t
Depa:zt1ment of Stati,sti.c8~ Univel'Sity of North CaztOtina at Chapel, Bil.'t.
This bJork bJaB supported in part by a g:rant from the Office of Naval Research~
Contract Number N00014-6?-A-0321-003 (NR04?-09S).
tt
Department of
MathematiC8~
University of North Carotina at Chapel, Bi'/,1,.
OPTItW.ITY CctmITIOOS PMJ CamRAINT
(bu.IFICAn<16 IN BNooi SPACE
by
F. J. Gould t and Jon W. Tollett
Department of Statistics
Unive1'8ity of North Carotina at Chapel Bil.'t
Institute of Statistics Mimeo Series No. 707
AugU6t., 7910
Condi ti.Cr-.3 u!":,l
C~t i~;~ity
t
~~
Gc~ld
F. J.
Q-..;;alificat io~s in Bana.ch Space
Con::~.raint
and Jon W. Tolle"
!n th:;~ pe;.per neces~~l..7 optimality conditions for nonin Banach spaces ar.<i constraint qualifica.tions for their
a?~:~cability are ccr.sidered.
1~ ue~ uptl~ality condition is introduced
.(:-5~:··a~:'.
...J..... ..
r
lj.:-:.,~~:r ~::-Ct;rai'ns
cnnstrai:1t qualification enslU'jx16 the validity of this condition
is giver.. lrnen the do~ai~ space is a reflexive space, it is shown that
t~e qu~lif~cation is the weakest possible.
If a certain convexity
ass~ption is made, then this opti~elity condition is shown to reduce to
the well-known extension of the Kuhn-Tucker conditions to Banach spaces.
In this case the constraint qualification is weaker than those previously
given.
e'.l1d ~
Intro2uction. In this paper we will be concerned with a general
"".......,,""',.....,....,,"""".....,.'""'"
co~ztrained optimization problem in Banac~ space.
Let X and Y be
3~ach
Axc X
A
spaces and let
~~d
fUnct~n
cor.strair.ed
g: X ~ Y be a differentiable map.
Ay~ Y define the constraint set, S,
f: X
~
R
naximu~
by S
= Ax n g-l( Ay ,J.
will be called an objective function
at
x
-oF'
J..
o
f
x
S and is differentiable at
o
For subsets
~~
local
is continuous on an open set containing
(; S where
x
0
is a local solution to
tee constrained optimization problem
naximize rex), subject to x
(1.1)
i. e. there exists an
£ >
xeS n {x: I Ix - xoll
<
0
£}.
such that
by
DF(x).
o
Thus
DF(x)
0
t(x ) > f(x)
o -
for all
The set of all objective functions Which
have a local constrained maximum at
the set of all derivatives at
~ S,
X
o
.
will be denoted by F(x )
o
o
X
of elements in F(x )
o
is a subset of
and
will be denoted
X·, the topological dual of X.
TDepartment of Statistics, University of North Carolina at Chapel Hill.
work wa.s s~pported in part by a grant froJ:1 the Office of Naval Rese~ch,
contract number N00014-67-A-0321-003 (NR041-095).
T~is
TTnepertment of Mathematics, University of North Carolina at Chapel Hill.
2
The optimization problem (1.1) is said to sa.tisfy an optimality
condition at
that
xo if there is a 5pecified subset of X*, sa.y A*, such
Df(x)
o £ A* for every objective function f with a local con-
strained maximum at
x; i. e. DF(x ) C A*. The JI'loat familiar optimality
o
0
"
condition for a problem of the type (1.1) is the KUhn-Tucker condition [6]
which was originally stated for the case when A = X = Rn , Y :I J:f!, and
x
m
Ay:l {y £ R : Y ~ 0, j l, ••••m}. For this condition the set A* is
=
j
defined by
A*
= {yoDg(Xo ):
~
Yj
0,
j
:I
l, ••• ,m, and yog(xo):I OJ.
The Kuhn-Tucker condition has since been extended to apply to a more general
problem which includes equality constraints and with A a proper subset
x
of X. This extension usually necessitates a change in the set A*; for
details see Mangasarian and Fromovitz [7] or Gould and Tolle [3].
In
addition considerable progress has been made in extending the Kuhn-Tucker
condition to nonlinear programming problems in infinite-dimensional spaces.
For results in this area as well as for lists of related references, the
reader is directed to the papers ot Guignard [4], Hurvicz [5], Ritter [8].
Russell [9], and Varaiya (11].
In general, the approach ot the above authors
is to attempt to determine sets
Df(x )
o
for some y*
£
B*
and x*
£
C* c: X*
= y*oDg(x0 )
C*.
and
B* c: y*
such that tor each
+ x*
In the cases where
C*
is the trivial
map these optimality conditions are of the Kuhn-Tucker type.
In order that an optimality condition be valid at
X
o
for the problem
(1.1), certain restrictions, called constraint qualifications, must usually
be placed on x, A , A • and g. In Euclidean ~pace the development ot
o x y
successively weaker cont;traint qUl'1.11ficat1nn~ for which the Kuhn-Tucker
3
co:·,': :. ~~.::'or: iz vs.~.:;..;. !1a~ led too a ~:.La.lificatior. wl.ic11 if; the weakest I~ossible t
i. ~:. tot:-: ne:::es;:;ary
al~d. ~-..:.fficicr!t ~'or
:'c:~ f:ver-j o'~;j e~t:''Ie
function
hese ceer.
spaces
~sed.
The constraint qualif'ications whicn
for <)l:r.::a.r o!'ti!:lality-
cOlldi~:ions
h~~~ g~n0~~1:y o~e~ en~e~tially
fi:.itt~·-jir.e::3io:lal caSES.
a.
[~J.
the Kuh.l'l.-Tucker condition to held.
C*
non~rivial
is
cation than does' a
Herein
~e
One
~ha~
in infinite-di:nensiona.l
the same as those applied in the
of
a.dvanto.g~
thf~
optimality condition with
often it requires a weaker constraint qualifi-
Ku1h~-Tuc~er
introduce a r.ev
and a constrai:lt qualification
type of condition.
uptL~ality
is sufficient to guarantee that the
wh~ch
optimality cor.dition is s&tisfiec.
condition for the problem (1.1)
Y.
In the case ween
is reflexive) it
;01111 be shewn ths.J~ the proble~ (1.1) satisfies the optimality condition
if and only if the
co~straint ~ualification
will be provec tr.at if Ay
earliertextensions ef the
is convex
Kuhn-Tuc~er
~he
is
sa~isfied.
Moreover, it
optimality condition reduces to
condition and that the constraint
qualification is weaker than those previously applied in this case.
~hese res~lt5 gene~alize
spaces (3]
~1d
the previous work of the authors in Euclidean
extend the investigations recently carried out by Gui6nard [4].
Ritter [8]) and Veraiya [11].
2.
D~~initions
and terminology.
In tnis section we will i:ltroduce
.~"""-"""'-"""""."""""'....J'I." .... _-",,-.., ,,~......-...
so~e
terminology vhich viII be necessary for the statement of our results.
DEFINI~!ON
1.
Let
L be a linear topological space and let
its topologicr-.l dual given the weak* topology.
of L, the 'Polat
3'
The
~
= {t*
follo~ing
For
or 13, B') is the subset of
€
L*: t*(2)
~
0
for every
L*
be
B a nonempty subset
L*
defined by
i £
B}.
properties of polar cones will be important:
2 s. Bi·
(i)
It
(ii)
B' • (closed convex hull ot B)'.
(iii)
B'
(iv)
B oS (B')'
~ So B2 ,
then B
"'" ..
is a closed convex cone.
with equality it and only if' B is a closed
convex cone.
DEFINITIOlf 2.
Let
Z be
&
Banach space, B c Z, and
-
z
(B.
the ~P1. tangents!2. B .!i zo' T(B,zo)' is the set ot all
Yhich there exists a nonnegative real sequence,
Then
0
{~
n
z c Z tor
}, and a sequence in
B, {zn }, such tpat
Z
(ii)
n + I0 ,
An (zn - 1 0 )
DEnNITION 3.
+
Por
z.
Z, B, Zo
&8
in definition 2, the
tane;ents!2. B .!1 zo , 'rY (B. z0 ), is the Bet ot all
exists a nonnegative real sequence,
{).n}~
z c Z
~~
2!
for which there
and a sequence in B, {In}' .such
that
I
n +
A (z
n
I. C
n
Z
0
- z )
0
+ Z
wea1c.l7; i.e.
Z*.
We shall denote by R(B,.)
and R (B,zo) the closed convex hulls
w
respectively. The properties ot tangent
o
ot 'l'(B,s) and T (B,z)
o
w
0
cones which will be useful in this paper are listed below.
properties are eaaily shown and no proofs are given.
Most of these
However, properties
iv and v are leaa obvious and ere dealt with more tully in section 6.
(1)
T{B,s ). T (B,z ). R(B,s ), and R
.
0
woo
V
(B.Zo ) are DOnempty cones.
and' R(B.z ) c: R (B,I) with equality
o-y
0
o-v
0
"
....
holdina in both cue. if Z ia tinite-d1mensional..
(1i)
T(B,s ) c: '1' (B,z)
'....
5
e
(iv)
If
B i8 convex. then
T(B.Zo )' Ty(B.z )' R(B,zo)' and
o
R (B,z ) . are all the 8ame closed convex cone.
v
0
(v)
R~(B.zo)
s. R' (B.zo )
vith equality
!!2i holding in general.
In the following, ve 8hall emploY' the same notation given in 8eC'tion 1
in stating pro~le1ll (1.1).
The dual space8
X-
and
Y-
vill be given
their respective veak- topologies.
The pseudollnearizing ~.!i x '
o
K(X )' and the veak pseudolinearizing cone at x, K (x ), are 8uboets o~
o
---0
v 0
X detined b7
DEFINITIOW 4•. Let
X
o
~ S.
and
It (x ) • {x « X: Dg(x )(x)
YO
0
E
R (A ,g(x
VY'
0
»l.
B7 usilll the properties ot the tangent cones, it can be easi17 veritied
that
K(X )
o
and ICy{Xo )
IC{x ) c I: (x)
o-vo
Ay
with e"ualltv holding it y
....
"
X and that
is tinite-dimensional or it
18 convex.
DEFINITIOW 5.
X*
are cl08ed conv. cone8 in
Let
XES.
o
The 8ubsets, B-{x)
0
and B*{x )' ot
v o
are detined b7
a*(xo ) • {x* c X*: x* • y*oD8(~o) tor acae
1*
E
T'(Ay.g(X
o
»}
and
a-(x
v 0 ) • {x*
E
X-: x* • y*oDg(Xo ) tor some
»}.
1*
c·T'(A ,g(x
v l'
0
with equality holding it Ay
B-(x) c a-(x)
"
v 0 ,0
Y is tinite-d1Jllensional. The 8ets
Clearly
but need not be closed.
i8 convex or
"
B-(x )
o
and
B;(Xo )
are convex conn
In tbe classical tinite-d1Jaensional nonlinear
progr.-.1q probl., D-(x ) • ~(xo)
o
i.
&
closed convex po17hedral cone.
6
For a discussion ot the cases in which B*(x)
o
is closed, the reader is
referred to Varaiya [10].
3. Statement of main results. We shall be concerned with an
optimality condition that establishes the set
~, the weak* closure of
101
A
y
101
o
as a subset of
Varaiya [11], in the case where
B*(x).
0
DF(x)
0
is convex, and Guignard [4], more generally, have shown that the
following relations are valid:
a*Cx). K'{x) CT'(S,x),
o
0
-
0
nf{x0
) CT'{S.x
},
0
where
B*CX '
o
refers to the weak* closure ot B*(Xo ).
Thus if the con-
.'.
straint qUalification
(3.3)
T'(S,x ) c K'(x )
o 0
holds, it follows that the optimality condition
DF(x ) c B*(x )
o
18 true.
-
0
B*(x )
o
is closed
and (3.3) is satisfied, it follows from (3.4) that for any
f ~ F(xo )
there is a
It should be noted that in the case when
y* ( T'(A ,g(x»
y
0
such that
Df(x)
0
= y*oDg(x).
0
This is a
direct extension of the Kuhn-Tucker condition in Euclidean spaces.
OUr first theorem gives a result similar to (3.1) and (3.2) utilizing
the notioD ot the weak cone of tangents.
THEOREM 1.
!h!. following
relations
.h2M.:
a-Cx
) • K'1010-101
(x ) c T' (S tX0 ).
1010
DF(x } c T'(S,x ).
o
-
101
0
As a consequence ot this theorem we have
COROLLARY 1.
~
optimality condition
~(x ) c B*(X ,
0-101
0
7
e
holds
.!!. ~ ~ constraint
(3.8)
qUal iflcation
T'(S,x )
v
0
K'(x )
c
w
-
0
!!. satistied.
Since B:(xof =. B.(X )' our specification of DF(x ) as a subset
o
o
of B:(xo' is a sharper result than that given by (3.4) above. The
corresponding constraint qualifications, (3.3) and (3.8), are not comparable
in general.
In the important case in which Ay
property iv ot the tangent cones that
is convex, it tollows :trom
K' ex ) • K' (x)
v
0
0
and B*(x). B*(x ).
v 0
0
Thus we obtain
COROLLARY 2. .
e
!!.
Ay
.!!. convex
~
!h!. constraint Qualification
(3.9)
TI(S,X ) c K'(x )
o w
0
is satisfied, then the opt.imality condition (3.4) holds.
For this case the optimality conditions ().4) and (3.7) are the same.
however, the constraint qualifications (3.3) and (3.9) are not equiValent.
Prom (3.5) and the fact that
T' (S,x ) c T' CS,x)
v
0 0
it is evident that the
constraint qualification (3.9) 111 less stringent than (3.3).
an example i. siven which illustrates that T' (S,x)
v
.0
in seneral, not equivalent
80
In section 6
and T' (S,x)
0
are,
that (3.9) and (3.3) are indeed ditferent.
It is natural to ask if the constraint qualification (3.8) is the
veakest qualification which will ensure the validity of the opt1mality
condition. (3.7).
In the event that
following theorem allovs
THEOREM 2.
equivalent.
!!.
X
UI
X 1s a renexlve Banach space, the
to answer this question in the affirmative.
.!!. reflexive,
~
8
.u.
COROLLARY 3.
(3.7)
.!.!.
X
!!. satisfied 1! ~
reflexive
.!Jl!.!! ~ o'Ptimality condition
1! lli
constraint qualification (3.8)
only
---
!! valid. In
this case
DF(xo ) • B*(x ).
w 0
Por
A ' convex we obtain the following extension
X reflexive and
01" corollary
y
2:
!!.
COROLLARY 4.
X
!!. reflexive
~
Ay .!!. convex,
~~
optimality condition (3.4) holds j:t' and on1:v:.lf.. ~bA cons..:tJ".!'!.~p..:t Qua1.j,ticati~
(3.9) holds.
!!!!!!
DF(x ) • B*(x ).
o
Final.ly' we note that if
Ay • {y (Y:
".1 .:i
0
X and
Y are tinite-dimensional and
B*(Xo ) == B*(xo ) • (I>'j0DgJ(Xo ): )..1
O). then
~
oi.
In
this case we obtain fran corollary 4 an earlier result 01" the authors [3].
COROLLARY
5.
X and
Let
-
---.
Y
be tinlte-dimensional
and let
i
...---
-
--
-
Then In order tor
Dr(xo ) • J: )..,.Dgj (Xo )'
Aj ~ O.
)..1 ogj (Xo ) .' 0
to hold fgr every-objective f'unction
separation theorem:
space.
~
~
IS. .!:!!!
~
K
2
Z
such that
with.!. local constrained maximum
locally convex linear topological
E!. disjoint
compact ~ )(2 !. £2!:!!..
--
~!.
t
closed convex subsets g! Z
~ there exist!.
z* (z*
~
~!!! CI > .0
9
e
z*(zl) ~
&1 ~
!2!:.!li.
IS.
C&
>
0 ~ z.(z2)
~ &2 ~ K2 •
Theorem 1 vill result :rrom the'tollowing three lemmas.
It should
be pointed out that the methods of proof ot these lemmas will be essentiall7
the lame a8 those used by 'Varaiya (11] end Guignard [4].
The only changes
are thole necessary to allow the use ot the veak cone ot tangents rather
than the cone ot tangents.
The importance ot this moditication is con-
talned in the statement ot theorem 2.
LEMMA 4.1. B.(x)
c K'(x ).
vow 0
Proot.
then
We tirst verifY that
•••••oDg(x)
o
eones
,*
where
R~(Ay,g(Xo».
€
,.
B·(x] c
y
x E:Ky(X ).
o
.·(x) • ,*oDg(x lex) • •• ('1) < 0
o
-,
Thus
•• ( X' (x)
y
and hence
0
trom the taet that
To show that
not hold.
K ' (x)
v
0
...
... -
e
henee
.*.
E
By property Hi
B*(x ).
€
v
0
ot polar
Then
0
w
0
-
v
lex) (
R (A ,g{x
v '1
0
».
The desired. result tollows
0
x* (K~(Xo)
i:
E
~eps.ration
>
y.(
y.(y)
E
and
X2 • B:<X ).
o
0 ~ .*(x)
theorem again with
we obtain a
7.oDg(Xo )
= {x·},
We
such that
X
R,,(Ay,g(Xo
B:(X,).
vith x·,
Z: X·, Xl
.* ( B·(x ). Clearly i:' K ex )
vow 0
tor all y
••
is closed.
x·Ci)
IC · • R (A ,S(x »
2
y
'1
0
(4.2)
SUppose
K' (x ) c B*(x ), we assume that theincluslon does
(4.1)
Applying the
».
since y. Dg(x
nov app~ the separation theorem with
tor all
x'y (x0 ).
B.(x) c K' (x ).
v 0 - v 0
Then there exists an
Thus there is an
-
T' (A ,g(x
v '1
0
f:
Let
0
».
Thus
B*(x
).
v 0
therefore (4.1) 1s contradicted.
y*
so
1.
Z • Y,
Dg(x
o
)(x) , R (A ,s(x
K1 • {f},
v '1
and
such that
> O·~ 1'.('1)
»•
R;(Ay'S(Xo
T~(\r,g(xo» and
But •• (x). y.(y) > 0 by (4.2) and
y. (
0
».
10
K' (x ) c: T' (S,x ).
LEH!If.A 4.2.
Proot.
0
'W
-
W
We show T (S,x )
w
wee.kly.
and
{An}
n = g(xn ).
Set
g.
where
tti1r . .
as
0
Let
Since
>. n(yn - g(x0 »
(4.3)
W
x
Then
Y
the continuity of
-
T (S,x ).
E:
v
III
Then there are sequences
0
Ilxn
- x IIo
.... 0 and
>. (x
n n
Y
and
Ay
E
n
g
The result then tollows from
0
such that
R+
€
K (x ).
c:
0
property i of polar cones.
{x } t: 8
n
0
llYn - g(xo ) II
is differentiable at
...
0
x0 ) ... x
by
we have
X
o
A 'Dg(x )(x - x ) + A ·&(x - x )
non
n
0
no
Ilzll'" O.
From (4. 3) we have tha.t
y.
for
Y•
E:
A 'y.{y - g(x » • A 'Y*oDg(x )(x - x )
n
non
0
no
. (4.4)
+ A ·y.o£(x - x ).
n
n
0
Iy·odx - x ) I
. .
As
n ... ·,
itA n .y*odicn
- xo
)11=n
II>. (x - x >I1'~
n
0
1 IX... - x.... J I
n
tends to zero since
hence bounded.
{A (x - x )}
n n
0
0
is a weakly convergent sequence and
Therefore from (4.4)
limy·b'n(Y - g(xo }»
n
n...-
= liJlly.oDg(xo)(An(xn
- xo)
n'"
= y·oDg(x )
o
Thus we have shown that for
llYn - g(x ) II
o
...
0, and
Dg(x lex) c T (A ,g(x
o
wy
0
LEMMA 4.3.
Proof.
{A }
n
t:
R+
t t: F(x ).
o
(4.5)
»
{Yn}
An{Yn - g(x
o
0
and
Ay
{An}
» ... Dg(xo){x)
c R (A ,g(x»
-vy
€
and so
x
E:
+
R ,
weakly.
E:
Therefore
K (x ).
vo
DF(x) c T' (S,x ).
o - w
0
Let
x E: T (8,x ).
Then there are sequences {x} t: S
v o n
such that
Since
t
II xn
- x II .... 0 and A (x
on n
1s d1t1'erent1able at
x
0
- x ) .... x
0
weakly.
and
Let
and has a local constrained
x , we have
o
A (t(x ) - t(x » • >. 'Pf(x }(x - x ) + >. ·£(x - x ) < 0
n
non
0
non
n
0-
11
e
for
n
sufficiently large.
Usinp; the same argument a.s given in '"he
Proof of the preceding lemma. we have
Therefore since
Df(x lex)
o
J>f(x)
as
+ 0
n
+
llD.
X*, we obtain the result
E:
o
II>..'
x - x ) I 'I
n £('n
0' I
= lim Df(x0 )(An (xn
- xO
» a lim>..
(rexn ) n
nBut then, from (4. 5) t we have
r{x0 ».
n-+o>
Dt(x )(x) < 0 which implies that
o
-
Df(Xo } ~ T~(S,xo)·
5.
if
Proof of theorem 2.
X is reflexive then for every
such that
Df(x ) · x*.
o
null vector of
norm
By theorem 1, it. 13 cuf'!'lc1Cllt. t.v ahvw 1.ohat
~ ~ . . ~. ..." ........ .--.J
II x·11
~e
*
J.IT£if
X2
x
• 1.
one-dimensional subspace of
x
E:
~ • X*(x)·x
2
defined
by'
For
sets of
E
p(x) •
> 0
l'
X
E:
o
x* (Tt(S,e)
w
F(x )
o
is the
has
is a cone.
X.(X ) . 1
2
defined by N
X
and
~
and
~.
N are closed
x - x2
E M.
and let
= {ax2 :
X has a unique decomposition, x
E N
there is an
0
This latter assumption causes no loss
Bow M and
the kernel of x..
and. each
w
and that the given
be chosen such that
X
E
T t (S,x)
XI
of generality since T'(S,e)
w
Let
E:
For convenience ve shall assume that
X, denoted by a,
• sup
x*
a
~inear
E:
be the
N
Let
R}.
subsp8ces of
M be
X
= Xl + x2 , where
The projection map
p: X .. M
1s then a continuous linear operator.
k
a positive integer, we define the following 8ub-
X:
BE • S
n {x t:
X:
Ilxll
<
d
and
Ck • {x
Clear~
C c: C +
k l
k
LEMMA 5.1.
f:
and
X: x
C
k
!2!: every
U
~
e, Arccos
{S}
(TI*)
~~
-
k :
2} •
is a cone.
'POsitive inteser
k
tr.ere
i!..!E.
e(k} > 0
12
e
In order to prove this leJ'!'lJ"l8.
!. subset 2!..!. renexive
[2]:
!t' ~
only
Proof'
!!.
~
€
employ the following result from
!2. weakly sequentially compact
Bo.nach snace
it!!. bOunded.
!2!. lemma g.
Suppose otherwise.
k
would exist a sequence
If
VI'O'
0Ia
{xm}m=l
£
Ck
k
with
xm
k
~
1
1
·x.(xk ) >
k
Ilx II
m -
cos(~
[
The set
x:
1IIT.l
t
At
that
C
k
k +
Consequently there 1s a
},-
Ilxn II
> O.
.
X; thus it is weakly sequentially' compact.
:~
In
a bounded subset of: the refiexive s_e
is
lllxm
subseque.nce,
there
w 2) > O.
-
2
m
k
SlIm for each
E
C , then it follows :from the definition of
(5.1)
any
Then for every
i
. ' , which converges weakly to some
n-1
€
X.
For
x* ( X·
0
x.( 1k
(5.2)
o Ilx II
n
Moreover since
follows that, as
{ikn }
(ik
n
-
e») . . x. (x)
as
n ... -.
0
is a 6ubsequence of'
{xk }
m
and
k
~
£
SlIm' i t
n"'Ak
x .... 9.
n
Bow (5.2) and (5.3) imply'that i
and (5.2) imply' that
€
Tw(S,6). On the other hand, (5.1)
x·(x) ~ cos(; - k : 2)
tradic1.1on cmd 1ibW:J comp1et.ee 'the proof.
We now define
and 8et
>
0 which yields the con-
13
k • 1,
to obtain a sequence,
decrease to
(5.lI)
0
as
.,.
Moreover, for all
k ... -.
.
net-
St:
< t: ,
f:
k
+.
Let L. be the smallest positive integer such that
1
1r
sin L + 1
and
~efine the sequence {Clt}:'L
<
II~II
by
11'
°•
k
The sequence
sin k + 1
1 -
Ilx2"·sin It : 1
.
•
{ok}JtwL' is strictly deer.s1ng and tends to
0
&8
It + -.
We nov detine the. tunction P: M ... R by
P(~) •
tor It ~ L and £k+l ~ II~ II < ek~
o
tor xl· 81 •
For k ~ L and £k+1 ~ II~II < £k' we mq rewrite P(~) as tollon:
(5.5)
P("J.) ••
I
1.
"'k (II"J.II
ck - £)[+1'1
-
<1<+1) + u k+1 '1<+1('k -
I1"J.1 I} •
low, in (5.5), we tirst replace Ok with CI k+l ' and then replace 0k+1
by
o
Qt.
estimates
Since the lequence
{CIt}
11 decreasing, this 7i81d8 the
e
~+l ~ Ilxlll < €k' k ~ L.
tor
that
P
It now tollows from (5.5) and (5.6)
M and differentiable at
is continuous on
xl. 9
with
1
being the trivial map.
DP(8 )
1
t:
Let
be defined by
X +'R
f(x) • x*(x) - (Pop)(x).
Since
x*
and p
are continuous linear maps, it is apparent trQID the
properties of P that
x •
e
f
is continuous on
with' Dt(S) • x*.
desired derivative.
maximum
at
X
o
Thus
f
1s a functional on
It remains to shovthat
' i.e. that
and differentiable at
X
t €F( x ) •
f
X having the
has a local constrained
The folloving lemma will ;yield this
o
result.
e
There exists ~
LEMMA 5.2.
E
> 0
~!!!!l
f(x) ~ 0
m~
x ESe' x ; 8.
Let
Proot.
XES..
Then by (5 ..4), x I. C
•
L
EL+l
x*Coo)
< cos(; - L
~
2) •
~in
L
and so
~
2 •
......
'!'hus
x*(x) < sin
Writina
X.
(5.7)
~ + x*(x) .~, we have
X*(X)
It x*(x) ~ 0, then
x
E
X.
L': 2 ·llxll·
<
sin L: 2(11~11 + Ix*(x)1
t(x) ~ 0
since
'1I x2 11)'
P(X1). (Pop)(x) ~ 0 tor all
It xtt(x) > 0, then fran (5"7) we obtain
..
sin t + 2
x*(x) <
1 -
II ~II
. It tollova f'nB (5.6) that
II~II·
IIp(x>l I
< £t'
sin t : 2
'11 XlII '. at+l" II ~II·
p(~). (Pop)(z) ~ aL+l" II ~ II
It we choose
..
E
E •
JIlin (CL+l '
L
llPTT),
it
15
then tor
x
€
S , x ;
e
e, we have
and hence
x-ex) ~ au.1·11~11 .i P(~) • (Pop)(x).
Thus
rex) ~ 0 which completes the proot ot the lemma.
Since
at
fCe). 0, it is clear that
has a local constrained JD&Ximua
t
x ' thus completing the proof of theorem 2.
o
It X i8 a Banach space such that the f'unction
differentiable except at
P, and hence the tunction
eYerphere.
x.
e.
x..
IlxII
i.
e.g. a Hilbert space, then the tunction
f, can be modified 80 as to be differentiable
In this case the point
the class of objective tunctions.
X
o
need not be specified to detine
The· objective functions will be the
smaller clus ot functions which are differentiable in an open set containlng S.
For a discussion of when the:; norm function is differentiable,
see [1].
6.
s~
SUpp1ement&17 results.
..
~ ~~
In this section we will clarify and prove
01' the statements concerning tangent cones made ee.rlier.
First we
shall prove propert7 iv 01' tangent cones.
PROPOSITION 6.1.
_and
z
o
€
!!
B
!!.!. convex
subset
!!!.!. Banach 8pace Z
B, then T(B,z ) • T (B,z ).
0
y
0
To prove this proposition we shall empl07 the tollowing result trCII
{x } is ~ sequence 01' elements .!A!. Banach space X £2!!.n
verging weakly!g. x f: X, !h!!!..!2!!!. seguence 01' convex combinations 9.! ~
. [1]:
(x )
n
.!!
converges!2. x.
Proof
2!.. proposition~.
From property i i ot taDgent cones it is
eVident that we need 01117 shoy that
Ty(B,B ) ~ T(B,B ).
O
o
.
-.
16
Let
j ( T (B,z ) •. Then there exist sequences (z) E B
V O n ·
{A } (R*
n
IIzn
- z IIo
... 0 and
~ (z
n n
such that
Por each ... > 0
let N... be chosen such that for all
r Izn - z0 II
(6.1)
Denote by
11
k
k
{A }
and
{zk}
n
z0 ) ...
-
<
~.
i
~ N ,
k
!k .
the sequences
n
terms removed and set
n
and
{z}
n
and
{A}
n
'11th the first
xnk = Ank( zkn - z0 ) •
For each fixed It, xk ... Z'" .weakly.
n
that each
Ak
n
Thus there is
8.
sequence
1s a convex combination ot elements in
X
It
{x }
n
<xkn }
and
such
as
n ....,
Ililtn - ill ... o.
(6.2)
Since the
itn are convex combina.tions ot the elements
..k
. it can be shown that each x
n
can be written as
Ak
xn •
where the
i kn
and
respectively.
{zlt}
n
and
('k(Ak
A n Zn - . Z 0
since each
trom
.~
(6.1) that
•
'"
~
~
n
there Is an ~
k
<
("'k
z
- z ) ... '"z
~
0
~
11.:It~ -.0 II ...
0
&8
such that
•
in the norm topology.
is a convex combination ot elements in
definition of T(B,z)
{~k}
n
Thus thei lt are in B.
II i~ - ill
"'k
It .... , x
~
",k
)
are convex combina.tions of the elements in
zk
n
From (6.2), we have that tor each
'l'hu8 as
x... - Ak( zk - s ),
n
n n
0
It ... ••
Moreover,
k
{sn} , it follows
It then tollows
z(
trom
the
that
T(B,z) vhichproves the proposition.
o
0
,
The tollowing example shows that in general T (B,. ) to R(B,z) for
v
0
0
B an a:rbitr&17 SUbset of a Banach space
and
B-
set
Z.
•
o
•
e
and
a positive integer} u (e).
17
denotes ~he
II~
- e II
Thus
80
e
l
that
~
k
~ 0
~h
uni t vector.
as
k ....
011
~
Taking
and that
=
k'1{ e1
k(~ -
+ e ) t we see tha~
k
~ e
e)
weakly as
l
k ... -.
Tw(B t 8). It is easily verified that T(B,O). R(Bte) • {e},
Ty (B,8)! R(B,e).
This example
lI'lIq
be JIOc1ified to give an exaaple
Ty (B,e) • Z and R(B,e). {e}.
tor which
R(B,zo ) 1. Tw(B,z)
~ be easi11' constructed
0
in finite-dimensional spaces where Ty (B,z0 ) • T(B,z0 ).
Examples 'tor which
That equal!ty does not hold in general in properly v of tangent
cones follows :trOll the above example, properly 1i i of polar cones. and
the following proposition.
T' (B,z ) • T' (B,z) if and only if
woo --
PROPOSITION 6.2.
T (B,z )
Y
0
R(B,. ).
c
-
0
Proot.
Suppose T (B,s ) c R(B,z).
Y
0
-
R'(B,zO
) c-T'(B.z
) c T'(B,z).
y'O0
0
Then
But R'CB,z).
T'(B,s0 ),
0
.0
T'(B,z
) • T'(B,z0 ).
y
0
Converse'''',
sup,ftt'Ilse
TV (B,z0
);. R(B,z0 ).
-J
Ir'"
Z€
T (B,s)
~
0
i I R{B,z).
0
such that
Since
Then there is a
R(B,z)
is a closed convex
0
cone the separation theorem of section 4 can be applied.
a
s. c Z. .ucb that
z·(z)
tor all
. But
Thus there is
z« R(B,s).
~. ~ T'(B,s)
Y
0
o
Therefore
since
> 0 ~
z*
z*(z) > O.
~
z·(z)
R'{B,z)
and hence .*( T'(B,. ).
0
0
Thus
T'(D,z) ~ T'CD,. ).
Y
0
0
t·,., ..
...
18
References
[1]
M. M.
[2J
If. Du.n1'ord, J.
~,
Normed Linear Spaces, Springer-Verlag, Berlin, 1958.
T. Schwartz, Linear Operators, Part I, Interac1ence
Publishers, New York,
[3]
1964.
.,. J. Gould, J. W. Tolle, A necessary and sutticient qualification
for constrained optimization, SIAM J. Appl. Math. ,to appear.
[4]
M. GUignard, Generalized Kuhn-Tucker conditions for mathematieal
progremmi~
problems in a Banach space, SIAM J .Control,
7 (1969), pp. 232-241.
[5]
L. Hurw1cz, Programming in linear spaces, Studies in Linear and
Nonlinear Progremm.ing, K. J. Arrow, L. Hurwicz, H. Uzawa, eds.,
Stantord University Press, Stanford,
[6]
1958.
H. Kuhn, A. Tucker, Nonlinear programming, Proceedings of the
Second Berkeley Symposium, on 'Mathematical Statistics and
Probability, J. Neyman, ed., University of California Press,
Berkeley,
[7]
o.
1951, pp. 481-492.
Mangasarian, S. Fromovitz, The Fritz John necessar)" optilllalit7
conditions in the presence of equality and inequality constraint.,
J. Math. Anal. Appl., 17 (1967), pp. 37-47.
[8J
K. Ritter, Optimization theory in linear spaces, Math. Ann., 184
(1970), pp. 133-154.
[9]
D. L. Russell, The Kuhn-Tucker conditions in Banach space with an
application to control theory, J. Math. Anal. Apple, 15 (1966),
pp. 200-212.
[10]
P. P. V&rai7&, Nonlinear programming and opti1ll8ol control, Eat
Tech. Memo. M-129, University of California, Berke1e;T, 1965.
19
[11]
P. P. Varaiy&, Nonlinear pr~ramming in Banach space, SIAM J.
App1. Math •• 15 (1967), pp. 284-293.
-i • "
atlo"
DOCUMENT CONTROL DATA. R "D
). "11:"0
TI 'H. It
Optimality Conditions and Constraint Qualifications in Banach Space
•. D.IC"'''TI''. NOTa. (7)0". .11."." . .d laclve'"" . ,••)
"'
.,. . . ,.. ... 1
-
'0 AU THO".') ("HI _ _• • 'fIIII. ""''''. I••'
--J
F. J. Gould and Jon W. Tolle
'0 "I£PO"T OAT.
, •• 1'01'10" NO. 0 .... AGK'
Augus t , 1970
I'.'
NO. 0" ... ".
?1
"
... CON T ....C l' 0" ."ANT NO.
N00014-67-A-032l-003 (NR047-095)
Institute of Statistics Mimeo Series
Number 707
AI. ""OoIIlC T NO.
c.
tL
10. ~T"'."TION .TAT . . . .N1'
The distribution of this report is unlimited.
II ••u
NTA"Y NOTU
11• .-oeuo"'NG MlL.ITA"Y ACTIYITY
Operations Research Program (Code 434)
Office of Naval Research
Washington, D.C. 20360
In this paper necessary optimality conditions for non-linear programs in Banach
spaces and constraint qualifications for their applicability are considered.
A new
optimality condition is introduced and a constraint qualification ensuring the validity of this condition is given.
When the domain space is a reflexive space, it
'.
is shown that the qualification is the weakest possible. If a certain convexity
•
assumption is made, then this optimality condition is shown to reduce to the wellknown extension of the Kuhn-Tucker conditions to Banach spaces.
In this case the
constraint qualification is t-leaker than those previously given.
e~
DOl '~0R"M •• 1473
1
.
ecuritylu" ilic:ahon
u.
.
1'01.1:
"'NI'!
LINK.
L' .. K ..
Key WO"OI
"o .. £
VII"
"T
"'01.«
t
"T
Kuhn-Tucker conditions
Optimality conditions
Constraint qualifications
Nonlinear programming
Banach spaces
.
.
.
.
•
.. .
,
,
\
I
,
I
!I
~
1
j
·-e
,,
I
Security Cl•• lIlficahon