level l), estimate

up (s,) =

and also

S1-_IxmX[z>1.52,

z>2.48)

1.18y, + 1.34y, + y, = 1.34,

“,“y”

(level l), estimate ua (S,) = me; 2.5~~ $. 2y, + ys = 2.5, W,
for x*=3, y’. w.‘==1, stop: w.d.74, z.-2.48.

w.=1.74,

4. Numerical experiments. calculating the upper estimates a simple analytic solution:

(S*)

;

2.-52.48

The algorithm and calculating

we have

S,,,=(z~X1~11.52,

xB2.48,

x(2.48)=0

(level 2);

for finding w, was written in ALGOL in two modifications: without the estimates vs. Its efficiency was verified in a test problem which has

Table 1 E-O.01

S==O.I

e-=0 I

z
1.~168, w.‘=i.lll

w,'=i

i.t%
l.S
The guaranteed

233cz.

~.~=1.67

1.52C.v. ~Z2.48, eQ=1.74

w.“=i

2.48-z+, w.s=t

result of the first game here equals w=

when using the e-optimum

were chosen

(n-Z/2) ( n(n-1)/2,

(Z-l), if

if

lGl
Pn,

strategy x*=

Two vectors

z41.52. Lo.*==i.oz

x1==i+f,,

l
l
xi=t/n,i=i,2

x,=0,

l
if if

in.

,..., n, where t=

and the total number The results of the experiments when n=5, e=O.mi shown in Table 2. The computer time indicated is for a BESM-6 computer.

of extreme

points

2” = 32 are

13

Table 2

w.=6.993 I _

2.9 3.1

_

2.5 1.8

4 5

3.1 5.8

n&=8.991

7.5 6.3

3 5

;:;

F

11.4 10.6

2

I.8 2.0

6 _

12.5 13.6

iu,=9.990

In conclusion

the author

thanks

F. I. Ereshko

and V. V. Fedorov

for useful advice

and comments.

REFERENCES 1.

2. 3.

4. 5. 6. 7.

8.

GERMEIER U. B. Games with Non-opposed Interests. Moscow, Nauka, 1976. PSHENICHNYI B. N. Convex Analysis and Extremum problems. Moscow, Nauka, 1980. ASHMANOV S. A. Linear Programming. Moscow: Nauka, 1981. MOLODTSOV D. A. Methods of solving one class of game with non-opposed interests: Dis. na soiskanie uch. st kand. fiz.-matem. n. MOSCOW: MGU, 1974. ERESHKO F. I. and ZLOBIN A. S. An algorithm of the centralized distribution of a resource between active subsystems. Ekonomika i matem. metody, 13, no. 4, 703--715, 1977. IVANILOV U. P. and MUKHAMEDTSEV B. M. Methods of solving linear two-person games or with noncoinciding interests. Ekonomika imatem. metody, 14, no. 3, 552-561, 1978. GOL’SHTEIN E.G. and UDIN D. B. Linear Programming, Theory, Methods and Applications. Moscow: Nauka, 1969. MUKHAMEDTSEV B. M. The solution of the problem of bilinear programming and of finding all the situations of equilibrium in bimatrix games. Zh. vychisl. Mat. mat. Fiz. 18, no. 2, 351-359, 1978. Translated

by H. Z.