- Email: [email protected]
An algorithm for solving a class of discrete multicriterion problems
39
aQ(~,,) =q(:c~
-q(:c*) =[T(@, :c~ :c~ + [T(~,:CO)--T(vO,:C~ ]~+ w~F(tt+ah l )
[ Y (v', z~ - - r (12~, 2.O) ] = C (z~ C (z*, 7",' (v*, :c*) ) +ct~< T," (v ~, :c~
r,' (t~0, z*) ) -
g>,
where O
a~
aC
(x~ ~ j~z (:c~ r , ' (v~ ~~ )+~ ~C
,
9
0
0
,
0
~
m a x [ ~--(z ,T, (v ,:c ))+~, Using Lemma 7, the theorem follows from this last inequality. Notice in conclusion that the present paper is a development of the ideas of /i/ and contains a more general statement of the problem. By using a more general definition of the set F(u ~ , we are able to avoid the assumption, essential in /i/, that intF(z)~Z. A result similar to our Theorem 2 is also proved in /2,3/, under the assumption that the mapping F is convex. REFERENCES i. DEM'YANOV V.F., Minimax: directional differentiability(Minimaks: differentsiruemost' po napravleniyam), Izd-vo LGU, Leningrad, 1974. 2. BERESNEV V.V., and PSHENICHIqYI B.N., On the differentiable properties of minimum functions, Zh. vych. Mat. i mat. Fiz., 14, No.3, 639-651, 1974. 3. PSHENICHNYI B.N., Convex analysis and extremal problems (Vypuklyi analiz i ekstremal'nye zadachi), Nauka, Moscow, 1980. 4. ROCKAFELLAR R., Convex analysis, Princeton D.P., 1970. 5. DEM'YANOV V.F. and MALOZEMOV V.N., Introduction to minimax (Vvedenie v minimaks), Nauka, Moscow, 1972. 6. BLAGODATSKIKH V.I., Sufficient conditions for optimality for differential inclusions, Izv. Akad. Nauk SSSR, Ser. matem., 33, No.3, 615-624, 1974. Translated by D.E.B.
U.S.S.R. Comput.Maths.Math.Phys. Printed in Great Britain
OO41-5553/83 $iO.00+O.00 9 1984 Pergamon Press Ltd.
Vol.23,No.3,pp.39-47,1983
AN ALGORITHM FOR SOLVING A CLASS OF DISCRETE MULTICRITERION PROBLEMS M.G. KREINES and N.M. NOVIKOVA
An algorithm is given for solving problems of multicriterion discrete optimization; it is based on the use of the minimum function when partial criteria are convoluted. In the case of monotonic constraints, the algorithm guarantees that a complete inspection isavoided. i. Discrete optimization problems in general involve the inspection of many different versions, so that, when solving a given problem, the completest account needs to be taken of its specific features. In the present paper we give a method for effectively solving the discrete maximization problem for a minimum function with monotonic constraints, and we show how a full inspection of the different versions can be avoided. Such problems occur in certain cases of multicriterion discrete optimization, if the convolution of criteria is taken to be their weighted minimum /i/. We consider the class of problems of seeking
max
l(z),
~(~I(Xl) .--.,~.(X .} ) ( ~ 0
z=(z, ..... x ~ ) ~ - [ X,,
(1)
ii I
where the sets X, are discrete, the function y is monotonically non-decreasing with respect to all its arguments, and f can be written as the superposition of minimum functions and certain other functions~ the total number of variables in the functions, differing from min and de,.pendent on a non-unique argument, being small (much less than n). For instance, in problems *Zh.v~chisl.Mat.mat.Fiz.,23,3,576-589,1983
40 of the optimal synthesis of a three-level system for data gathering and processing we have /(x)=mingi(xj(,), rain hj(xh(n, min ek(zk))), (2) where
U( U K ( j ) U { h ( j ) } ) U { j ( i ) } = { i , 2
. . . . . . n},
while the functions g~, hj are monotonic with respect to all their arguments. At the lowest level, measurements are made of certain observed quantities ~ . At the next level, the measurement data are used to find certain characteristics m I of the observed obj act. The accuracy zj of finding each characteristic m~ depends both on the accuracy xho~ of calculating m~, and o n the accuracy x~ of measuring each of the u~ determining the rnj, k~K(]). If the accuracy is interpreted as unity minus the error probability and the errors are assumed independent, then, provided that the measurement errors do not add up, we can take zj ~ rain X~{jlZk=ZA{h mln XA . AwZ(J}
kmE(.f91
In conditions when the system efficiency criterion is time (the system operates in real time), if we take x to be the operating time of the respective element with the minus sign, then
zj=zAcj~+min
x~.
k~E($)
It is assumed in general that the efficiency of findingrns is
z,=h~(xs, cj~, rain X~z~)= rain hj(z~{l~,~.,z~,) -, here, ~.k are weighting factors, and the functionhjis non-decreasing with respect to each variable. The characteristics of the observed object provide the basis for defining the set of its descriptive parameters p=(pili~/) (according to which the system makes its decisions), As in the case of the previous level, the efficiency x{ of finding each parameter pl is equal to
z,=g,(x~,~, m i n
~ / z j ) = rain
g,(zm,,~./z~)
and the efficiency of the system (e.g., the probability of making the correct decision or the time required for this) namely ~ _-~(x~,z~(0, x*(91~K(]), ]~(~), ~/)=~eff{z).is given by (2). The system elements, calculating p~e~" oz measuring ~ Vk~K(]), ]~](~), i~f, can be chosen from the set V o f different collections of elements. Each chosen element is characterized by its efficiency x a n d its resource consumption ~supplied with the index of the element. Elements having lower efficiency and greater resource consumption are naturally replaced by better types when available. The problem of synthesizing the optimal system is the problem of maximizing the function / in the set V u n d e r the constraints eff where the function ~resiS non-decreasing with respect to each argument (e.g., it is the sum of its arguments in the case of a monetary resource, or is the superposition of sums and maxima if the resource is time). Hence problem (i), (2) is obtained. In the general case, the functions g{, h~ may depend on many arguments (minimum functions), or the number of levels (superpositions) may be increased. To avoid inspecting all the versions in such problems, we propose a method of solution based on the following fairly simple algorithms for ordering the elements. We first have to arrange in decreasing order of any two functions ~,, g{ the sequence {x/}7__i x of elements of the set X{, for which there is no x~X~ with a greater value of gs(x{) and a greater value of {~(x~) , or with a lower value of ~(xl) and a greater or equal value of ~(z~), than the values of the corresponding functions of the chosen element (if two elements have the same values of ~;, ~{ , then either element is chosen) ; for this we use: Al~orithm
i.
Step
i.
Denote by Xti the set of samples of max
X?
=
],(xl)"
Arg max/,(z,).. x~
As the first element of the required sequence we take any
2',' ~ Arg rain y, (x,). z,, Step Put
k+t.
Assume that we have obtained the first kelements of the sequence: xts,...,x,~. If X ~ = ~ , then the construction ends and m,=k. Otherwise, put
X~={z,~Xi[~,(z~)<~l,(z~)).
X,~+'=Arg ma x l, (z,) X~
and choose any
z,~+~ Arg rainre(z,)
41 On repeating the last step at most IX,/times with the required sequence {xi')yl,.
k=l, 2,..., we can obviously construct
In this way we can order all the sets X, which make up X, so that the search process (i) can be simplified. Next, we omit the index ~ in the signs of the product, sum, max, and rain, if the relevant operation is performed with respect to i=|, 2,...,n. We introduce the notation x(l,,...,l,)for the vector (x l,,....z~.) with the relevant component superscripts. To construct the sequence {z'}K=, of elements xErI{z,i}~, ~ realizing all the possible different values of max min/,(z,), (=Iv(y,(=,),....w n(=,))~yeJ
(3)
in decreasing order of yo from +oo to --oo , where the sequences ing algorithm 1 for l=|, 2,.9 we use: Algorithm
2.
Step
i.
We choose the element
{ziJ}Tlt are obtained by apply-
x'=z(| ..... I)=(/,' .... ,z,').
S t e p I. Assume that the first k--I terms of the sequence have been constructed: x',..., /~-', 1~.k~l, and at the previous ( l--1)-th step we choose the element x=/(k, .... , k,), i~-ki~;nt. E If k,=r~, Vi=l, 2,..., n,then the construction of {zt}t=, ends and XE=Xk=I- Otherwise, we find the number j realizing m a x {f~(z~"i+t )[k,~lnl} , and we choose the element z'=z(k+ ....,kj_,,Icj+i, kj+i,..., ]r If then min],(z,')
Y.(r~,--i) times with /=|, 2,..., we end the construc-
tion. It is easily seen that the resulting sequence of elements has the required properties, since every new element is chosen in such a way that the value of mhl~(z,) is reduced as little as possible. The estimate K~i~Z(ra~--l) , following from algorithm 2, for the number of different values (3), cannot be reduced a priori. For, putting y=~z,, ~(z,)=xl, zi'=i4~(ln~--i) Vi=|, 2 .... , n, and (assuming that m , > ...>ra,) z,i=z~ -* --i V]r 8 ..... nz,,where / = m a x {|i;;~-;~k--~}, x~i=zki_, --i Vi= 2, 8~ .... ~, we find that each value z,' can be the value (3) with suitable ye; this gives 1+Z(m~--i) different values. Hence algorithm 2 does not involve any steps that are clearly superfluous. 2. In the case when, in all X,, i=|, 2 ..... n, sequences
{xi0j~=i, are chosen
(algorithm i),
we can use algorithm 2 to seek the element ze~lqX,, giving the value min~,(/,e), equal to (3) with fixed y0 (see /2/). For this, at each step of algorithm 2 we check if the constraints are satisfied, and the first element z', for which the constraints are satisfied, is naturally the required z e. However, the task of seeking z ~ knowing {z/}~t , can be performed more efficiently (see below), and the main merit of algorithm 2 is that it can be used in parallel with algorithm i, in order to avoid the complete construction of sequences {x,j}j=t m~ for v e r y small values of ye.
In short, to seek ze~I]~/,, for which min~(zi")is equal to (3), we use:
Algorithm 3. S t e p i. we construct zt~ from algorithm i (step i) choose z' from algorithm 2 (step i).
Vi=l, 2,...,n.
We
S t e p k+i. Assume that we have obtained at the previous step all the kc-th elements of the sequence { z ~ 1 , and that we have chosen z~=z(k,,...,k,). If y(~,(z~),..., y.(z.~))<~l e, the construction ends. Otherwise, we perform the (ki+|)-th step of algorithm 1 for all i for which ~.+, z~* is not constructed, and we choose the next element from algorithm 2. In the worst version (no solution, [ X d = n h ) we require ~ + Z ( [ X d - i ) steps. The problem of seeking z ~ realizing (3), is a particular case of problem (i) (with [(z)= Inin/,(z~)), which admits of quite rapid solution, since the number of different values of (3) for arbitrary ya is not large: i+Y~(rr~--i) as compared with [[;n~ in the general case of any (albeit for [(z)=~f~(z~)). However, if satisfaction of the constraints is laborious to check, e.g., in the case of many vector constraints, then it is vitally important to minimize the number of these checks during the search for the solution. To this end, under the preliminary assumption that all the sets X~ are ordered in accordance with algorithm l, we give the following for seeking
z~
1 ,at which min~i(/ra) is equal to (3):
Algorithm 4. S t e p i. we leave out of consideration the components/ of vector z,for which m~=i, and accordingly reduce the dimensionality n. Put
l,=ra,-- [ t If
/,>0
for all
t=l, 2 ..... n, we choose
such a way that the superscript of each
ra i z'=z(l~,...,l~). Otherwise. we choose z'~II{z~}~=,, in
~-th component is equal to ;~+i--min {ra,[/,~0}, i=i, 2 ..... n.
Step k+i. Assume that z'=z(ri,...,r~,.:.,r.) , have been obtained for r=i, 2,...,k . Then, if the vector z~=z(k,,...,k~) does not satisfy the constraints in (3), on reducing the dimensionality n b y ignoring components ] with ~1=;nj (if ~ = ; ~ V]=i, 2, .... n , there is no solution) , we find for each ~=i, 2,..., n the number p~, equal to the least of them~ and of all r~>~ such
42 that x{' is the
i-th ce=ponent of vectorx'with
that the superscript of each
t-th component,
r
and we choose
in such a way
t=f, 2, .... n, is equal to
dt=p,-[n~Z(p,-k,)], if
or to
x~+i~H{x~7l,
V i = I , 2 ..... n,
d~>0
dt=pt+i--min{p,--k,[d,~O},
~ildl<'~0.
if
If the vector x~=z(k,,..., k.) satisfies the constraints in (3), we find an index ], realizing min/,(x,ht),and for it we find the number qj , equal to the maximum of 1 and all the ~ k i such that ~ q is the ]-th component of some x', r
z~+'=z(k . . . . . . ~-,, q,, ~ . . . . . . . k.). On repeating
step k+l
with
k=l, 2,...,
we inspect at most n + m a x m ,
different elements
x~R{x~}7~,, and we find that either x~+'=z ~ (in which case, if the vector satisfies the constraints, it is the desired x~ or if it does not, there is no solution), or else that zh+i=z% r
(3) for any values of y,(x,) and
~(z~)
in ,Xi,[=l, 2 .....n.
Proof. Each z ~ X may not satisfy the constraints in (3). Assume that a vector x(k,, 9..,k~)~{zi~?~, is chosen and we have checked if it satisfies the constraints in (3). If it does not, then it follows from algorithm 2 that, on only considering the elements z(l,,...,Z,) with numbers of components li~k~, i=I, 2,...,n, we shall not lose the solution x ~ and moreover, regardless of the specific values of the components x~i, any such element x(l,....,l=), different from the z(kl,...,k,) considered, may be the solution. *) If the element considered satisfies the constraints, then we have to find its component ~, realizing min~i(x,hl), and the solut i o n ~ c a n only be among the x(r,,...,~) for which ~(x~i)'~(z~$) Vi=i, 2 ....,no But since it is not possible to find the specific value of ],(xt'i) from the value of ~,(x~') , then in the worst case the solution can be given by any x(r,,...,~) for which rj~k~ Since, on choosing the first vector, we cannot know a priori whether it satisfies the constraints in (3), the optimal vector will be that which minimizes the maximum of the sets containing the solution in each of the cases considered. Hence, since the number of the component realizing min~(z,) for the chosen z is not known in advance, we obtain the condition that all the m~--l~, i=I, 2,..., n, be equal, if the first chosen vector is z'=z(l,,...,ln). Now, using the estimate of the maximum number of different solutions as a function of the problem dimensionality, given in algorithm 2, we equate these numbers for the cases when the vector z': 2 ( m ~ - l ) - ( m j - ~ ) = Z ( m i - ? i - 1 ) satisfies and does not satisfy the constraints in (3), i.e., Zmi=(n+i)(m~--~) V]=l,2,...,n. Since the problem is integer-valued, this method is the same as method 2. Given an optimal choice of the next element from the set found at the Ist step (~fter checking x'), we also have to minimize the maximum set of possible localization of the solution, and if x'does not satisfy the constraints, we proceed in the same way as in the previous step in accordance with algorithm 4; or if it does satisfy the constraints, then we decrease as much as possible the superscript of component ], realizing min~(zlh), since IF-l< m,--l~ for n>1. On continuing similar arguments, we find that, if we deviate at some point from algorithm 4, the specific parameters of the problem may be such that the set of possible localization of the solution (after obtaining all the information about the chosen element) is not less than that given by algorithm 4, and will require at least as much inspection of different elements. **) The division of the solution localization set ends when there remain in it only components with adjacent superscripts, and then, the solution is found by algorithm 3 in at most (or at least, in the worst case) n steps. The time taken to obtain the final set by algorithm 4 is not more than can be guaranteed by any other method. 3. Algorithm 4 can be used to reduce the inspection of different when function / has the form
/(x)=minh(xql~5),
x~Xt
in problem
(i)
U L = { I , 2 . . . . . n},
under the constraints y(y,,...,y,)=z(z~(yg I/,~L)[I~J), where the functions z, z~ are not decreasing. We first construct ordered sequences of vectors (x,] ] i ~ ) (for all ]~J), on which the values of the functions % and ~ decrease (see algorithm i), i.e., the inspection is made separately with respect to smaller sets of indices [~ and then algorithm 4 is used. If some of the functions ~ are in turn superpositions of minimum functions, e.g.,
~(xsli~eI,)= ~'~minht(xhllt~L,), "~RIt~T~
0
U L,=I,~
reRj t~T~
*) For example, it is sufficient to take the case mentioned after algorithm 2, putting y=aZx~, where the parameter a is unknown. **) Algorithm 4 undoubtedly admits of improvements in specific cases but the improvements are insignificant for the class of problems as a whole.
43 and the constraints break down further: z~=u~(u,(w~(Ytt [tt~Lt)ll~Tr) Ir~RJ), then, when obtaining the ordered sequences of vectors (zo[i,~lj) , we can construct by a l g o r i t h m 2 for each r~/~j decreasing sequences of appropriate minima and later order the values of their sums. This last operation requires in the w o r s t case inspection of all the p o s s i b l e versions {(xq ll,~Lt, t~T,) [rERj}, though the over-all inspection is reduced due to the construction of sequences of vectors (ztt[l,~L,, tET,) by a l g o r i t h m 2. Later reduction is possible in the inspection if one of the functions ht can be written in terms of m i n i m u m functions and o t h e r functions with fewer variables than Lt (the sum in the p r e s e n t example of ]~ can also be replaced by another function) w i t h a suitable division of the constraints, etc. However, if all the functions ]j,hi/ etc., different from rain when expanding [, are monotonic, then, to Obtain the next (x,jlij~l~), we do n o t need to know all the o r d e r e d v e c t o r sequences (xq If,ELi, t~T,), r~B~, but only parts of them (counting from the start or the end), and all the ordering processes can be combined into a single process. Then, at the n e x t step of seeking the solution, it m a k e s sense to use algorithm 3 rather than 4 (or even a l g o r i t h m 2); though this m a y mean a greater inspection in the w o r s t case, it offers a considerable economy in memory. B e l o w we write out the algorithm for ordering the values of a monotonic function, and we give a method for solving p r o b l e m (i) with function (2) (optimal synthesis of 3-1evel data gathering and p r o c e s s i n g systems) , combining algorithms 1 and 2 and u s i n g parallel o r d e r i n g of the values of the functions g;, hi.. There is no great d i f f i c u l t y in extending the m e t h o d to other functions of the class considered, i n accordance with our above remarks. Assume that we have function F(a,,..., an),. n o n ' d e c r e a s i n g with respect to all its arguments a~ b e l o n g i n g to d e c r e a s i n g sequences (a~}~,~,, i=J, 2,...,n. To construct a d e c r e a s i n g sequence of values of the n o n - d e c r e a s i n g function F i n the Cartesian p r o d u c t of d e c r e a s i n g sequences
(~/q;:~,, i=t, 2. . . . .
.,
we use : Algorithm
5. Denote b y
a(],..... ],) the v e c t o r
F'=F(a,' ..... a,')=F(a(i ..... I)).
Step
i.
Step
S+I.
Assume
that we have c o n s t r u c t e d FS=['(a(jd
j.'))=
.....
a t the S - t h
step
max F(a(j,' ..... j.')). i,r
If
(a,~',.... a,&).
8
]/=n Vi=i, 2,..., n, then S=N
and the construction ends; otherwise, we p u t FS~"=max{F(a(lt'+l,j2 t . . . . . j . ' ) ) , F ( a ( j t t , j , ' + J , j , z. . . . . 1 . ' ) ) . . . .
,F(a(j,'. . . . .
.! .! 1 ....1~ +I)),
F(a(j/ .....jr))},
max
leaving out of consideration the sets a(],,..., j=) which do n o t b e l o n g to the domain of definition of the function F, i.e., j~>r~. on repeating the last step w i t h S=I, 2,... , less than [[re times, we construct the sequence {FS}~=1. Let us assume for simplicity that e,(xh)~-x~, that the function g~, hj are strictlv increasing with respect to each argument, and that the function y=Y.Yi, where xt=/=x,' %rx,,x,'~X,; then to solve p r o b l e m (1)--(2), we use: Algorithm
6.
Step
i. we choose
x'=(x,' ..... x,'), where
x,'=maxx, x$
V t = i , 2 ..... n.
S t e p l+i. Assume that at step [ we have constructed the v e c t o r xt=(x/', .--, x,'~)=x( l,.... , l,). If this v e c t o r satisfies the p r o b l e m constraints, then it is the solution. Otherwise, we construct x ~+~ as follows. For all t=J, 2, . .., n, for which zl*+I has not been found previously, we find in accordance w i t h a l g o r i t h m i, -
max
-
X t\
putting
x t'§ t
=Xt ,, , if
{x/L....
=:t }
Yt (x,5}
(xt I'Jr(xt)<
~,(=,)~>y,(=,") Vx,~{=,' ..... =,"}.
Next, we find the index k~ Viii Vje](i) , realizing rnax{x~ § [k~Kj} (see a l g o r i t h m 2). Assume that the next value hjs of function hj has been obtained at step Z b y means of algorithm 5 from the expression S
.
.
IkO )
Ik
h i ~ n j t x A - O > , m i n x~ ) ~ k~KO)
then we denote by
t
max
l~t~
hj(x~(i),
dk
rain
xk ) ;
kEK(i)
(x~'~,x'~(~)Ik~K(j)) the vector realizing hS§
max
hi(x~o), nfin =~),
l ~ t ~ t l , tglk(j)
k~K(j)
l<.dk~dk|, dkr k Vk~A'(j) 9
9
lk(j)
Ik.+l
" ' h:O)+' rain xt~), nj tx~,()) , m i n { x ~ / ,~J ~Xk(j)
vj~j(O
t k~KO')
vi~t
,
lk
lnin x/~ })} k~K(.i)\lki}
44 (see a l g o r i t h m 5). If
then we find (in the same w a y as above) x~~2 for k~K(])U{k(])} and p e r f o r m the n e x t step of a l g o r i t h m 5 for seeking h~ ~, while the r e a l i z a t i o n of the v a l u e is d e n o t e d b y t h e v e c t o r (z~i,, z~I~Ik~-K(])) . Next, we a g a i n check (4), and if the i n e q u a l i t y is satisfied, we c o n s t r u c t h f ~,, etc., u n t i l we find e i t h e r that (4) is false, o r that the n e x t (z', , z ~ t ~ I k ~ K ~ ) ) is the same &(J) as the previous, in which case we p u t
(~,,, z ~ ' " l k ~ K ( ] ) )
Now,
Vi~l
w e find index
= (z,'.lk~K(]) U {k(]) }).
1~, r e a l i z i n g
h~(xst. , rain xS~(~)),.
max
,l.'(I)
and a s s u m i n g that we o b t a i n e d a t step
~tO) lt,~[J)
It
Z
tr u ~
xll(t)
as
max g~(z ' m ) ' max h,(z'~tn,
min z , $ ' ) ) ,
j~d(|)
heX(j)
we find f r o m the set o f indices
R={r=t,2,...,r,, the vector
"i ztb) ~'(~) , (z/0~,
Vie](i), p , = t , 2 . . . . , p , , , V k ~ J~:{I) U K(/)},
q,=t,2,...,qv
1~/(0) realizing
z~tlk~K(i),
,
:to3 '~z,'U) min zx-tt )), gi(z/~ i), g['*=max{gt(xm)l~(1) +1 , rain --nltz~u) 9
iel(i)
rain{
t, 1 ,-I~0) " ~ 0 ) , min
min
jc-d(i)\
k~K(j)
{jil
max g, (zxi ,,r min R\T
where
S].
x~), hli(zt(ji ),
.
mm k~A'(J t)
SkUi)x~ ~
x~r
HI'
hj (xt~,),r mm" x~))}, k~K0)
i~JH)
T={r=lm, qi=hu, p,=lA VkeK(]) V]"=l(i)}.
j. , if the r e a l i z a t i o n g~* r e q u i r e s a g r e a t e r In the same w a y as w h e n c o n s t r u c t i n g h s§ total r e s o u r c e then the r e a l i z a t i o n of gi", i.e.,
+
+ I lJ(lh ~j(i)~ZRi)1 §
~
[
>
Ik(,)%
'
jejti)
(2~k)]i k~EK(j)
we seek the r e a l i z a t i o n g~ ,g, , ..., until the n e x t r e a l i z a t i o n c o n s u m e s n o less r e s o u r c e than the r e a l i z a t i o n g~* ; we denote this b y the v e c t o r
(z~
9 /i)
, i
:~(,) , x~ ~ I k ~ K (I),Ie] ( O ),
otherwise, we denote by this v e c t o r the r e a l i z a t i o n W e f i n d the index i0~[ w h i c h r e a l i z e s vi . . , t'/(I)
maxgl(x~(i), m m
g,'. 't, t
nl(x,b-) , min x~t))~
a n d we p u t
z"=CCz:~Ik~ IGl\{i.} IJ
U {kO)]U KO)),
jGJ(i)U {.~i)l
ZjO0, (Z~O) , z~k" I k ~ K 0), ] ~ .r Cto))). On r e p e a t i n g this (/+i)-th step w i t h
I=|, 2, ..., less than
'~, (IXjvd ~ (lx..,I ~ IX.I)) .~J[O
ImZ
(5)
texiD
times, we e i t h e r o b t a i n the solution, o r w e o b t a i n s o l u t i o n (the c o n s t r a i n t s are n o t satlsfled).
.zt+i=:zt,
w h i c h m e a n s that there is
4. In the g e n e r a l case, in o r d e r to use our m e t h o d s to solve p r o b l e m w r i t e the function / in the f o r m ](z):min/i.(min/i,(...( rain /it. ,( rnin [il (xlk)1/~ ~ i,~l,
t*~ f~"t
J,t-x) I ],,-* ~
U
U
U
~--I, i,~Ji, it~I],
:k-l~/']k~t
Ju--=) ] 9 9 ") I ]t. ~ "'" i
U
IkG ]k
J~.),
U
k_tC- IJ.k_ I JkEJi~:_1
I ~ = { 1 , 2 ..... .},
no
(i), we have to (6)
45 where ~! are non-decreasing (with l
t,c:l, ~.CLT/, 4a : 9a "
/~-1~
Jif-I
JkEJi~-I I k ~ ' l J k
different elements ~eX. The upper bound (7) of the number of different vectors zEIIXt, which are checked for satisfaction of the constraints of problem (i) when seeking the solution, shows the extent to which the presence of rain in the writing of function ] "reduces the total inspection ll]X~I of vectors, made in the general case of solving discrete optimization problems, even when the target function is monotonic and the constraints y=Zy,, ]=~j; y,, ], are arbitrary, are monotonic. It can be concluded from our results that, in problems of discrete multicriterion maximization, when the value of each criterion depends on its own group of arguments, though all the arguments are linked by joint constraints, the convolution of criteria of the minimum type with weights is preferable, from considerations of actual computation, to the weighted sum (while the convolutions are otherwise equivalent /i/). As a practical example of writing function ] in form (6) , we consider the problem of maximizing the multiproduct flow in a network. We aim to transmit over the network as large flows as possible of several types of product under the condition that, as the total flow over each rib increases, there is a non-linear increase in rental cost, and the total expenditure is bounded. Maximization of multiproduct flow is a multicriterion problem, and in accordance with what has been said, we shall maximize the minimumof the flows, possibly with a weight. The weighting factors may be numbers proportional to the demands on the flows of the respective products. Assume that, in (6), all the functions
9fl r (SIr+ t [ Jr+ 1 ~ JIr),
zl,+,~-
rain
where
]irr247
zjk -~- ~j~, r = 0 , 9 4.....k--2,
Ir+t~IJr+l are sums:
li,=
~,
zj~+~,
r = O , 1. . . . . k - - t,
Ilk =zlk.
Jr§ Then function
](z) will be fully defined by the collection of sets of indices
{L, ( A l t o ~ h ) ,
(Z,,Ih ~ A ,
i~
.....
(1,~_,lt~-,~;,~_,,..~,
tt~X,,,
],~l~, io~I,), (l~[]~l%_t, i~-,ElJk_,, .... i,~l,, ],~l~, 1oG/,)}, where the min has to be taken with respect to indices of Ih, and the sum of the respective minima with respect to indices of .dq_z , l=|, 2 ..... k. We assign to arguments x~t the values of the number of flow units of each product over each rib. The number n is equal to the sum over all types of product of the number of ribs participating in the particular product flow. The problem constraints can be written either in the form )~y,(x,)~y~ where yi{z,)VI=|, 2, ..., n are the devices moving each product flow over each rib, or else as constraints on the sum over all ribs of the rib rental function, dependent on the sum over all products of the flows over the rib. We can also consider separately constraints on each rib (the vector of contraints). In all these cases, the constraint function increases as the flow increases. When stating the problem, the most laborious part is construction of the collection of sets of indices. Let It be the set of all types of product sent over the network. The further construction is performed separately for each product JoEl,. Consider all the paths from the source to the sink of product f0 9 We isolate the paths which do not have a rib in common with any other path, and assign them indices ],'.....it", r,>~0. Then Ij,,,l=l,2,...,r,, consists of all ribs of path ],',d,,={iJ} for i,~Ij,, and does not divide further. The remaining (not yet indexed) paths are divided into minimal groups in such a way that paths of one group have no ribs in common with paths of other groups, such groups are given indices ]~'*x,...,],S',ss~l;Jt.=~{],',..., ],"} are different for each product io~lo. Now consider a group of paths ],t,r,|. Then, l~,,={itt,...,l, "'} for each respective l=rt-i-1,...,s,. In turn, J,,={i,'} for i,'e[j,,,r~ti,l>r,. For r>t, the subgroups it" are again divided into minimal groups of s"~bpaths (the parts of a path corresponding to the subgroup), having no ribs in common with subpaths of other groups. And so on, until oply unexpanded minimal groups of ribs remain. In order to finalize the expansion of the groups containing more than one rib, we have to increase the number of problem variables, assigning to certain ribs of unexpanded groups different values of the argument, depending on the different unexpanded subpaths going through them, while the connection of these split arguments is taken
46 into account in the constraints. This naturally does not destroy the monotonicity of the constraints. The subpaths divided after this procedure are again divided as described above, until only one rib each in each unexpanded group remains, after which the resulting problem (i), (6) can be solved by the methods of Para.3. If the problem dimensionality is too large for these methods, we recommend using algorithms 3, 4 in order to improve some version of the flow distribution over the network. This is done as follows. Assume that we have distributed all products 1,2,...,n, except i, over the network paths; then for the t-th product there are several paths and for each, by algorithm 4, we solve the problem of maximizing the min over all product types of the ratio of the product flow to the demand for it, under the constraints on the total resource. Hence we choose the best path, from the point of view of the system operating criterion, for the i-th product when the paths of the other products are fixed. Since algorithm 4 works quite fast, we can repeat this procedure for different i=|,2,...,n , several times, thereby in general improving the structure of the flow distribution, though ovbiously we can only be sure of obtaining the global maximum by reduction to problem (i), (6), as described above. 5. In all our methods for solving problem (i), essential use has been made of the fact that the function specifying the constraints can be written in the form y(y,(x,),...,y~(zn)), where y is non-decreasing with respect to each argument. If problem (i) has vector constraints, then, to utilize our methods, they must be assigned functions y' , non-decreasing with respect to each argument, in the form y'(y1(x,),...,y.(x,))~yo',t=l,2,...,N. Then, when defining the complexity of the algorithms for solving problem (3), the number of different z ~ X inspected has to be multiplied by n + N , since, at each step, we make a check of the constraints, and in general compute min~(xl). If the problem constraints cannot be written in this way, then, for the function ] = m i n ~ , the maximum number of inspected x ~ X may be close to IX[ , e.g., when we have the lexicographic problem of optimization and the constraint is the set of all x ~ X realizing the max of function y. Hence, in order to avoid a complete inspection in such problems, supplementary information is used about the properties of the maximized function and the set of constraints. In certain cases, when quite a large inspection of different x ~ X is still needed to obtain the solution of problem (i), while the allotted time is short (the problem is solved in real time), it may be sufficient to find a vector x, satisfying the constraints, which is "not too bad" in the sense of criterion I" For instance, consider the problem of the optimal distribution of elements of a system (say a group of radio stations /3/). Assume that we have a set of objects I={I,2 ..... n} , and a set of locations ]={I, 2 ..... m ~ then, for any iE[, denoting by x~ the location of the ~-th object, we obtain Xi=], X=~Xf, m~n. The distribution of the objects over the locations is characterized by the fact that two different objects cannot occupy the same location. We are given the matrix II~(])I] (of values of the functions ]i in sets Ms ) of characteristics of allocation of each object 6~I, and we know the function F(1,(x,)..... 1,(z.)) (where the function F is non-decreasing with respect to each variable), the maximization of which is the aim of optimal distribution of the objects over the locations. We thus obtain the problem max
{x~X[~iCzI VIii}
F (I* (z,) . . . . . I . (z.)).
(8)
In the same way as above, we can order the values of F b y algorithm 5 (without cancelling the repeated elements) and seek the vector x~X, satisfying the constraints of problem (8), which is first in order of non-increasing F. In the worst case, however, we have to inspect all the elements that do not satisfy the constraints, before the solution can be found. It is therefore sensible to reduce the inspection as compared with that of algorithm 5 in order to hit the constraints faster at the expense of incurring a loss in the maximized criterion. To construct the vector x , satisfying the constraints of problem (8), and acceptable in the sense of criterion F , we give: Algorithm 7. S t e p i. w e arrange sets X~ in non-increasing order of values of.~(z,), and obtain Xi={z~ t.....xC} Viii (see algorithm i). S t e p 2. We renumber objects i~I in non-increasing order of their priorities are priorities) in accordance with the function F.
(if there
S t e p 3. We choose x,' and store components x,',i=2,3,...,/, lI, checking for satisfaction of the inequalities ~'r n and x~'~x,i Vi<], i~l. If an inequality is not satisfied, we proceed with the relevant ]-th component in the same way as with the ~ t h . When all components iEI have thus been inspected, the vector (zi".... , x,")~X, satisfying the constraints of problem (8), will be stored. S t e p 4. If the constructed (z~',,...,x,'.) is unacceptable from practical considerations from the point of view of criterion F of the system operation, we reduce the sets X~, by discarding from them all the x~, giving unacceptable values of 1~(x,). After this we repeat step 3 with the following modifications.
47 When xt'=zi' and, for any xt t left in set Xt, there exists irt-,, for which x~1_~=x,' Vi
i
= , and so on, comparing them,
not
with
xlt ,'-,~
but
with
~T ~
z-, -
If the values
are the same, we proceed as described above. If x~_, is unacceptable, we put r~-~=| and increase the superscript of the ([--2)-th component to the minimum extent needed for it not to be the same as the values of the previous components, and so on. If even the first component goes outside the acceptable range, it remains either to widen the acceptable range or to return to the vector obtained at step 3. S t e p 5. step 4 can be repeated several times with different acceptable ranges, e.g., discarding all the time the values of components realizing min;.~[~(xl), as long as it is possible to construct in the relevant X~ a vector x satisfying the constraints. Obviously, if, for any i~[ , all the values of /t(x,) in Xf are distinct, then algorithm 7 at step 3 gives the Pareto-optimal vector with respect to the system of criteria (~i(z,)li~I), while otherwise (~(x,")]i~]) is weakly Pareto (in the set of constraints of problem 8)). Thus) if we interpret function F a s a convolution of criteria ~ then, even at the first iteration, algorithm 7 gives not too bad a result, which is certainly important in actual problems. The above methods of solving problem (1)--(3), (8) enable us to deal with optimization or at least improvement of a whole range of complex problems with different practical applications. The isolation in discrete optimization, and in particular, multicriterion, problems with monotonic constraints, of the rain (max) functions, when the criteria are written in accordance with relation (6), immediately reduces the inspection of possible versions in accordance with (7), thus speeding the search for the solution. REFERENCES i. GERMEIER YU.B., Introduction to the theory of operations research (Vvedenie v teoriyu issledovaniya operatsii), Nauka, Moscow, 1969. 2. KREINES M.G. and ZHUKOVSKII V.D., Foundations of the optimal choice of diagnostic methods, in: Mathematical methods of optimization and applications to large economic and technical systems (Matem. metody optimizatsii i ikh prilozheniya v bol'shikh ekonomich, i technich, sistemakh), TsEMI Akad. NAUK SSSR, Moscow, 1980. 3. NOVIKOVA N.M., The control of a system of radio stations, in: Mathematical methods in operations research (Matem. metody v issl. operatsii), Izd-vo MGU, Moscow, 1981.
Translated by D.E.B. U.S.S.R. Comput.Maths.Math.Phys. Vol.23,No.3,pp.47-55,1983 Printed in Great Britain
OO41-5553/83 $10.OO40.OO 9 1984 Pergamon Press Ltd.
EVALUATION OF THE CONDITIONALMINIMUM OF A FUNCTION WITH GIVEN ACCURACY* V.N. NEFEDOV Methods for finding with given accuracy the minimum value of a continuous function in a compactum 3re described. The problem often arises of finding the minimum ~ of a continuous function y(x) in a compactumX. Due to computing errors, and possible inaccuracy in specifying the function y(z) , or the functions specifiying the set X , our actual task is merely to find the number 9', differing from the required ~ by at most the chosen accuracy 6of solving the problem. In the present paper we give methods for finding such a number ~' for a preassigned accuracy 6 > 0 under certain assumptions regarding the disturbances of the set X and of the function y(x). i. We shall require some notation. Let ~ be the collection of all sets of m-dimensional Euclidean space E'; fc(m) is the class of functions defined in E~, and np(m) is the class of continuous functions defined in ~'. We introduce the sets
A r g : " " [ Zo, m (x) . . . . . y, (x), yo (x) ] = (x~Yoly,(x)~s,, i = i , 2 , . . . , x, ya(z)~ zinf yo(x')+~} '~Y where ...,
Y={z~Yoly,(z)~O, i = l , 2. . . . . ~}#~, Yo~.~E~,y,(z)~fc (m), i=0, l ..... % ~FE', j = t , 2, '~>~t; Q6[y(x), Y] ={~(x)~np (m) l W ~ r
*Zh.v~chisl.Mat.mat.Fiz.,23,3,590-6ol,1983
Ig(z)-y(z) I<~),
aQ(~,,) =q(:c~
-q(:c*) =[T(@, :c~ :c~ + [T(~,:CO)--T(vO,:C~ ]~
[ Y (v', z~ - - r (12~, 2.O) ] = C (z~ C (z*, 7",' (v*, :c*) ) +ct~< T," (v ~, :c~
r,' (t~0, z*) ) -
g>,
where O
a~
aC
(x~ ~ j~z (:c~ r , ' (v~ ~~ )+
,
9
0
0
,
0
~
m a x [ ~--(z ,T, (v ,:c ))+
U.S.S.R. Comput.Maths.Math.Phys. Printed in Great Britain
OO41-5553/83 $iO.00+O.00 9 1984 Pergamon Press Ltd.
Vol.23,No.3,pp.39-47,1983
AN ALGORITHM FOR SOLVING A CLASS OF DISCRETE MULTICRITERION PROBLEMS M.G. KREINES and N.M. NOVIKOVA
An algorithm is given for solving problems of multicriterion discrete optimization; it is based on the use of the minimum function when partial criteria are convoluted. In the case of monotonic constraints, the algorithm guarantees that a complete inspection isavoided. i. Discrete optimization problems in general involve the inspection of many different versions, so that, when solving a given problem, the completest account needs to be taken of its specific features. In the present paper we give a method for effectively solving the discrete maximization problem for a minimum function with monotonic constraints, and we show how a full inspection of the different versions can be avoided. Such problems occur in certain cases of multicriterion discrete optimization, if the convolution of criteria is taken to be their weighted minimum /i/. We consider the class of problems of seeking
max
l(z),
~(~I(Xl) .--.,~.(X .} ) ( ~ 0
z=(z, ..... x ~ ) ~ - [ X,,
(1)
ii I
where the sets X, are discrete, the function y is monotonically non-decreasing with respect to all its arguments, and f can be written as the superposition of minimum functions and certain other functions~ the total number of variables in the functions, differing from min and de,.pendent on a non-unique argument, being small (much less than n). For instance, in problems *Zh.v~chisl.Mat.mat.Fiz.,23,3,576-589,1983
40 of the optimal synthesis of a three-level system for data gathering and processing we have /(x)=mingi(xj(,), rain hj(xh(n, min ek(zk))), (2) where
U( U K ( j ) U { h ( j ) } ) U { j ( i ) } = { i , 2
. . . . . . n},
while the functions g~, hj are monotonic with respect to all their arguments. At the lowest level, measurements are made of certain observed quantities ~ . At the next level, the measurement data are used to find certain characteristics m I of the observed obj act. The accuracy zj of finding each characteristic m~ depends both on the accuracy xho~ of calculating m~, and o n the accuracy x~ of measuring each of the u~ determining the rnj, k~K(]). If the accuracy is interpreted as unity minus the error probability and the errors are assumed independent, then, provided that the measurement errors do not add up, we can take zj ~ rain X~{jlZk=ZA{h mln XA . AwZ(J}
kmE(.f91
In conditions when the system efficiency criterion is time (the system operates in real time), if we take x to be the operating time of the respective element with the minus sign, then
zj=zAcj~+min
x~.
k~E($)
It is assumed in general that the efficiency of findingrns is
z,=h~(xs, cj~, rain X~z~)= rain hj(z~{l~,~.,z~,) -, here, ~.k are weighting factors, and the functionhjis non-decreasing with respect to each variable. The characteristics of the observed object provide the basis for defining the set of its descriptive parameters p=(pili~/) (according to which the system makes its decisions), As in the case of the previous level, the efficiency x{ of finding each parameter pl is equal to
z,=g,(x~,~, m i n
~ / z j ) = rain
g,(zm,,~./z~)
and the efficiency of the system (e.g., the probability of making the correct decision or the time required for this) namely ~ _-~(x~,z~(0, x*(91~K(]), ]~(~), ~/)=~eff{z).is given by (2). The system elements, calculating p~e~" oz measuring ~ Vk~K(]), ]~](~), i~f, can be chosen from the set V o f different collections of elements. Each chosen element is characterized by its efficiency x a n d its resource consumption ~supplied with the index of the element. Elements having lower efficiency and greater resource consumption are naturally replaced by better types when available. The problem of synthesizing the optimal system is the problem of maximizing the function / in the set V u n d e r the constraints eff where the function ~resiS non-decreasing with respect to each argument (e.g., it is the sum of its arguments in the case of a monetary resource, or is the superposition of sums and maxima if the resource is time). Hence problem (i), (2) is obtained. In the general case, the functions g{, h~ may depend on many arguments (minimum functions), or the number of levels (superpositions) may be increased. To avoid inspecting all the versions in such problems, we propose a method of solution based on the following fairly simple algorithms for ordering the elements. We first have to arrange in decreasing order of any two functions ~,, g{ the sequence {x/}7__i x of elements of the set X{, for which there is no x~X~ with a greater value of gs(x{) and a greater value of {~(x~) , or with a lower value of ~(xl) and a greater or equal value of ~(z~), than the values of the corresponding functions of the chosen element (if two elements have the same values of ~;, ~{ , then either element is chosen) ; for this we use: Al~orithm
i.
Step
i.
Denote by Xti the set of samples of max
X?
=
],(xl)"
Arg max/,(z,).. x~
As the first element of the required sequence we take any
2',' ~ Arg rain y, (x,). z,, Step Put
k+t.
Assume that we have obtained the first kelements of the sequence: xts,...,x,~. If X ~ = ~ , then the construction ends and m,=k. Otherwise, put
X~={z,~Xi[~,(z~)<~l,(z~)).
X,~+'=Arg ma x l, (z,) X~
and choose any
z,~+~ Arg rainre(z,)
41 On repeating the last step at most IX,/times with the required sequence {xi')yl,.
k=l, 2,..., we can obviously construct
In this way we can order all the sets X, which make up X, so that the search process (i) can be simplified. Next, we omit the index ~ in the signs of the product, sum, max, and rain, if the relevant operation is performed with respect to i=|, 2,...,n. We introduce the notation x(l,,...,l,)for the vector (x l,,....z~.) with the relevant component superscripts. To construct the sequence {z'}K=, of elements xErI{z,i}~, ~ realizing all the possible different values of max min/,(z,), (=Iv(y,(=,),....w n(=,))~yeJ
(3)
in decreasing order of yo from +oo to --oo , where the sequences ing algorithm 1 for l=|, 2,.9 we use: Algorithm
2.
Step
i.
We choose the element
{ziJ}Tlt are obtained by apply-
x'=z(| ..... I)=(/,' .... ,z,').
S t e p I. Assume that the first k--I terms of the sequence have been constructed: x',..., /~-', 1~.k~l, and at the previous ( l--1)-th step we choose the element x=/(k, .... , k,), i~-ki~;nt. E If k,=r~, Vi=l, 2,..., n,then the construction of {zt}t=, ends and XE=Xk=I- Otherwise, we find the number j realizing m a x {f~(z~"i+t )[k,~lnl} , and we choose the element z'=z(k+ ....,kj_,,Icj+i, kj+i,..., ]r If then min],(z,')
Y.(r~,--i) times with /=|, 2,..., we end the construc-
tion. It is easily seen that the resulting sequence of elements has the required properties, since every new element is chosen in such a way that the value of mhl~(z,) is reduced as little as possible. The estimate K~i~Z(ra~--l) , following from algorithm 2, for the number of different values (3), cannot be reduced a priori. For, putting y=~z,, ~(z,)=xl, zi'=i4~(ln~--i) Vi=|, 2 .... , n, and (assuming that m , > ...>ra,) z,i=z~ -* --i V]r 8 ..... nz,,where / = m a x {|i;;~-;~k--~}, x~i=zki_, --i Vi= 2, 8~ .... ~, we find that each value z,' can be the value (3) with suitable ye; this gives 1+Z(m~--i) different values. Hence algorithm 2 does not involve any steps that are clearly superfluous. 2. In the case when, in all X,, i=|, 2 ..... n, sequences
{xi0j~=i, are chosen
(algorithm i),
we can use algorithm 2 to seek the element ze~lqX,, giving the value min~,(/,e), equal to (3) with fixed y0 (see /2/). For this, at each step of algorithm 2 we check if the constraints are satisfied, and the first element z', for which the constraints are satisfied, is naturally the required z e. However, the task of seeking z ~ knowing {z/}~t , can be performed more efficiently (see below), and the main merit of algorithm 2 is that it can be used in parallel with algorithm i, in order to avoid the complete construction of sequences {x,j}j=t m~ for v e r y small values of ye.
In short, to seek ze~I]~/,, for which min~(zi")is equal to (3), we use:
Algorithm 3. S t e p i. we construct zt~ from algorithm i (step i) choose z' from algorithm 2 (step i).
Vi=l, 2,...,n.
We
S t e p k+i. Assume that we have obtained at the previous step all the kc-th elements of the sequence { z ~ 1 , and that we have chosen z~=z(k,,...,k,). If y(~,(z~),..., y.(z.~))<~l e, the construction ends. Otherwise, we perform the (ki+|)-th step of algorithm 1 for all i for which ~.+, z~* is not constructed, and we choose the next element from algorithm 2. In the worst version (no solution, [ X d = n h ) we require ~ + Z ( [ X d - i ) steps. The problem of seeking z ~ realizing (3), is a particular case of problem (i) (with [(z)= Inin/,(z~)), which admits of quite rapid solution, since the number of different values of (3) for arbitrary ya is not large: i+Y~(rr~--i) as compared with [[;n~ in the general case of any (albeit for [(z)=~f~(z~)). However, if satisfaction of the constraints is laborious to check, e.g., in the case of many vector constraints, then it is vitally important to minimize the number of these checks during the search for the solution. To this end, under the preliminary assumption that all the sets X~ are ordered in accordance with algorithm l, we give the following for seeking
z~
1 ,at which min~i(/ra) is equal to (3):
Algorithm 4. S t e p i. we leave out of consideration the components/ of vector z,for which m~=i, and accordingly reduce the dimensionality n. Put
l,=ra,-- [ t If
/,>0
for all
t=l, 2 ..... n, we choose
such a way that the superscript of each
ra i z'=z(l~,...,l~). Otherwise. we choose z'~II{z~}~=,, in
~-th component is equal to ;~+i--min {ra,[/,~0}, i=i, 2 ..... n.
Step k+i. Assume that z'=z(ri,...,r~,.:.,r.) , have been obtained for r=i, 2,...,k . Then, if the vector z~=z(k,,...,k~) does not satisfy the constraints in (3), on reducing the dimensionality n b y ignoring components ] with ~1=;nj (if ~ = ; ~ V]=i, 2, .... n , there is no solution) , we find for each ~=i, 2,..., n the number p~, equal to the least of them~ and of all r~>~ such
42 that x{' is the
i-th ce=ponent of vectorx'with
that the superscript of each
t-th component,
r
and we choose
in such a way
t=f, 2, .... n, is equal to
dt=p,-[n~Z(p,-k,)], if
or to
x~+i~H{x~7l,
V i = I , 2 ..... n,
d~>0
dt=pt+i--min{p,--k,[d,~O},
~ildl<'~0.
if
If the vector x~=z(k,,..., k.) satisfies the constraints in (3), we find an index ], realizing min/,(x,ht),and for it we find the number qj , equal to the maximum of 1 and all the ~ k i such that ~ q is the ]-th component of some x', r
z~+'=z(k . . . . . . ~-,, q,, ~ . . . . . . . k.). On repeating
step k+l
with
k=l, 2,...,
we inspect at most n + m a x m ,
different elements
x~R{x~}7~,, and we find that either x~+'=z ~ (in which case, if the vector satisfies the constraints, it is the desired x~ or if it does not, there is no solution), or else that zh+i=z% r
(3) for any values of y,(x,) and
~(z~)
in ,Xi,[=l, 2 .....n.
Proof. Each z ~ X may not satisfy the constraints in (3). Assume that a vector x(k,, 9..,k~)~{zi~?~, is chosen and we have checked if it satisfies the constraints in (3). If it does not, then it follows from algorithm 2 that, on only considering the elements z(l,,...,Z,) with numbers of components li~k~, i=I, 2,...,n, we shall not lose the solution x ~ and moreover, regardless of the specific values of the components x~i, any such element x(l,....,l=), different from the z(kl,...,k,) considered, may be the solution. *) If the element considered satisfies the constraints, then we have to find its component ~, realizing min~i(x,hl), and the solut i o n ~ c a n only be among the x(r,,...,~) for which ~(x~i)'~(z~$) Vi=i, 2 ....,no But since it is not possible to find the specific value of ],(xt'i) from the value of ~,(x~') , then in the worst case the solution can be given by any x(r,,...,~) for which rj~k~ Since, on choosing the first vector, we cannot know a priori whether it satisfies the constraints in (3), the optimal vector will be that which minimizes the maximum of the sets containing the solution in each of the cases considered. Hence, since the number of the component realizing min~(z,) for the chosen z is not known in advance, we obtain the condition that all the m~--l~, i=I, 2,..., n, be equal, if the first chosen vector is z'=z(l,,...,ln). Now, using the estimate of the maximum number of different solutions as a function of the problem dimensionality, given in algorithm 2, we equate these numbers for the cases when the vector z': 2 ( m ~ - l ) - ( m j - ~ ) = Z ( m i - ? i - 1 ) satisfies and does not satisfy the constraints in (3), i.e., Zmi=(n+i)(m~--~) V]=l,2,...,n. Since the problem is integer-valued, this method is the same as method 2. Given an optimal choice of the next element from the set found at the Ist step (~fter checking x'), we also have to minimize the maximum set of possible localization of the solution, and if x'does not satisfy the constraints, we proceed in the same way as in the previous step in accordance with algorithm 4; or if it does satisfy the constraints, then we decrease as much as possible the superscript of component ], realizing min~(zlh), since IF-l< m,--l~ for n>1. On continuing similar arguments, we find that, if we deviate at some point from algorithm 4, the specific parameters of the problem may be such that the set of possible localization of the solution (after obtaining all the information about the chosen element) is not less than that given by algorithm 4, and will require at least as much inspection of different elements. **) The division of the solution localization set ends when there remain in it only components with adjacent superscripts, and then, the solution is found by algorithm 3 in at most (or at least, in the worst case) n steps. The time taken to obtain the final set by algorithm 4 is not more than can be guaranteed by any other method. 3. Algorithm 4 can be used to reduce the inspection of different when function / has the form
/(x)=minh(xql~5),
x~Xt
in problem
(i)
U L = { I , 2 . . . . . n},
under the constraints y(y,,...,y,)=z(z~(yg I/,~L)[I~J), where the functions z, z~ are not decreasing. We first construct ordered sequences of vectors (x,] ] i ~ ) (for all ]~J), on which the values of the functions % and ~ decrease (see algorithm i), i.e., the inspection is made separately with respect to smaller sets of indices [~ and then algorithm 4 is used. If some of the functions ~ are in turn superpositions of minimum functions, e.g.,
~(xsli~eI,)= ~'~minht(xhllt~L,), "~RIt~T~
0
U L,=I,~
reRj t~T~
*) For example, it is sufficient to take the case mentioned after algorithm 2, putting y=aZx~, where the parameter a is unknown. **) Algorithm 4 undoubtedly admits of improvements in specific cases but the improvements are insignificant for the class of problems as a whole.
43 and the constraints break down further: z~=u~(u,(w~(Ytt [tt~Lt)ll~Tr) Ir~RJ), then, when obtaining the ordered sequences of vectors (zo[i,~lj) , we can construct by a l g o r i t h m 2 for each r~/~j decreasing sequences of appropriate minima and later order the values of their sums. This last operation requires in the w o r s t case inspection of all the p o s s i b l e versions {(xq ll,~Lt, t~T,) [rERj}, though the over-all inspection is reduced due to the construction of sequences of vectors (ztt[l,~L,, tET,) by a l g o r i t h m 2. Later reduction is possible in the inspection if one of the functions ht can be written in terms of m i n i m u m functions and o t h e r functions with fewer variables than Lt (the sum in the p r e s e n t example of ]~ can also be replaced by another function) w i t h a suitable division of the constraints, etc. However, if all the functions ]j,hi/ etc., different from rain when expanding [, are monotonic, then, to Obtain the next (x,jlij~l~), we do n o t need to know all the o r d e r e d v e c t o r sequences (xq If,ELi, t~T,), r~B~, but only parts of them (counting from the start or the end), and all the ordering processes can be combined into a single process. Then, at the n e x t step of seeking the solution, it m a k e s sense to use algorithm 3 rather than 4 (or even a l g o r i t h m 2); though this m a y mean a greater inspection in the w o r s t case, it offers a considerable economy in memory. B e l o w we write out the algorithm for ordering the values of a monotonic function, and we give a method for solving p r o b l e m (i) with function (2) (optimal synthesis of 3-1evel data gathering and p r o c e s s i n g systems) , combining algorithms 1 and 2 and u s i n g parallel o r d e r i n g of the values of the functions g;, hi.. There is no great d i f f i c u l t y in extending the m e t h o d to other functions of the class considered, i n accordance with our above remarks. Assume that we have function F(a,,..., an),. n o n ' d e c r e a s i n g with respect to all its arguments a~ b e l o n g i n g to d e c r e a s i n g sequences (a~}~,~,, i=J, 2,...,n. To construct a d e c r e a s i n g sequence of values of the n o n - d e c r e a s i n g function F i n the Cartesian p r o d u c t of d e c r e a s i n g sequences
(~/q;:~,, i=t, 2. . . . .
.,
we use : Algorithm
5. Denote b y
a(],..... ],) the v e c t o r
F'=F(a,' ..... a,')=F(a(i ..... I)).
Step
i.
Step
S+I.
Assume
that we have c o n s t r u c t e d FS=['(a(jd
j.'))=
.....
a t the S - t h
step
max F(a(j,' ..... j.')). i,r
If
(a,~',.... a,&).
8
]/=n Vi=i, 2,..., n, then S=N
and the construction ends; otherwise, we p u t FS~"=max{F(a(lt'+l,j2 t . . . . . j . ' ) ) , F ( a ( j t t , j , ' + J , j , z. . . . . 1 . ' ) ) . . . .
,F(a(j,'. . . . .
.! .! 1 ....1~ +I)),
F(a(j/ .....jr))},
max
leaving out of consideration the sets a(],,..., j=) which do n o t b e l o n g to the domain of definition of the function F, i.e., j~>r~. on repeating the last step w i t h S=I, 2,... , less than [[re times, we construct the sequence {FS}~=1. Let us assume for simplicity that e,(xh)~-x~, that the function g~, hj are strictlv increasing with respect to each argument, and that the function y=Y.Yi, where xt=/=x,' %rx,,x,'~X,; then to solve p r o b l e m (1)--(2), we use: Algorithm
6.
Step
i. we choose
x'=(x,' ..... x,'), where
x,'=maxx, x$
V t = i , 2 ..... n.
S t e p l+i. Assume that at step [ we have constructed the v e c t o r xt=(x/', .--, x,'~)=x( l,.... , l,). If this v e c t o r satisfies the p r o b l e m constraints, then it is the solution. Otherwise, we construct x ~+~ as follows. For all t=J, 2, . .., n, for which zl*+I has not been found previously, we find in accordance w i t h a l g o r i t h m i, -
max
-
X t\
putting
x t'§ t
=Xt ,, , if
{x/L....
=:t }
Yt (x,5}
(xt I'Jr(xt)<
~,(=,)~>y,(=,") Vx,~{=,' ..... =,"}.
Next, we find the index k~ Viii Vje](i) , realizing rnax{x~ § [k~Kj} (see a l g o r i t h m 2). Assume that the next value hjs of function hj has been obtained at step Z b y means of algorithm 5 from the expression S
.
.
IkO )
Ik
h i ~ n j t x A - O > , m i n x~ ) ~ k~KO)
then we denote by
t
max
l~t~
hj(x~(i),
dk
rain
xk ) ;
kEK(i)
(x~'~,x'~(~)Ik~K(j)) the vector realizing hS§
max
hi(x~o), nfin =~),
l ~ t ~ t l , tglk(j)
k~K(j)
l<.dk~dk|, dkr k Vk~A'(j) 9
9
lk(j)
Ik.+l
" ' h:O)+' rain xt~), nj tx~,()) , m i n { x ~ / ,~J ~Xk(j)
vj~j(O
t k~KO')
vi~t
,
lk
lnin x/~ })} k~K(.i)\lki}
44 (see a l g o r i t h m 5). If
then we find (in the same w a y as above) x~~2 for k~K(])U{k(])} and p e r f o r m the n e x t step of a l g o r i t h m 5 for seeking h~ ~, while the r e a l i z a t i o n of the v a l u e is d e n o t e d b y t h e v e c t o r (z~i,, z~I~Ik~-K(])) . Next, we a g a i n check (4), and if the i n e q u a l i t y is satisfied, we c o n s t r u c t h f ~,, etc., u n t i l we find e i t h e r that (4) is false, o r that the n e x t (z', , z ~ t ~ I k ~ K ~ ) ) is the same &(J) as the previous, in which case we p u t
(~,,, z ~ ' " l k ~ K ( ] ) )
Now,
Vi~l
w e find index
= (z,'.lk~K(]) U {k(]) }).
1~, r e a l i z i n g
h~(xst. , rain xS~(~)),.
max
,l.'(I)
and a s s u m i n g that we o b t a i n e d a t step
~tO) lt,~[J)
It
Z
tr u ~
xll(t)
as
max g~(z ' m ) ' max h,(z'~tn,
min z , $ ' ) ) ,
j~d(|)
heX(j)
we find f r o m the set o f indices
R={r=t,2,...,r,, the vector
"i ztb) ~'(~) , (z/0~,
Vie](i), p , = t , 2 . . . . , p , , , V k ~ J~:{I) U K(/)},
q,=t,2,...,qv
1~/(0) realizing
z~tlk~K(i),
,
:to3 '~z,'U) min zx-tt )), gi(z/~ i), g['*=max{gt(xm)l~(1) +1 , rain --nltz~u) 9
iel(i)
rain{
t, 1 ,-I~0) " ~ 0 ) , min
min
jc-d(i)\
k~K(j)
{jil
max g, (zxi ,,r min R\T
where
S].
x~), hli(zt(ji ),
.
mm k~A'(J t)
SkUi)x~ ~
x~r
HI'
hj (xt~,),r mm" x~))}, k~K0)
i~JH)
T={r=lm, qi=hu, p,=lA VkeK(]) V]"=l(i)}.
j. , if the r e a l i z a t i o n g~* r e q u i r e s a g r e a t e r In the same w a y as w h e n c o n s t r u c t i n g h s§ total r e s o u r c e then the r e a l i z a t i o n of gi", i.e.,
+
+ I lJ(lh ~j(i)~ZRi)1 §
~
[
>
Ik(,)%
'
jejti)
(2~k)]i k~EK(j)
we seek the r e a l i z a t i o n g~ ,g, , ..., until the n e x t r e a l i z a t i o n c o n s u m e s n o less r e s o u r c e than the r e a l i z a t i o n g~* ; we denote this b y the v e c t o r
(z~
9 /i)
, i
:~(,) , x~ ~ I k ~ K (I),Ie] ( O ),
otherwise, we denote by this v e c t o r the r e a l i z a t i o n W e f i n d the index i0~[ w h i c h r e a l i z e s vi . . , t'/(I)
maxgl(x~(i), m m
g,'. 't, t
nl(x,b-) , min x~t))~
a n d we p u t
z"=CCz:~Ik~ IGl\{i.} IJ
U {kO)]U KO)),
jGJ(i)U {.~i)l
ZjO0, (Z~O) , z~k" I k ~ K 0), ] ~ .r Cto))). On r e p e a t i n g this (/+i)-th step w i t h
I=|, 2, ..., less than
'~, (IXjvd ~ (lx..,I ~ IX.I)) .~J[O
ImZ
(5)
texiD
times, we e i t h e r o b t a i n the solution, o r w e o b t a i n s o l u t i o n (the c o n s t r a i n t s are n o t satlsfled).
.zt+i=:zt,
w h i c h m e a n s that there is
4. In the g e n e r a l case, in o r d e r to use our m e t h o d s to solve p r o b l e m w r i t e the function / in the f o r m ](z):min/i.(min/i,(...( rain /it. ,( rnin [il (xlk)1/~ ~ i,~l,
t*~ f~"t
J,t-x) I ],,-* ~
U
U
U
~--I, i,~Ji, it~I],
:k-l~/']k~t
Ju--=) ] 9 9 ") I ]t. ~ "'" i
U
IkG ]k
J~.),
U
k_tC- IJ.k_ I JkEJi~:_1
I ~ = { 1 , 2 ..... .},
no
(i), we have to (6)
45 where ~! are non-decreasing (with l
t,c:l, ~.CLT/, 4a : 9a "
/~-1~
Jif-I
JkEJi~-I I k ~ ' l J k
different elements ~eX. The upper bound (7) of the number of different vectors zEIIXt, which are checked for satisfaction of the constraints of problem (i) when seeking the solution, shows the extent to which the presence of rain in the writing of function ] "reduces the total inspection ll]X~I of vectors, made in the general case of solving discrete optimization problems, even when the target function is monotonic and the constraints y=Zy,, ]=~j; y,, ], are arbitrary, are monotonic. It can be concluded from our results that, in problems of discrete multicriterion maximization, when the value of each criterion depends on its own group of arguments, though all the arguments are linked by joint constraints, the convolution of criteria of the minimum type with weights is preferable, from considerations of actual computation, to the weighted sum (while the convolutions are otherwise equivalent /i/). As a practical example of writing function ] in form (6) , we consider the problem of maximizing the multiproduct flow in a network. We aim to transmit over the network as large flows as possible of several types of product under the condition that, as the total flow over each rib increases, there is a non-linear increase in rental cost, and the total expenditure is bounded. Maximization of multiproduct flow is a multicriterion problem, and in accordance with what has been said, we shall maximize the minimumof the flows, possibly with a weight. The weighting factors may be numbers proportional to the demands on the flows of the respective products. Assume that, in (6), all the functions
9fl r (SIr+ t [ Jr+ 1 ~ JIr),
zl,+,~-
rain
where
]irr247
zjk -~- ~j~, r = 0 , 9 4.....k--2,
Ir+t~IJr+l are sums:
li,=
~,
zj~+~,
r = O , 1. . . . . k - - t,
Ilk =zlk.
Jr§ Then function
](z) will be fully defined by the collection of sets of indices
{L, ( A l t o ~ h ) ,
(Z,,Ih ~ A ,
i~
.....
(1,~_,lt~-,~;,~_,,..~,
tt~X,,,
],~l~, io~I,), (l~[]~l%_t, i~-,ElJk_,, .... i,~l,, ],~l~, 1oG/,)}, where the min has to be taken with respect to indices of Ih, and the sum of the respective minima with respect to indices of .dq_z , l=|, 2 ..... k. We assign to arguments x~t the values of the number of flow units of each product over each rib. The number n is equal to the sum over all types of product of the number of ribs participating in the particular product flow. The problem constraints can be written either in the form )~y,(x,)~y~ where yi{z,)VI=|, 2, ..., n are the devices moving each product flow over each rib, or else as constraints on the sum over all ribs of the rib rental function, dependent on the sum over all products of the flows over the rib. We can also consider separately constraints on each rib (the vector of contraints). In all these cases, the constraint function increases as the flow increases. When stating the problem, the most laborious part is construction of the collection of sets of indices. Let It be the set of all types of product sent over the network. The further construction is performed separately for each product JoEl,. Consider all the paths from the source to the sink of product f0 9 We isolate the paths which do not have a rib in common with any other path, and assign them indices ],'.....it", r,>~0. Then Ij,,,l=l,2,...,r,, consists of all ribs of path ],',d,,={iJ} for i,~Ij,, and does not divide further. The remaining (not yet indexed) paths are divided into minimal groups in such a way that paths of one group have no ribs in common with paths of other groups, such groups are given indices ]~'*x,...,],S',ss~l;Jt.=~{],',..., ],"} are different for each product io~lo. Now consider a group of paths ],t,r,
46 into account in the constraints. This naturally does not destroy the monotonicity of the constraints. The subpaths divided after this procedure are again divided as described above, until only one rib each in each unexpanded group remains, after which the resulting problem (i), (6) can be solved by the methods of Para.3. If the problem dimensionality is too large for these methods, we recommend using algorithms 3, 4 in order to improve some version of the flow distribution over the network. This is done as follows. Assume that we have distributed all products 1,2,...,n, except i, over the network paths; then for the t-th product there are several paths and for each, by algorithm 4, we solve the problem of maximizing the min over all product types of the ratio of the product flow to the demand for it, under the constraints on the total resource. Hence we choose the best path, from the point of view of the system operating criterion, for the i-th product when the paths of the other products are fixed. Since algorithm 4 works quite fast, we can repeat this procedure for different i=|,2,...,n , several times, thereby in general improving the structure of the flow distribution, though ovbiously we can only be sure of obtaining the global maximum by reduction to problem (i), (6), as described above. 5. In all our methods for solving problem (i), essential use has been made of the fact that the function specifying the constraints can be written in the form y(y,(x,),...,y~(zn)), where y is non-decreasing with respect to each argument. If problem (i) has vector constraints, then, to utilize our methods, they must be assigned functions y' , non-decreasing with respect to each argument, in the form y'(y1(x,),...,y.(x,))~yo',t=l,2,...,N. Then, when defining the complexity of the algorithms for solving problem (3), the number of different z ~ X inspected has to be multiplied by n + N , since, at each step, we make a check of the constraints, and in general compute min~(xl). If the problem constraints cannot be written in this way, then, for the function ] = m i n ~ , the maximum number of inspected x ~ X may be close to IX[ , e.g., when we have the lexicographic problem of optimization and the constraint is the set of all x ~ X realizing the max of function y. Hence, in order to avoid a complete inspection in such problems, supplementary information is used about the properties of the maximized function and the set of constraints. In certain cases, when quite a large inspection of different x ~ X is still needed to obtain the solution of problem (i), while the allotted time is short (the problem is solved in real time), it may be sufficient to find a vector x, satisfying the constraints, which is "not too bad" in the sense of criterion I" For instance, consider the problem of the optimal distribution of elements of a system (say a group of radio stations /3/). Assume that we have a set of objects I={I,2 ..... n} , and a set of locations ]={I, 2 ..... m ~ then, for any iE[, denoting by x~ the location of the ~-th object, we obtain Xi=], X=~Xf, m~n. The distribution of the objects over the locations is characterized by the fact that two different objects cannot occupy the same location. We are given the matrix II~(])I] (of values of the functions ]i in sets Ms ) of characteristics of allocation of each object 6~I, and we know the function F(1,(x,)..... 1,(z.)) (where the function F is non-decreasing with respect to each variable), the maximization of which is the aim of optimal distribution of the objects over the locations. We thus obtain the problem max
{x~X[~iCzI VIii}
F (I* (z,) . . . . . I . (z.)).
(8)
In the same way as above, we can order the values of F b y algorithm 5 (without cancelling the repeated elements) and seek the vector x~X, satisfying the constraints of problem (8), which is first in order of non-increasing F. In the worst case, however, we have to inspect all the elements that do not satisfy the constraints, before the solution can be found. It is therefore sensible to reduce the inspection as compared with that of algorithm 5 in order to hit the constraints faster at the expense of incurring a loss in the maximized criterion. To construct the vector x , satisfying the constraints of problem (8), and acceptable in the sense of criterion F , we give: Algorithm 7. S t e p i. w e arrange sets X~ in non-increasing order of values of.~(z,), and obtain Xi={z~ t.....xC} Viii (see algorithm i). S t e p 2. We renumber objects i~I in non-increasing order of their priorities are priorities) in accordance with the function F.
(if there
S t e p 3. We choose x,' and store components x,',i=2,3,...,/, l
47 When xt'=zi' and, for any xt t left in set Xt, there exists i
i
= , and so on, comparing them,
not
with
xlt ,'-,~
but
with
~T ~
z-, -
If the values
are the same, we proceed as described above. If x~_, is unacceptable, we put r~-~=| and increase the superscript of the ([--2)-th component to the minimum extent needed for it not to be the same as the values of the previous components, and so on. If even the first component goes outside the acceptable range, it remains either to widen the acceptable range or to return to the vector obtained at step 3. S t e p 5. step 4 can be repeated several times with different acceptable ranges, e.g., discarding all the time the values of components realizing min;.~[~(xl), as long as it is possible to construct in the relevant X~ a vector x satisfying the constraints. Obviously, if, for any i~[ , all the values of /t(x,) in Xf are distinct, then algorithm 7 at step 3 gives the Pareto-optimal vector with respect to the system of criteria (~i(z,)li~I), while otherwise (~(x,")]i~]) is weakly Pareto (in the set of constraints of problem 8)). Thus) if we interpret function F a s a convolution of criteria ~ then, even at the first iteration, algorithm 7 gives not too bad a result, which is certainly important in actual problems. The above methods of solving problem (1)--(3), (8) enable us to deal with optimization or at least improvement of a whole range of complex problems with different practical applications. The isolation in discrete optimization, and in particular, multicriterion, problems with monotonic constraints, of the rain (max) functions, when the criteria are written in accordance with relation (6), immediately reduces the inspection of possible versions in accordance with (7), thus speeding the search for the solution. REFERENCES i. GERMEIER YU.B., Introduction to the theory of operations research (Vvedenie v teoriyu issledovaniya operatsii), Nauka, Moscow, 1969. 2. KREINES M.G. and ZHUKOVSKII V.D., Foundations of the optimal choice of diagnostic methods, in: Mathematical methods of optimization and applications to large economic and technical systems (Matem. metody optimizatsii i ikh prilozheniya v bol'shikh ekonomich, i technich, sistemakh), TsEMI Akad. NAUK SSSR, Moscow, 1980. 3. NOVIKOVA N.M., The control of a system of radio stations, in: Mathematical methods in operations research (Matem. metody v issl. operatsii), Izd-vo MGU, Moscow, 1981.
Translated by D.E.B. U.S.S.R. Comput.Maths.Math.Phys. Vol.23,No.3,pp.47-55,1983 Printed in Great Britain
OO41-5553/83 $10.OO40.OO 9 1984 Pergamon Press Ltd.
EVALUATION OF THE CONDITIONALMINIMUM OF A FUNCTION WITH GIVEN ACCURACY* V.N. NEFEDOV Methods for finding with given accuracy the minimum value of a continuous function in a compactum 3re described. The problem often arises of finding the minimum ~ of a continuous function y(x) in a compactumX. Due to computing errors, and possible inaccuracy in specifying the function y(z) , or the functions specifiying the set X , our actual task is merely to find the number 9', differing from the required ~ by at most the chosen accuracy 6of solving the problem. In the present paper we give methods for finding such a number ~' for a preassigned accuracy 6 > 0 under certain assumptions regarding the disturbances of the set X and of the function y(x). i. We shall require some notation. Let ~ be the collection of all sets of m-dimensional Euclidean space E'; fc(m) is the class of functions defined in E~, and np(m) is the class of continuous functions defined in ~'. We introduce the sets
A r g : " " [ Zo, m (x) . . . . . y, (x), yo (x) ] = (x~Yoly,(x)~s,, i = i , 2 , . . . , x, ya(z)~ zinf yo(x')+~} '~Y where ...,
Y={z~Yoly,(z)~O, i = l , 2. . . . . ~}#~, Yo~.~E~,y,(z)~fc (m), i=0, l ..... % ~FE', j = t , 2, '~>~t; Q6[y(x), Y] ={~(x)~np (m) l W ~ r
*Zh.v~chisl.Mat.mat.Fiz.,23,3,590-6ol,1983
Ig(z)-y(z) I<~),