•
Yugoslav J ournal of Operation s Research
8 (1998), Number 1, 111-128
J ovan Petrie memorial volum e
•
PSEUDO-DELTA FUNCTIONS AND SEQUENCES
IN THE OPTIMIZATION THEORY
Endre PAP, Nenad TEOFANOV
institute of Mathematics, University of Noui Sad
Trg D. Obrodo oica 4, 21000 Nooi Sad, Yugoslavia
Dedicated to the memory ofProfessor Jovan Petrie
Abstract: Pseudo-delta sequences an d pseudo-delta fun ction s in the fram ewo rk of
pseudo-analysis are presented. In this paper a n importan t property of pseudo-delta
sequences with respect to pseudo-convolu tion is proved. The st u dy of such sequences is
of interest sin ce we can use them in approx imaton of a class of operato rs that appear in
opt im iza t ion problems.
Key words: P seudo-addition , p eu do-mu lt iplicatio n, deco mposable mea u re, p eudo-integral,
pseudo-convol ut ion, pseudo-delta seque nce.
1. INTRODUCTION
Classica l mathematical analysis is based on t he field of reals (R, +,.). Th is has
implications on t he corresponding linear algebra, measure theory and integration
theory , which are corner stones in many applications on ordinary and partial
differential equat ions and difference equatio ns (most ly linear). Many types of nonadditive measures and cor responding integrals (1 6 1, !8], 118], 129 ]) have been
investigated, stim u lated by many different problems in practice, mostly by modeling
different uncertainties in the theory of fuzzy systems, which are bases for the decision
theory a nd artific ial intelligence, t he system theory, an d the gam e theory. An important
su bclass of non-additive measures con tains decomposable measures with respect to
some semir ing ( [ a , b ).G7 . I8i) . - oo < a < b < +<x:J (15 1, [7 ]) . There are many different
applications of a nalysis based on such sem ir ings (usually called pseudo-analysis) in
optim ization theory, nonlinear equa tions of different types, 'decision theory, etc. ([ 21,
181 ,1 121, 1181, 1201, 125]).
•
11 ~
E. P ap, N. Teo fanov/Pseudo-delta Functi ons a IHI Seque nces in th e Opt.irni za ti on Theory
In the second Section we present the basic definitions and exa m ples related to
sem ir ings on r eeds. In Section Three we briefly present the corresponding measure and
integration theory and introduce the notion of pseudo-convolutions , a ge nera lization uf
the classical convolution of functions . Section Four contains the basic results on
pseudo-delta function s and pseudo-delta sequen ces and in Section Five we prove a
theorem on the pseudo-convolution of pseudo-delta se que nces. In Section Six. we
present some applications in the optimization theory.
2. SEMIRINGS ON AN INTERVAL OF EXTENDED REALS
Let
[a ,b 1be a closed su binterval of [
- CX) ,
1(in some cases we will also take
+ CI)
sem iclosed su binter vals). The partial order on [a ,b 1 will be denoted by: < . The sym bol
•
< has the usual meaning: for any x , Y
E [
a , b ], x < y if and only if
X
< y and x +-
.v .
The str u ctu re ([ a.b ], EEl ,0 ) or [a ,b 1m,c,) is a sem ir ing in which the
operations EEl and 0 have the following properties:
The operation EEl (pseu do-addit ion) is a fun ction EEl : [ a,b 1x [a,b 1~ [a,b ]
which is commutative , nondecreasing (with respect to :::; ) a ssociative and eit her
is a zero elemen t, denoted by 0, i.e., for each x E [ a, b ] 0 EEl x = x holds.
The pseudo-addition of n eleme nts is defined by
define further: EEl;' I
Xi
= lim EEl;' 1
Xi .
asn
d) 1. I Xi
=
or b
(J
=(=11 - 1
XII w d) 1. I Xi ) '
We
•
1l - ) {.fI
Pseudo-addition EEl is idempotent if for any x
Let [a ,b
L = [ x : x E [a ,b ], x > 0
E
[0 ,b ], x EEl x
=
x holds.
).
The operation 0 (pseudo-m u lt iplica t ion) is a function 0: [a ,b ]x[a ,b ]
is commutative , positively non decreasing, i.e. x :::; y implies x 0 z < y 0
[a ,b] which
Z, Z E
associative and for which there exists a unit elemen t 1 E [ a , b 1' i.e., for each x
[a,b
E [
L'
a ,b ]
1 0 x =x .
We su ppose, further, 0 0 x = 0 and that 0 is a distributive pseudomultiplication with respect to EEl , i.e., x 0 (y EEl z) = ( x 0 y) EEl (x 0 z).
Let rand h be functions defined on X and with values in a sem ir ing
[a,b ] $ ,<-) .
Then, we define, for any x E X , ( f EEl h )(x ) = (( x) EEl h i x ), (f 0 h. t i.x ) = [ i x ) 0 h ix ) and
for any A E
[
a.b ] (A 0
( )( x ) =
A0
j '(x ) .
In this paper we will consider the following sern ir ings:
E Pap.. ' Teofanov P eudo-delta Functi on. a nd eq ue nce in th e Optimizati on Theory
I a) ( i ) T he scrn ir ing (- 'Y..,
x
]rn in,+ :
'l;;
y = min lx.y :'
11 3
x 0 y =x + y , x . y e(- ':I) . + oo ] .
We have 0 = +rf and 1 = O. The id empote nt operatio n min indu ce s a partial ifu ll) o r de r
the followin g way : x ::: y if and only if min Lr, y) = y , wh ich is the opposite o f the
111
u u al o r d
' I'
un the in terval (_rr. , rr.
[-f.,
I a) ( i i ) The enu r ing
.,•
~
W" have 0 urd
' I'
. - '<Y
.0.
~
Y = .•" +..
."
X' y
.•
E [ -,'f). ,
'r: )
•
and J = O . The idempotent operation m ax in d uces a partial (fu ll)
r
in the followin g way : x $ y if and on ly if max (x,y) = y . H ence t h is or der is the
usual o rde r u n till' in terv al [I b ) (i) T h e se m ir ing ( 0 , +
y
x
Wl' have 0
e
'r
-f
r.
.
-r )
]rnin .:
mi n i x.y }, x 0y =x -y, x,ye ( O,+oo l.
and I - ) . T h e idempo ten t operation min induce s a partial tfu ll) or der
th ' lolluwin g way : x ~ y if and o n ly if min (x, y) = y , which is the opposite o f the
111
1I .
l"": :
f.
\' _ III ax' II ~, - • Y I, .
I.
ual u rdr- r un the in terval (O , + 'f ].
) h) (ii) '1'111' , l' lIl in n g [ 0,
lila]'
w,III
have 0
II r ill'
-u u r rng [ I),
W.. haw 0
In
r
).
v if and unly if m ax (x, y) = y, which is the u sua l orde r o n the
•
t
f
JUIIIl
1I .
up"rllt lll n
.11.1."
.flu
() . '1'111' id nn pote nt ope ra t io n min indu ce s a partia l (fu ll) ord ' I'
\' if and only if m in (x,y) - y . l lenco this urder is the
h .. Iollnwr ru; way: .r
t
U, ~ r
) 'I' IH' id empo ten t o peration max induces a partial (fu ll) order
nd I
r
(t
.\ v , .\ , y - [
).
uppu. Il .. uf t lu-
III
•
I
th« followu u; w ay : x
I 0. ~
) IJlUX . .
1.\ v ,.
0 and )
ui u -rval
f
u al orch- r un th« in t.. rvul
n un uh-m po u -n t
a nd
C)
' I t' 111
cunt rn u uu . I-(l' n..r ItA lr ~
10,
r
I. W.. al so co nsidu r a
sem ir ing with
III III
wh ich t h« Jl .. udo -operations an' d.. finod by munotnn« and
1151,1 ) I
114
E. Pap, N. TeofanovlPseu do-delta Functi on s a nd Seque nces in t he Optimizat ion Theory
By Aczel's representation theorem for each st r ict pseudo-addition E!3 (i.e.
continu ous and monotone on its domain), there exists a monotone function g
(generator for E!3), g : [a ,b] ~ [0 ,00] su ch that g (O) = 0 and
If the zero element for the pseudo-addition is a , we will consider increasing
generators. Then gl a) = 0 and g (b ) = 00 . If the zero elemen t for the pseudo-addition is
b, we will consider decreasing generators. Then g(a) = 00 and g( b) = 0 . Hence g is an
,
isomorphism of sem igrou p ( [a ,b ], E!3 ) with the semigr ou p ( [0 ,00 ], + ) .
Using a generator g of st rict pseudo-addition E!3, we ca n define pseudo•
multiplication 0 :
•
u0 v
=
g - l 19lu) glv».
This is the only way to define pseudo-multiplication 0, which is distributive
with respect to E!3 generated by the function g (see [13 J). The operatio n 0 has all t he
properties of the pseudo-multiplication from the previous Section. It can be easily seen
that g O) = 1 .
We will denote by S the domain of a sem ir ing of the type I - III.
3. MEASURES, INTEGRALS AND CONVOLUTION
IN PSEUDO-ANALYSIS
,
The goal of this Section is to define the integrals based on (J - E!3 decomposable
measures (see the definition below). For that purpose we omit the details and refer t he
reader to [18] for a more detailed stu dy of the subject (see also 12], Ill] , 112], [24 J).
Let X be a non-empty set . Let L be a
(J -
algebra of subsets of X.
Defirrit.ion 1. A set [unction. m: L ~ [a ,b] (or semiclosed interval) is a
E!3 -
d ecomposable m easure if it holds that m (0 ) = 0 (if E!3 is ide mpoten t we do n ot always
suppose this cond ition); m (A v B ) = m (A ) E!3 m (B ) for A , B E L such that A r-. B = 0 .
A E!3 - d ecomposable measure m is (J - E!3 - d ecomposable if
00
m ( UA , ) =
i=l
00
EB m (A i )
i=l
holds for any sequence {A i } ofpairwise disjoint sets from L
•
E . Pap, N'. T eo tanov/Pseudo-del ta Function s a n d Seque nces in t h e Optimizatio n T heory
11 5
Remark that in the case when 67 is idempotent it is possible that m. is not
defined on an em pty set. Let m. be a a - 67 - decomposable measure. Some function
{ : X ~ [ a , b ] is measurable if for any C E [ a , b ] the set { x : (( x ) <: C } belongs to L
We su ppose further that ([a ,b],67 ) and ([a .b],0 ) are complete lattice
[a ,b]
ordered sem igr ou ps. A complete lattice means t hat for each set A c
bounded
from above (below) there exists sup A (in f A) . Further, we su ppose that [a ,b] is
endowed with a metric d compatible with su p and inf, i.e., lim sup x"
=
x and
lim in!' x " = x imply lim d t» ; , x ) = 0 , and which sat isfies at least one of the following
/l - >lf)
con dition s:
(a)
d i x 67 y , x ' 67 y' ) < d i x , x ' ) + d ry , y' )
(b)
d (x 67 y , x ' 67y' ) < m ux { d (x , x ' ), d (y , y' )} .
We su ppose further t he m onotonicity uf the metric d , i.e.,
x < z < y
implies
d t:x , y ) > max { d ry , z ), d t:x , z) } .
For
example,
on
t he
•
•
sernm ng
(- etJ, + etJ
]rni n,+
t he
metric ' d (x ,y ) =
sa tisfies all uf t he preceding conditions.
e maxt r ,y ) - e - m ln \x ,y )
We define the characteristic function with values in a semir ing by
0, x eo A
x E A.
1,
The mapping
e:
X ~ [a ,b] is an elemen tary (measu rable) function if it has the
fullowing representation
•
<f J
e=EB
. a;0 XA, fora ; E[a ,b]
1= 1
a nd A I
E
2: is disjoint if 67 is nut idempotent .
•
The pseudo-integral of an elementary function is defined by
•
ffi
W
Ix e18> dm = EB a, 18> m CA
j ) •
1= 1
For any measurable function {: X ~
[a,b]
one can construct a sequence { tp /I
}
of
elemen tary functions su ch that, for ea ch x E X, d (tp /l (x), { (x » ~ 0 uniformly as
ti
~ co
(see [18 1!. Using this fact we give the following
11 6
E. Pap, N. Teol'a novlPseudo-delta Functions and Seque nces in th e Optimizati on Th eory
Definition 2. The pseudo-integral o] a bounded measurable [unct ion
r:X
[a , b] is
defined by
fEll f
X
where { lfJ 1/
}
0 el m = lim
"
C
lfJl/ (x ) 0
dm ,
) (f
is the sequence of elemen tary [unctions constructed in the above-mentioned
theorem.
Some elementary properties (e.g. , linearity ) and applications of the introduced pseudointegral can be found in 1141, 1181. 1161, 1171, 1191·
Definition 3. L et B( X ,8 ) d enote the semimodule o] all [unctions [rom X into the
•
. sem iring (8,$,0) such that
•
C
j"(x) d m
where m is a
(T -
E
8 ,
$ - d ecomposable measure .
•
In t he rest of t he paper we take that X = R .
Definition 4. The pseudo-conooluiion o] two [unctions
[: R
[a ,b] and
h:
R - ) [a,b ] with respect to a $ -dcco mposable m easure m. is given in the [ollouung way
•
where
m il = m.
in the case of su p-dec.:o mposable measure m (A ) = sup x u
\
h sx ) (an d in
the case o] inf-decomposable measure m (A ) = infxEi\ h lx ) ), and dm i; = h. 0 dm in the
case of $ - decomposable measure m , where $ has an additive generator g and g o m. is
the Lebesgue m easure tg-calculus ).
it is easy to check that a pseudo-convolution is a commutative operation. This
follows by the equality
U * h ) m ( x) =
•
f:
j"(x - t ) 0 dm j, =
f:
h ex - t ) 0 dm f = ( h
* f')( x)
.
•
I t is also an associative operation .
The following exam ples show the explicit forms of pseudo-integral and pseudoconvolut ion for specia l important cases (see also 1201, 121 1).
E. P ap, N. Teofa nov/Pseudo-delta F unctions a nd Sequences in t he Optimi zati on T heory
11 7
I a) (i) For any real valu ed function h bounded from below we can define a 0" - su p decomposable measure m eA ) = infxEA h ex ) (A c R ) . By taking EEl = min = inf , 181 = +,
we obtain
JR ( i81dm = ninfoR (f(x ) +h(x » ,
for ( bounded below. We will denote by B(X,(- oo, + oo
r
in
.+ )
t he sem iring of all
fun ctions bounded from below (wit h respect to the u sual order). In t his case t he pseudo
convolution becomes
(f* h )/II (x ) =
fR
•
JR h (x - l ) i81 d m /
El1
{ (x -l ) i81dm" =
= inf (f(l ) + h (x - l » .
tER
I a) (ii) For any real valued function li bounded from above we can define a
decomposable measure m eA ) = sUPxEA h ex ) (A c R ) , Taking EEl = max
•
0" -
= SUp,
181
su p -
= +,
we obtain
JR f 181 dm = sup(f(x ) + h (x » ,
XER
for ( bounded above. We will denote by B (X , [- 00, + 00
ynax.+ )
the sem ir ing of a ll
functions bounded from above.
(f
*h)
/II
(x)
=
J: lC
x - t ) 181 d m j,
=sup h (x -
t ) 181 dm. /
R
= su P (f (l) + h.( x -
t ») .
tER
•
Cases I b) (i) and I b) (ii) are t reated sim ila rly, taking in stead of +.
II For bounded fun ctions ( with values in semir ing
[0,+00] min ,max
the pseudo-integral
based on inf-decomposable measure m , m eA ) = inf xEA h ex ) , is given by
•
r'
R
( i81dm = inf (max(f(x ), h (x» ) ,
x <, R
a nd t he pseudo-convolu tion of t he functions rand h will be
•
(f* h )lII( x ) = fEB { (x -l ) i81dm" = i n f ( m ax (f(l), h( x - l») .
R
tER
III If EEl is a st r ict pseudo-addition with a monotone generato r g, g om : ~ -) [ 0, g ee ) ]
with
C E
[G,b ]
is a measure and !"is a measurable function, we have.
•
118
E. Pap, N. Teofanov/P seud o-delta Fu nctio ns a nd Sequences in t he Optimizati on T heory
= di g
where dx
0
m)
is the Lebesgue m easure and u 181 v
= g - llglu) · glv».
In the
sequel, ge nerator s g will be m onot one and contin uou s fun ctions , so g - llgls» =
glg
1 ls»
=s .
The pseudo-convolu tion in t h e se nse of the g -inte/:"Ta l is give n by
(I' * h )lx l = g - l (f g( I'U» · g( h( x - t n dt ),
0)
Remark. Using contin uously differentiable gene rato rs g, the g -der iva t ive of a
differen t iable fun ct ion ( ca n be defined. This definition has local characte r and the
fu nctio n ( m ust have the same m onotonicity as the gene rato r g. The prope rties of the g deriva tive and g -in tegr a l and the corresponding applications can be found, e .g . in 118 I,
1151. 114 1, 122 1, 1201 ·
•
4. PSEUDO-DELTA SEQUENCES
4.1. The 'd e lta function' and delta convergent sequences - the classical case
Abou t sixty yea r s ago , P aul Dirac in troduced the famous 'delta fu nction ' while
solving prob lem s in quantum-mechanics . Although t he formal u se of th is object has
become an e fficien t to ol in quantum mechanics ca lcu lu s, t h is 'fu n ction' h as
contra d ict ory pro perties: it differs from zero in on ly on e point , bu t its int egral equ a ls
one. Th is contradiction has been overcome by the co nstruction of variou s delta
seque nces which in a certa in sense converge to a 'delt a. fun ction' a n d a lso by defming a
'delta fu n ction' as a fun ctional. T he classical 'delta function' is introdu ced as a 'fu nct ion'
defin ed on the real line as follows
o:
if x = 0 ,
o
if x
~
0,
with a n a dditiona l pro perty
f ol x) dx = 1 . We have fur ther t hat for a
test fun ction r/!
(in fin it ely differentiable funct ion wit h compact su ppor t)
f0([ - x )r/! U )dt = ¢(x)
ho lds.
R
A classical del ta seque nce
( () II
l is a sequen ce of fun ctions wh ich converges to a 'delta
fun ction' in t he following sense
lim
n -> (f l
f (~'II U - x)¢U) d t = ¢(x).
R
Eo Pap, N , T enran nv/P sellon-d elt a FII net in ns a nd Seque nces in t hl· (lpl iIlli za l inn T hl 'nry
11!l
a nd at th e same ti me the equa t ion
o
b
lim
II
)
Ji i
ll
(t - .r)
if b < x or x
a,
dt =
( ~)
(t
1
rl
ifa s x sb .
is sa t is fied.
By t 2) it fo llows that lim
11
) r ll
J
H.
S II
(x) dx = 1 , which ex pla in s the equa t ion
Il
orne well known exam ples (see 19 1.
1 sin
/I X
1
,
/I
•J
I}
12 :~ I) of de lta
,
-
1/
l'
/IX
~
seque nces are :
"
•
Jr l +/I " x ~
X
Jr
Jii( x) (Lr = 1 .
4.2. Idempotent analysis and g-ealculus
If the pseudo -additio n EEl is e ithe r max or min , t he pseudo-delta fun ction will
be (see also 1121. 121 D
•
•
c)'
1
, - ( x)
if x = 0 ,
•
=
t a)
0
if x 7'O .
•
As u sual 0 is the zero e lemen t fo r pse udo-additio n , and I is the unit ilemen t
for pseudo-mu ltiplication . The pseudo-de lta function defined by t3) is th ' uni t e le me nt
for the 'co n volu t ion ' in the se nse of a pseudo-int '6'1'a l. As in the classical th 'ory, we
have
S ince
o- H) ,(')
may no t be a function in t h e u sual se nse, we have investigated t he
existe nce of the sequences of fun ctions which converge to c)" t),
D efinition 5. Pseudo-delta sequence It
H ( X , [a,b 1<D ,C')
1.
with the properties:
lim (8,~ '( )
Il -
2,
)
* !' Hx ) = [ i x ) ,
) (JJ
D
.,.(D,<')(
)
d
=
1
, JE
Irm
X V II
X
X
n
) Cf)
,
s: (f) (.)
v /I '
I I.S.
.
.
()• /I(i) '
a seque/lce o/ ' ./ 'II net /.Ons
E
120
E. Pap, N . Teofanov/Pseudo-delta Functi on s and Seque nces in th e Optimi zati on Theory
For sim plicity we su ppose that the e lements of delta sequen ces are eve n
functions . We ga ve the followin g characterizations q21 \) :
I a) (i)
For the som iring (- 00, +
S
.
rm n .:, ( x)
1
(=
0
(=
lmin ,. , from ( 3) it follows :
0)
if x
=
=
0,
ifx :;c O.
)
The pseudo-delta seque nce will be a seque nce which con ver ges to t h is pseudo-delta
fun ction which is t he unit e lemen t for the pseudo-convolution. H ence for the pseudo-
l the following equ at ion mu st hold
delta sequence : S,',nin"
lim (f' * 6'/'i1in. , )(x) = lim f ) s/~n i n . T ( X- I)0 d m r =
•
II - ) ( / l
R
ll - ) ( f )
•
lim inf «(t ) + S /',m n,+ (x /1 -
t€R
)0 (.1)
t» = [t .x )
.
There fo re we have :
Theorem I . The sequence of [unctions { (Y,',n in,T } is a pseudo-delta sequence
•
In
H ( R , (- 00 , + 00 lmin" ) i( the (allowing three con ditions are satisfied
•
n
S/', m , , (0) = 0 ;
1.
•
J m m ,,(x»
.)
~ .
O
/I
i( x :;c 0 ;
•
J/1,n m,t (x)
3.
An exam ple of
nin
{ 6',',
wh en n
lor x :;c 0 .
l is t he sequence
,+
Ill ' x
2m
l, for an arbitrary but fixed In E N .
,+ 00 )lllllX, . , the pseudo-delta function has the following
I a) (ii) For the se m iring [-
form
if x
1
(=
0)
o
( = - <X:\ )
=
0,
S IllUX, ' Lr) =
Ps m do- de lta seq uence
J
I
ifx e O.
o· mllx, . \
is a sequence fur which t he followin g equa t ion holds
J
II
t
lim (/' * S,~nllx . 1 )( x) = lim
11
11
. (1
)0 (1' 1
lim s u p( ( lf) + O'/~n llx, T ( x
11
) 1
We now have :
I
R
R
O·,~nllx. , ( x -
- I )) =
[ tx) :
I ) 0 tlm
r=
. Pap,.' T eo ta n o v P ell do-della Funci inn and 'eYlience _ in t he Opt inuzat ron T heory
Theorem 2. T he .equencc of [u ncttons
[olluu. tug three conditions are sat tsfied
1
o mil.
Il) )
'J
-
o rn ux
tX )
:1
s:
/I
/I
c5 max,
:
/I
I ~ a p seudo-delta sequence i( th e
= () ;
U I(X-;c.O;
uihen
- x,
(x)
fI
f
t
1~1
Y;
Il
(or
X
-;c.
0.
t ake the following -xam ple o' :~UX. (x) = - n . x '2111
II en',
WI' Q U I
( '(\
I II I I i) a n d I II I ( ii) can he t reated sim ilurly, taking into account that, in the se
WI' have I
1 . 'a • II i a na logo us to Ct se I a ) ( i ) .
CH
'
III
w«
Ill'
I
,
m EN .
tart with a cou nt erexam pl • which h ows that definition (:3) is nut adequate if
ck-finod ltv H m ono tone a nd co nu nuou ge ne ra tur g.
E ump le J. L ·t u s a su i ue tha t the g -de lta fun ction c5
is defined by (a) , a nd le t the
'p lll'rul<H' g ill'
.l
011 . I
[IJ
.l
1
1
I
•
I' h-n w" hav«
I I · (,
J I(
j(.
f
I'(t)
I Igl
I'It) )
I
f Jig
,,'
I ..,.
u
gl, .
tI
dm
»
IX -l)))ell)
" bUillit'd th« g- lJIli·6n ·HI " I' fun ct ion r, \ luch,
••uuph- 1 ~'''~~Pl lor 11.1 1 .l we o hu u n
\\ t'
r .\
111 11
elm
•
11"lll' II
1"".
(
hat r..
I
ln t l
.\
III
wv
'1Vp
deltu [uncuon
lol )
U
((J
I
(fj ( ,
1
tilt' li,1I"W1IIg dc-firu u on
b II.l
III t lu
.\ 1
gp np ra l, differ: froin fun ction [. Fur
II .l
D efiu it ion fi • •\
,
XI
III
o] u
U
tl« II/UPP'I/g
I'
l'
f)
II
/1 b
I
, [ «.b
I el,'lil/.,el bv
('I )
()
U
It
th, idclto
IUlld1U1l I
.
122
E . Pap, N. TeofanovlP se udo-delta Fu nctions and Sequences in t he Optimization T heory
a
of.! (x) =
if x = 0 ,
(5)
b
i{ x
~
O.
fffi iiEf'J'(')(t)0 cim = 1 . Recall that the
We can easily show that we again have
convolu t ion is given by it> h )(x ) = g
I
pseudo-
(f g ({U » · g (h (x - t. n dt. ),
The g -delta seque nce ( i5 f, }, is a seque nce of fun ctions which converges to
..
i)
.
, I.e.,
~
lim g
/I
J
tf gt fU ». gto!/ tx -
t6)
t) dt ) = f tx ).
) (fl
An a lmost immediate consequence of the (6 ) is t he fo llowing
Theorem 3. A sequence { i)'f, } of functions is a g-delta sequence if and only if
g
10
i)'", where : 0" } is a classical delta-sequence with the property
Example 2. Let g U) = e - I
t E [ a,b ]
put
=[- 00, OC! ],
.
(~,,(x)
It
=- e
II x 'I.
g
,
J tu) =
(x E
R).
- In u . Generato r g is a decrea sing fun ction,
and b = 0 , 0 = 1 . From (3) it fo llows : Sf, (x)
, we obtain
s; ( x) 2:: 0
o,f =
=-
In (S" (x». If we
~
i)';; (x) = nx - In (n/ zr l .
<,
n
Example 3. Let gU)
fun ction , t
E
=-
In (I -
[ a , b ] = [ 0, 1 ], a nd a
t ),g
J (u)
=0 =0
=1-
, 1 - e-
eI
1/ .
Generator g is a n increasing
•
= 1. The g -delta seque nce
is
o/f (x) =
5. CONVOLUTION OF PSEUDO-DELTA SEQUENCES
.
We now introduce a s pe cia l s u bclass of pseu do -delta sequen ces.
Definition 7. A pseudo-della sequence
lim S,~ '
x .. 1-'l.I
f i-)al'(') }
I
"
.
is
asym ptotically admissible if
> 1 . /01' every n E N .
•
E xample 4 . P seudo-delta sequence
{ i)',I,nin .+ }
is asymptotically admissible if it
•
satisfies the fo llowing condition:
lim
i)',I,m n, .
(x) > s > 0 fo r som e c > O.
x ) .:t.l/ 1
Examp le 5 . P seudo-delta sequence
th e fo llowing conditio n: lim
X
) + 00
I)' :~ ux , +
is asymptotically admissible if it sa t isfies
o:,nux,+(x) < -c < 0
for some c > O.
. E. Pap, N. Teofanov/Pseudo-d elta Functions and S equ ence in th e Optimi zati on Th eory
12:3
Theorem 4. a) The pseudo-convolution [o r idempotent cases II and II ) o[ two
asy mp totically admissible pseudo-delta seque nces is an asymptotically adm issible
p seudo-delta sequence i[or cases I and II , respective ly ).
b ) Th e pseudo-con volut ion O(tLVO g-delto sequences is a g-deltc sequence.
Proof:
We sho w that the co nvo lutio n of two a symptotically a dmissible pseudo-de lta
sequ e n ces for Case I a ) (i) is again a n asymptotically a dmissib le pseu do- de lta sequ e n ce
for the Case I a ) (i ). The othe r idempote nt cas es can be proved in a n a nalogou s way.
a)
.
.
I r)'211ll Il
Let ( c')11I1In,+
}
a
nd
,II
,11
,T
be two pseudo delta seque nces. We s how that
}
..
their pseudo-co nvo lu tion l8 mm ,« * 8 mm ,+ Hx)
1, /1
2 , /1
. .
= I"infR lO- 1,mm
,+ u) + O· mm ,« lx /1
2,/1
t)
is a pseudo -
delt a sequence , i.e., it satisfies co n d ition s 1 - 3 in Theorem 1.
1.
.
we have inf lo· mm,T(O) + 8 11ll n (0 )
Le t x
= O . F or t = 0
If
mm
0 we have (y l ,II ,+
,T
If- R
.
t '"
.
.
(o' lI1 m,+ * 8 mm ,+ HO)
2 ,/1
] , /I
•
mce
( t)
I
I , /I
.
mm
+ r'5 2 ,11 ,+
1,/1
2,/1
(- t)
> 0 . H ence
. .
= inf lr'5 mm ,+(t) + 8,111111,+l-t))
If, R
(y mm,+ \ and J\
I
.
I
1,/1
~.
m m,«
( '2 ,11
2,/1
\
,
= O.
=0 .
are both a symptotically admissible pseudo-
s equ en ces, it is easy to see that the ir pseudo -convolution.
.
.
I11m
mm
inf l8. ,+ u) + 0'2 ,+ ( x - t» is s t r ict ly po sitive for a ll x '" 0 .
delta
t€R
3.
,11
,II
F or every a r bitrary but fixed x
.
.
we have lim inf (6 :~:n,+ (t ) + r).~n,:n,+ lx 1l -)wt E R '
I
I
\
We s h a II prove that
· min +
. min +
(~ I .n '*(~2 ,11 '
•
x
+00
x
for so me
5]
>0
for some
52
> O.
and
•
.
mm.« I x )
lim 8 2 II
1:(1"'
\
> 52 > 0
I
•
Now
•
•
] ,/1
2 ,/1
lim inf l6 mm ,» (t ) + 6 ,mm,+(x - t » =
/I - ' Cf!
IE R
min,+ I t ) + (j min,+ (x - t » > 8 min,+ (t ) + 5 > 0
lim (6
\
] /I
\ 0
2,11
0
I,ll
0
2
,
/1 - ) (1)
,
where to denote s the point in which t he infimum is r eached.
Thu s, t he first part of t he t heore m is proved.
OC) •
t
} is a n a symptotically admissible pseudo-
delt a sequence. By definition we have
lim 8 Imm
,+ (x
) >5I > 0
/I
\
t» =
1 ~4
E. Pap, N. T eofanov/Pseudo-de lta Functi on s a n d Seque nces in th e Optimization Theory
b) Let
()' IN
,II
g
()'I
.n
(x) and
() zg
,ll
(x) be pseudo-delta sequences. Their convolution is defined by
* 0zg,11 J( x) . So we have
[
I
(Jg( s f ) t)· g() L,( x- t » d t) = g - I (Jg ([
g - ' ( J(\ //( t) . <'>z, //( x - t) d t) =g - I Ui l ,//
where
[ <'\ //
I, and
: 02, // land :
s,
* <'>z ,//(x »
' (<'>1,// (t » ) . g(g - I ()z ,// (x - t ») d t) =
= g - I(61/(X » = <'>,f( x ),
l are classical nonnegative delta sequences. H ere
we used t he fact t hat the convolution of t wo delta sequ ences is a delta sequence (see II I
p. 117l.
•
6. PSEUDO-INTEGRAL REPRESENTATION:
APPLICATIONS IN THE OPTIMIZATION THEORY
In optimization problems the operator known as the Bellman operator often
occurs 1I3j). Let X and Y be arbitrary non-empty su bsets of R . Let !l E G( X x Y) . Then
the operator B : G(Y )
G( X ) is defined by
W( )(x ) = max (!l (x ,Y ) + ( (y » .
•"" y
Using t he pseudo-integral with respect to EB = max and 0 = + , we can rewrite
the preceding operator in the form
.
•
This non-linear operato r is pseudo-lineal' in the following sense
•
Definition 8. A mapping T : ElY ,S )
cond itions are satisfied
ElX ,S ) is pseudo-linear i[ the [olloioing
T(O ) =O ; T (( EBh ) =T(n EBT(h ) ; T (A 0 n = A0T(n
.
(AE S, ( ,hE El Y, S» .
The Bellman operator B can be extended over the whole B (Y ,Smax,+) by
wn (x ) = su p( hlX, Y)
( ly » .
y" y
In paper 1211 we found a condit ion which allows pseudo-integral representation for a
class of pseudo-linear operators. Namely , the followin g theorem is proved:
E. Pap, N. TeofanovlPseudo-delta Functions and Sequ ences in th e Optimization Theory
125
Theorem 5. Let S be one of the semirings of the type I-II, with the property that for
every bounded subset {aa } of S and A E S we have EBa (A i&I aa ) = A'i&I EBa aa' if
T : BeY ,S) ~ B(X,S ) is a p seudo-linear operator, then it satisfies the condition
if and only ifthe re exists a unique function h E B(X x Y ,S) such that
.
.
( T f")( x)= f ; h( x. y)i&l f( y) dy .
In the proof of the theorem (see 121 1) we u sed a pseudo-delta function to obtain the
kernel of the pseudo-integral representation.
We can now u se a pseudo-delta sequence Jl ((~. y(1) ' vl )" }, where
0. ,,(I) ' VJ (- - y ) , to construct a sequence
i
T
'l" (x, y ) = (-
{
h"
l of pseudo-kernels as follows
. (1) (x)
(e5 y ' )" )(x ) .
Then the sequence { Til f } of pseudo-lineal' operators defined by
( T ,J)( x) = f ; h" ( x. y)
e f ( y )dy
tends pseudo-weakly to T{ , i.e.,
tends to
•
as
t i ~ 00 .
In su ch a way we can approximate pseudo-lineal' operators.
The r epresentation of a pseudo-linear operator, in the form of a Bellman
operator, occurs in many fields. Let us give only two examples.
•
•
•
The Shortest Path. In discrete optimization, the pseudo-additive Bellman
operator occurs in trajectory problems (121 , [12]) with the variable time of motion.
Namely, let (V , A ) be a graph with the set of vertices V and the 'set of arcs A. The
problem of the shor test path is the construction of the shortest path from the fixed
point Xo E V to any point x E V , with the motion beginning at time to . If we denote
by h ta.t ) the time of motion on the arc a
E
A
beginning at time i , then the
corresponding Bellman operator B is defined on the set of function s
126
E. Pap, N. 'I'eofanov/P eudo-de lta F uncti on s a nd Seque nces in t he Op timiza tion T heory
(rl t . \1 ~ { t
E
R t > to } }
by
I B{ )( xi) = m in min ( r + h( a,r )),
a <-, A r .:(( x j
)
where min m,A is taken over all arcs a
E
enter ing xi ' Then the shortest t ime
A
[a ,b], for the motion from Xo to x sat isfie s the following Bellman
T l x ), T : V
equa t ion T = tB'I' ) Ef) (to 0
(\0) , where for
if x = Xo .
if x E V \ : x 0
1
+
}.
•
Antagonistic multistep games. Let X = : 1,2,.. . ,n
1 and
let M be the
metric space of t he strategies of two players. Let Pij la ,b) be the pro bability of
t ransition from the sta te i to the sta te j, if the first player chooses the st rategy a E M
and the second one the st rategy b E M . If we denote by l ij (a ,b) the in come of the fir st
player from the given transitio n , then the game is called a game wit h value if the
l
fo llowing equa lity holds for a ll x = lx ,x
"
2
, ... ,x ") E
R"
.
/1
.
min max . L Pij (c , bwx ' + 1ij (c, b») -. max min L Pij (o, bJlx ) + 1ij (c, b) .
a
b
a
) =1
b
j =]
•
T he opera tor B : R "
~
R " given by
B j (x ) = m in max
a
b
"L Pij la ,bJl x )
.
+ lijl a, b))
) =1
is t he Bellman operator for the game. It has the following properties
B (c + x ) =
C
+ Blx ) x
Blx ) - Bly ) < x -v
•
E
R" , c
X E
= (c,c,.. .c) E
R" ;
(7)
R" , Y E R " ,
(8)
I I= max i x i .
. where x
It ca n be proved, using dynamic programming 141 , t hat the value of t he r-step
game given by t he initial position i and the term inal in come x
E
R " of the first player
always exists and is given by Bj' (x) , The representation of any operator B which
sat isfies (7) and (8) has the form of t he Bellman operator of the game. For details see
[101·
E . Pap, N. TeofanovlP seudo-delta Functi ons and Sequences in th e Optimizati on Theory
127
7. CONCLUSIONS
We presented a part of even developing pseudo-analysis which serves as a
mathematical base for many different fields such as fuzzy logic, decision theory, system
theory, nonlinear equat ions, optimization, and control theory. Pseudo-delta functions
and pseudo-delta sequences in nonlinear analysis play an analogous role to deltafunction s and delta sequences in lin ear analysis.
REFERENCES
III
121
131
141
151
161
/7 1
I~ I
191
11 01
III J
11 2J
1131
1141
11 51
11 61
Antosik, P ., Miku sinski , J ., Sikorski , R. , Th eory of Distributions, Elsevier Sci. Publ , Comp. ,
Amsterdam , PWN-Poli sh Sci . Pub!. . Warszawa, 1973.
Baccelli , F ., Coh en , G., Olsder, G.J., and Quadrat, Synchronization and Linearity: an Algebra
fo r Discrete Event Systems , John-Wiley and Sons, New York, 1992.
Bellman, R. , Karu sh , W., "On a new functional transform in analysis: the Maximum
T ransform", Bull . Amer. Math. S oc., 67 (1961) 501-503.
Bellman , R., Dynam ic Program m ing , Princeton University Press, Princeton , NJ; 1960.
Cu ninghame-Gree n, R.A., Min imax Algebra, Lecture Notes in Economics and Math. Systems
166, S pr inge r-Verlag, Berlin-Heidelberg- New York , 1979.
Denn eb erg, D., N on-Additive Measure and Integral , Kluwer Academic Publi shers, Dordrecht,
1994 .
Golan , J .S., The Th eory of S emirings with Applications in Mathematics and Theoretical
Com puter S ciences , Pitman Monog. and S urveys in Pure and Appl. Maths. 54, Longm an , New
York, 1992.
Grabisch, M., . Nguyen , Il.T., Wal ker, E.A., Fundam entals of Uncertainty Calculi with
Applications to Fuzzy In ference, K1uwer Acad emic Publishers, Dordrecht-Boston-London,
1995.
Kanwal , R., Generalized Fun ctions: Theory and Technique, Academi c Press, 1983.
Kolokoltsov, V.N ., "O n linear, additive, and homogenou s operators in idempotent a na lysis",
in: V.P . Maslov and S.N. Samborskij (eds.), Idempotent Analysis, Advances in S oviet
Math ematics 13, Amer. Math . Soc., Providence, Rhode Island , 1992, 87-101.
Maslov, V.P ., "On a new super pos ition principle in optimization theory", Uspehi mat . n auh, 42
(1987) 39-48 .
Maslov, V.P., a nd Samborskij S.N. (eds .), Idempotent Analysis , Advances in S oviet
Mathematics 13, Amer. Math . Soc. , Providence, Rhode Island , 1992 .
Mes.ar, h. a n d Rybarik, J ., "Pseudo-arithmetical operations", Tatra Mountains Math . Publ. , 2
(1 993 ) 185-192.
P ap, E., "Integra l ge ne rated by decomposable measure", Un iv. u N ouom S adu Zb. Rad.
Prirod.-Mat. Fak. Ser. Mat ., 20 (1) (1990) 135-144.
P ap , E., "g -ca lcu lus", Univ. u N ouom Sad u Zb. Rad. Prirod.-Mat. Fall. Ser. Mat ., 23 (1 ) ( 1993)
145-150.
•
P ap, E ., "Solu tion of nonlin ear differential and difference equa t ions", Proc. of EUFIT '93, 1
W J93 ) 498-504 .
11 71 Pap , E ., "No nli nea r difference equat ions and neural nets", BUSEFAL, 60 (1994) 7-14 .
11 81 Pap, E., N ull-A dditive S et Fun ctions, K1uwer Acad emi c Publishers, Dordrecht-BostonLondon , 1995.
,
,
1 ~8
E. Pap" ' . 'I'eolu nnv/Pseudu-de lta Functions and Sequences in the Optimization Theory
II (II Pap, E., "Deco rn po a ble measures and nonlinear equations", Fu zzy S ets and Systems, 92
(1 9~17 ) ~ {)5-~~ 1.
12°1 Pap, E., Ral evic, . ' ., "P eudo-La place tran sform ", Nonlin ear Anulysis (to appear).
1211 Pap. E., Teofanov, . ' .. "P eudo-de lt a seque nce" (prepr int).
1221 Rulevicv , ' ., "Some new properti e of g-calculu ", Uni o. u N ooom Sadu Zb. Rad. Prtrod.sMat .
Fah. S er. su«; ~ 4, 1 (199 4) 139-1 57.
12.'1 Sc h war tz. 1.., T h corie des d istribut ion s t. n, Pari , 1950, 1951.
1241 Sugeno, M., a n d Murnlu hi, T .. "P eudo-additive measures and integral s", J , Math . Anal.
App!., 1 ~~ ( 198 7) 197 -~~ ~ .
12:' 1 Wan g Z., a nd Klir, (; ,J., Fu zzy M easure Th eory , Plenum Pre s. New York , 1992.
•
-