•
t
Depal'tment of Statistics~ University of North Caroz.ina_ at Chapel, HitZ.
This bJork bJQ8 supponed in pan by a grant from the Office of NavaZ ReseaPch~
Contract Numbep N00014-67-A-0321-003 (NR047-095J.
tt
Depazatment of
MathematiC6~
University of Nopth CaroUna at ChapeZ HiZZ.
OPrlrw..ITY CDmITIOOS PHD CamRAINT
Q.w.IFICAnOOS IN BANAei SPACE
by
F. J. Gould t and Jon W. Tollett
DepaPtrnent of Statistics
University of Nor'th CaroUna at ChapeZ HiZ1,
Institute of Statistics Mimeo Series No. 707
AugU6-t, 1910
C';>t i~;c.lity Condi t"tOY'.3 ar.:l
F" J.
.c~:~s~:
...,'. ...... ...a(;:'.
.. '
li~2~r ?rc~ra~s
-~
'. ~
Conr;~rr:dnt
t
Q~alifica'Cior.:s
in Banach Spa.ce
~~
Go~ld
and Jon W. Tolle"
In th.!.:.; pc..pel'" nec€s£:.l.."j' optimality Gonditions for non-
in Eanach spaces ar.u constraint qualifications for their
are cor.sidered" i~ lle~ uptl~ality condition is introduced
constrabt q,ualification emnll"jnB the validity of this condition
is given. ~rnen the do~ain space is a re!lexive space t it is sho~~ that
t~e qualification is the weakest possible"
If a certain convexity
ass~ption is made~ then this optima.lity 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.
appl~cability
E.l1cl. .3.
Intro~uction.
~"."",.."",.."VV"..,/\........,,.V'
co~ztrained
~~nach
In this paper we will be concerned with a general
optimization problem in
Banac~
space.
Let
X and Y be
spaces and let g: X. Y be a differentiable map.
For subsets
=
A c X and A ~ Y define the constraint set, St by S A n g-l(A ).
xyx
y
A tunct~n f: X. R will be called an objective function ~ ~ local
",p
::1. ...
S
f
is continuous on an open set containing
xo ' S where
and is-differentiable at
is a local solution to
X
o
tee constrained optimization problem
maximize f(x), SUbject to x
i. e. there exists an
X € S
hav~
n {x: I Ix -
€ >
0
xol' < €}.
Drex )'
o
Thus
DF(x )
o
S,
t(x ) > f(x)
o -
for all
The set of all objective functions which
a local constrained maximum at
the set of all derivatives at
by
such that
€
.
X
o
will be denoted by F(x )
o
and
of elements in F'x
\ 0 ) will be denoted
is a subset of X*t the topologica.l dual of X.
TDepartment of Statistics, University of North Carolina at Chapel Hill.
work was silpported in part by So gra.nt from the Office of Na.val Research t
contract number N00014-67-A-0321-003 (NRo47-095).
T~1s
ttnepartment of ~Athematicst University of North Carolina at Chapel Hill.
2
'.
The optimization problem (1.1) is said to sa.tisfy an optimality
condition at
that Df(x)
o
o if there is a Gpecified subset of X*. say A*. such
A* for every objective fUnction f with a local con-
X
€
strained maximum at x; Le. DF(x ) c A*.
o
0
-
The most familiar optimality
"
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 = If!, and
x
m
A;r = {y € R : Yj ~ 0, j = 1, ••••m}. For this condition the set A* is
defined by
A*
= {yoDg(xo ):
Yj ~ 0, j
=
= l, ••• ,m,
and yog(xo ) OJ.
The Kuhn-Tucker condition has since been exteJ:lded to apply to a more general
problem which includes equality constraints
and with Ax a proper subset
.
of X.
This extension usually necessitates a change in the set A*; for
details see Mangasarian and Fromovitz [7J or Gould and Tolle [3].
addition considerable progress has been JIl8,de in extending
t~e
In
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 of Guignard [4], Hurwicz [5], Ritter [8],
Russell [9J. and Varaiya [11].
In general, the approach of the above authors
is to attempt to determine sets C* £. X* and B* S. y* such that for each
f
f:
F(x ).
o
=
for some y.
€
Df(x ) y*oDg(x ) + x*
o
0
B* and x* f: C*. In the cases where C*
map these optimality conditions are of
t~e Kuhn-Tu~ker
is the trivial
type.
In order that an optimality condition be valid at Xo for the problem
(1.1). certain restrictions, called constraint qualifications, must usuallr
be placed on x, A t A , and g. In Euclidean space the developnent of
o x y
successively weaker cont;trai.nt que.1i:ficatinn!': for which the Kuhn-Tucker
3
CO;.: '.-.. ::'or:.
iz
has led ·co a
1. e. botr:
neces~ary
v::...J.i;~.
8,l;d
4
~$ed
speces br;,,-,=
for
a nontrivial
C*
;,'or the Kuhn-Tucker condition to hold
si~~:ar o~ti~ality COluli~ions
gt;:Zlv;."Q.lly bee:.
fi:-.ite-di!:.er.zional
cation than
:::u.ffidcr~t
which in the weakest IjOssible)
f'u.r.:ction [3 J• The constraint qualifications which
i'C: eYe!"'J O·~lj ect:'.ve
hcxe heer.
~"..La.lificatior.
One
caSES.
is
es.:'e~~tially
~hat
infinite-d~ensional
the same a.s those applied in the
a.dvanto.g~
of
thf~
optimality condition with
often it requires a weaker constraint qualifi-
does'aKuhn-Tuc~er type
Herein we introduce a
in
of condition.
~ew optL~&lity
condition for the problem (l.l)
and a constraint qualification which is sufficient to guarantee that the
optimality condition is satisfiec.
will be shewn that the
i t a..'ld
proble~
In the case ween X is reflexive 2 it
(1.1) satisfies the optimality condition
only. i1' the cor,straint qualifica.tion is satisfied.
~1oreover,
it
will be proved that if Ay is convex ~he optimality condition red.uces to
earlierrextensions of the Kuhn-Tucker condition and that the constraint
qualification is weaker than those previously applied in this case.
~hese res~lts
spaces [3}
~~d
generalize the previous work of the authors in Euclidean
extend the investigations recently carried out by Guignard
Ritter [8], and Varaiya [11].
2.
D~finitions
and terminology.
In this section we will introduce
.~""'-'.-.,-'.-"""""""'\i."""""-"'''''''-''''_'''~<''''''.,""~"",,-,,
so~e
terminology which will be necessary for the statement of our results.
DEFINI~ION
1.
Let
L be a linear topological space and let
its topological dual given the wea.k* topology.
of T., the llola.r
B'
~
= {!*
of B, B'
€
L*:
2
For :a
is the subset of L*
t*(~) ~
0 for every
! £
a,
t* be
nonempty subset
defined by
B}.
The following properties of polar cones will be important:
(4]~
4
(1)
It ~.s B2 , then
(11)
B'.
(lil)
D'
(iv)
B .: (B')'
2
B .5. Bi.
(closed convex hull ot :8)'.
.
is a closed convex cone.
with equality 1t, and only 1t B 1s a closed
convex cone.
I
DEFINITION 2.· Let
the ~ g! tangents
i.2.
,
Z be a Banach space, B
B
.s. Z,
and
Zo (B.
.!i zo' T(D,Zo)' 1s the set of
all
Then
z c Z tor
Yhich there eXists a nonnega'tive real sequence, O'n}' and a sequence in
a, {zn },
(i)
such that
zn"" zo'
(ii) An (zn - zo) .... z.
DEPINITION 3.
tangents!g. B
'or
.5 .0'
Z, B, Zo
as in definition 2, the Yeak
Ty(R,zo)' 1s the set ot all
£:.29!..2!.
z« Z tor Which there
exists a nonnegative real sequence, {An}~ and a sequence in B, {zn l , such
that
(i)
z + z
n
0
+.
(ii) An (zn
- a ) o
weakly;n
i.e. A z·(z - z )
n
0
We shall denote by R(B, z)
o
and R (B, z)
w
ot T(B,zo) and Tw(B,zo) respectively.
0
+ z*(z)
tor evert
the closed convex hulls
The properties
cones which will be usef'ul. in this paper are listed below.
properties are easiq shown and no proots are given.
ot tangent
Most, ot these
However, properties
iv and v are le88 obvious and are dealt with more tully in aection 6.
(1)
T(B,So)' Ty(B,zo)' R(B,Zo)' 'and Rw(D,.o)
(ii)
TCB,so
),c
T (B,z)
-y
0
are nonempty conea.
and "R(B,zo) cv
R (B,a)
with equality
0
t1
"
holdina in both casea it Z is tinite-d1mensional.
.'
~
.
.1
,
.
il convex, then T(a,.),! (1,& ), R(I,. ), and
o
woo
Ry(I,.o) 'are all the lame closed conyex cone.
(ly)
If I
(v)
R~(I,zo).5. III (15'.0)
In the following,
V,. .ball employ the same notation given in 8ection 1
in stating pro~lem (1.1).
their respective
with equality ~ holding in general.
The dual sPaCes' X- ~ Y* vill be siven
.eak* topolOlies.
DEFIIITI01'l 4. '~t
X
o
f:
S.
The pseudolinearizipg cone
K(xo )' and the X!!!:Ji pseudollnprizins S2S!.
X detined bJ'
n
5.
xo ' Ky{Xo )' are
x '
o
e.tib«H:'tS
.~
and
lC.(XO) • {x c X: Dg(Xo)(X) C Ry(A"g(Xo
»}.
B;r using the properties ot the tanaent cones, it can be easl17veritie4
that IC(x ) and Ky (Xo ) are cl08ed convex cones in X and. that '
o
lC(x ) c I: (x) with equallt7 holding it Y is tinite-dimensional or' it
o -
Ay
v
0
18 convex.
DIPIII'l'I01'l ,.
X*
X c S.
Let
o
The 8ub8ets, B.(X )
o
and
B:(X )' ot
o
are detined bJ'
B*(xo ) • {x* c 1*: xtt • '-'o1>8(x
~~
.0 ) tor 8cae
'
7*
f:
T'(Ay~g(Xo»}
and
1:(xO '
•
(zit c X*: ztt • 7*oDg(Xo ) tor some
7* C·T~(A"g(xO»}·
Clearly B.(x' c I.(x) nth equality holding it A". 18 convex or.
y
0
-
Y il t1n1te-<UJlensional.
but need not be cloled.
prosr-ina
,0
.,
B*(xo ) andB:(xo ) are COllTex con..
In the classical tinite-d1aenslonal nonlinear
The sets
probl_, 3·(xo ) • a:<xo )
ia a closeel conyex po17beclral
CODe.
"...
6
For a discussion of the cases in which B*(xo ) 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 DF(x)
o
a*(x
), the weak.
closure
of a*(x).
w
o
w
0
Ay
as a subset of
Varaiya [llJ, in the case where
is convex, and Guignard [4], more generally, have shown that the
following relations are valid:
a.exo )
= K' (x0 )
C
-
T' (8,x0 ),
Df(x0
) cT'(S,x
),
0
where
straint qualification
(3.3)
+'. .
T'(S,x0
) CK~(x )
0
holds; it follows that the.optimality condition
DF(x0
) c-a-(x
)
0
is true.
It should be noted that in the case when »*(xo ) is closed
and (3.3) is satisfied, it follows from (3.4) that for any f ~ F(x )
o
there is a 1* € T'(Ay ,g(x»
such that Df(x)
= y.oDg(x). This is a
0
0
0
direct extension of the Kuhn-Tucker condition in Euclidean spaces.
Our first theor_ gives a result similar
to (3.1) and (3.2) utilizina
the notion ot the weak cone of tangents.
THEOREM 1.
!h!. following relations
a*Cx )
wo
II
~:
Kt (x )
C T' (8.x ),
wo-w
0
DF(x ) c T'(S,Xo )'
o - w
As a consequence of this theorem we have
COROLLARY 1. The optimality condition
~(x ) c
o -
mx::r
w
0
7
."
holds
!!. ~ ~
constraint g,ua;! if'lcation
T'(S,x )
(3.8)
w
0
K'(x )
c
-
W
0
in general.
In the important case in which Ay
is convex, it tollows :trom
K~(xo). K' (X
property iv ot the tangent cones that
o
) and B;(X ) . B.(~).
o
'!'hus we obtain
COROLLARY 2.·
It
(3.9)
Ay
!! convex
'1"(S,x
)
w
0
~
the constraint qualification
c K'(x )
-
0
!!. ~atisti~, ~ the opt.im&lity condition (3.4) holds.
For this case the optimality condiM:ons (3.4) and (3.7) are the same;
however, the constraint qualifications (3.3) and (3.9) are not equivalent.
c: T'CS ,x) it is evident that the
w
0
constraint qualification (3.9) is less stringent than (3.3). In section 6
From (3.5) and the tact that
'1" (5,Xo )
an example is siven which illustrates that
'1'':'(8 '~o)
and '1" (8 ,xo )
are,
iu general, not equivalent so that (3.9) and (3.3) are iuc1eed Mtterent.
It 1s natural to ask it the constraint qualification (3.8) 1s the
weakest qualification which will ensure the validity of the opt1malit7
condition. (3.l).
In the event that
X is a. reflexive :Banach 8~ce, the
tollowing theorem allows us to answer this question in the affirmative.
THEOREM 2.
(3.10)
1!.
X
!!. reflexive,
DF(xo )
~
='r'(8,x
).
w
0
110 now follovs from (3.5) and (3.10) that for
X ret'lexive, the
weak constraint qualification (3.8) and the optimality condition (3.7) are
equivalent.
8
•
1l
COROLLARY 3.
(3.1)
X
.!!. reflexive
~.lh!.
optimality condition
!!. satisfied i!..!:!!! only .!! ~ constraint qualification
is valid.
---
In this case
-DF(x )
o
For
(3.8)
X reflexive and
= B:<xo )·
A ' convex we obtain the tollowing extension
y
ot corollary 2:
COROLLARY
4. !!. X 1!. reflexive ~ A.y .!!. convex, ~ ~
optimality condition (3.4) holds if and only].f tba ,SLonstrain...:..1;. qualiticatiOJl
-
(3.9) hold.. Then
DF(x )
o
= B*(x0 ).
Finallr we note that if X and
Y are finite-dimensional and
Y: 7.1 .:5. O}. then B*(Xo ):= B*(xo ) • (tA j ·Dg" (Xo ): >'j ~ oj. In
this case we obtain from corollary 4 an earlier result ot the authors [3].
Ay • {y
f:
COROLLARY
5. Let X and Y ~
be finite-dimensional and let
-
......,.
i
--
\,Dgj
- - -
Then in order for
Dt(Xo ) • ~
(Xo ) ,
>. 3 ~ 0,
A
j
to
!:!.
~!9I ev~r:c.
X
.gJ (x )
o
t
ob.1eQtive fUnction
o '!!.!!. necessary
.2 sufficient
T'(S,x )
C'
0
~~
local constrained maximum
~
= K'(x ).
.00
4. Proot ot theorem 1. We shall need the following well-known
separation theorem:
space.
IJ.
~
compact
-such that
IS. !!!
~
lC2
~
K2
z !!!.!. locally convex linear topological
!?!. dis,1oint closed convex subsets g!, Z
!.~.
!h!n there
exist!. z· c
z*
~!!!.
~
CI
> .0
9
•
Z*(Zl) ~ «
!!!!..!!!.
11 (
IS.
~
>
0 ~ z*(z2)
z2 ( X2 •
Theorem 1 vill result from -the 'following three lemmas.
It should
be pointed out that the methods ot proof of these lemmas vill be essential17
the same as those used by 'Varaiya [11] and Guigne.rd [4 J• The onl1 changes
are those necessary to allov the use ot the weak cone of tqents rather
than the cone of tangents.
The importance of this modification 1s con-
tained in the statement of theorem 2.
LEMMA 4.1.
a*(x)
v 0
We first verity that a;Cxo }':' I\,(xo )' SUppose .* c B;<xo )'
Proof.
then .*. ,*oDg(x}
o
cones
where
w*
T' (A ,g(x
v y
0
£
».
By property iii of polar
:B~(A')",g(Xo»'
Let x (Kv(Xo )' Then
.*(x) • ,*oDg(xo lex) • ",*(y) -< .0 since y. Dg(x0 lex)
Thus
",*
x'v (xo )'
c
.*
E:
f:
X'wo
ex)
and
hence
c: K' (x ).
..
. a*(x)
YO-YO
t
R (A ,g(x
-y
y
0
».
The desired result follows
trom the fact that X' (x) is closed.
w 0
To show thatK' (x ) c: k*(x ), we assume that the ·inclusion does
.
v 0 v 0
not hold.
'!'hen there exists an x*
E
x~(xo)
now app~ the separation theorem with
'l'hus there is an
i: (
X
x*(x)
€
Applying the
a*(x
w ·0 }.
Clearly
~epe.ration
Z: X*. Ie = {x*}, and
l
x2
We
• a;(xo ) •
such that
(4.1)
for all .*
with x*' a;(xo }'
> 0 ~
.*Cx>
i ' 1(,,(xo ) so 1. :og(xo Hx) , Rw(Ay ,Sex0 ».
theorem again with
z•
Y,
IS. • til,
and
2 • Rw(A'Y'S(Xo » ve obtain a y*( y* such that
IC
y*(y) > O'~ y*(y}
(4.2)
.*.
tor all 'Y
••. "of'
hence
C
R.v(\,g(xo
»'
Thus
y* ( R;{Ay'S(Xo
'Y*oDg(xo ) «a*(x
w o '. :But
theretore (4.1) is contradicted.
» = T.:,(\,g(xo »
.*(x). y*(y)
> 0
and
by (4.2) and
10
•
W~.A
4.2.
Proof.
K' (x ) c T' (S,x ).
w
0
-
W
We show T (S,x ) c iC (x ).
W
0
property i of polar cones.
{Xn}
€
S and
weakly.
Set
O}
~ (y
mtr
-+-
x
T (S,x ).
v
0
E:
Then Yn
Since. g
0
The result then follovs from
0
II -+- 0 and ~ (x - x0 ) .... x
non n
E
and
Ay
is differentiable at
n .... ·,
II).n ·Y·Ot(xn
for
y*
hence bounded.
..
x ) II =
0
ve have
y*
E:
since y*oDg (x ) € X*.
o
Thus we have shown that for
•
lim y*oDg(x )(An(X .. xo )
n
o
n--
y.oDg(x )
o
....
LEMMA 4.3.
Proof.
{). } €
R+
F(x )'
o
0, and
n
E
Ay
and
{~n} E
An (Yn .. g(xo
+
R ,
Therefore.
weakly.
E
Kw(xo )'
DF(x) c '1" (B,xo )'
o - w
Let
x
€
BUch that
Since
f
T (S,x ).
W
0
Then there are sequences
.
II xn
- x II .... 0 and ). (x
on n
1s differentiable at
o ve .have
). (r(x ) .. rex »
maximum at
{Y }
n
» . . Dg(xo)(x)
'l'w(Ay,g(Xo » =. Rv(Ay,g(Xo » and so x
llYn" g(xo ) II
C
0
1s a weakly convergent sequence and
1&
»&(xo)(x)
n
'!'herefore from (4.4)
lim y*(). (y .. g(x »)
o
n...n n
(4.5)
by
. .
Iy*odx .. x ) I
""n(xn .. xo ) II'~
II~-~II
{~(Xn" x )}
n
0
tends to zero since
E
0
.. x0 ) + An ·£(xn .. x0 )
n
t
X
o
....
An 'y*(1n
.. g(x
»n
• A 'Y*oDg(x0 )(xn- o
x )
o
+ A 'y*odx .. x· ).
. (4.4)
n
llYn" g(xo ) II
liz It .... o.
&8
From (4.3) we have tha.t
As
Then there are sequences
llx .. x
- g(x » • A 'Dg(x )(x
0
non
nn
where
W
Let
Yn· g(x )·
n
g.
-
such tha.t
R+
E
n
the continuity of
(4.3)
0
X
o
.. x )..... x
0
{x}
n
€
weakly.
1&
).
Let
and has a loeal constrained
X ,
non
S and
·nt(x0 )(xn
.. xo
) +n
A ·,(x .. x ) < 0
n
0-
11
tor n sufficiently large.
Using the same argument as given in '~he
proof of the preceding lemma we have
Therefore since
,
J>f(x)
E:
0
II AD • dxn
xo·) II
I
-10
0
n
as
+
OD.
X*, we obtain the result
».
- x » = lim 1 (r(x ) - rex
0
n
0
n-te
n- "'Jl
:But then, trom Ca.. 5), we have Dt(x )(x) < 0 which implies tha.t
o
Dt(xo)(x)
Dt(xo )
it
T'(S,x
).
W
0
~
5.
Proof' ot theorem 2.
By theorem 1, 1t is cuff1c1cl1t. tv Bhvw that
...,'W....--'
~~",...
X is reflexive then tor every
~UCh that
x*.
DfeXo)·
null vector ot
norm
=lim. Df(xo )(An (xn
Ilx*ll.
~e
W
*x
x,
X2
Let
the kernel ot x*.
x
£
x*
X
o
is the
Tt(S,e)
has
This latter assumption causes no
10S8
€
w
is a cone.
X be chosen such th~t X*(X ) . 1
2
f:
one-dimensional. subspace ot
and. each
there is an t ~ F(x )
w o o
a, and that the given
• 1-
T~(s,e)
ot generality since
~ T' (S,x)
For convenience we shall assume that
X, depoted by
sup
x*
X defined by
low M and 1'1
and let
N = {ax : a
2
are closed
X has a unique decomposition, x
~inear
€
N be the
R}.
Let
sub.paces ot
M be
X
= Xl + x 2 ' where
• x*Cx)'X £ 1'1 and Xl· x - x 2 £ M. The projection map p: X + M
2
2
de-tined by" p(x) • ~ is then a continuous linear operator.
X
For
t
> 0
and
k
a positive integer, we detine the tollowing sub-
sets ot X:
B
E:
• S
n {x £
X:
Ilxll
< e}
and
Ck • b:
Clearly
C c: C +
k l
k
LEMMA 5.1.
~::.!l!l
€
and
X: x
Ok u
!2!:. every
Bt (k) n Ck - ••
r
9, Arccoa(IT£Tt)
tel.
~i
-k :
aJ •
is a cone.
'POsitive inteser
k
ttere!!..!B. &(k) > 0
" ... ,
12
In, order to prove this lemma we employ the following resUlt from
(2):
!
!t' ~
2!..!. renexive
subset
only
Proof
!.!. .!l!!. bOunded.
2!. lemma.2.:.1.
~
E
Suppose otherwise.
k
would exist a sequence
It
Bana,ch snace ~ weakly sequentially compact
~
{xml1ne1
E:
C
k
with
Then for every k
x
k
m
E
SlIm
C , then it follows from the definition of
k
1
1 ~ Il~
[
The set
x=
1-
II
*( k) >'
(n'
·x xm -- cos 2'
is
-
for each
Ok
there
D1
> O.
that
n')
k + 2 > O.
a bounded subset of the reflexive space
lllx=IIJm=l
X; thus it is weakly sequentially compact.
sUbsequ~nce.
e.
any x*
o
E
t
Consequently there is a
Ak }-
::
x
I/ n "
" , which converges weakly to some
X
E
X.
For
n=l
X·
n .. -.
k
{x }
m
Moreover since
, follows that, as
and
k £ S11m' it
xm
n"...k
x .. 8.
n
Now (5.2) and (5.3) imply that
and (5.2) imply that
x€
Tw(S.6). On the other hand, (5.1)
x·Ci) ~ cos(i - k : 2) > 0 which yields the con-
tradic't.1on· 'end tbWJ' cOIIIlpJ.e1Ieo -the proof.
We now define
and set
13
•
k • 1,
to obtain a sequence,
,.
4ecrease to 0
(5.!J)
&8
t
~
••
Moreover, for all
'.
B£ n Ct •
f:
< £k t
+.
Let L. be the8lD&llest positive integer such that
1
'If
sin L + 1 < 11~J1
•
and stetine tbe sequence
{Clt}}tIIL
by
w
sin k + 1
'!'he sequence
• is strictly d~easing and tends
{Ot}P!.'
to 0 as k + -.
We nov detine the. tunction P: M + R by
e ...
PC;,) •
tor It ~ L and
o
Ct +1 ~ II;,II
tor xl· 81 •
For k ~ L and £k+l ~ II~ II < £k' we may rewrite P(;,)
(5.5)
P("J.l..
1.
111.£k<lI"J.l1 - £k+1)
c k - £k+l·t
+
&8
tollova:
~1£k+1(£k - 1I"l1l}'
Iov, in (5.5), we tirst replace Cl k witb CI k+l ' and then replace
by Ok. Since the sequence {Ot} is decreasing, this ,.ields the
. estiJllates
< CtP
~+l
~+l ~ J Ixlll < €k' k ~ L.
for
that
It now tollows from (5.5) and (5.6)
P is continuOu8on M and difterentiable at
DP(8 )
1
Xl· 8
1
with
being the trivial map.
Let t: X +'R be defined by
= x·{x)
t(x)
Since x· and p
are continuous linear maps, it is apparent from the
properties ot P that
x•
e
with'
is continuous on X and ditterentiable at
t
nt(e) • x*.
desired derivative.
maximum at
- (Pop)(x).
Thus
18 a tunctional on
t
It remains to show .that
x ' i.e. that
f
o
o )'
t: 'F(x
X having the
has a local constrained
t
The folloving lemma will 1ield thie
result.
Proot.
/
Let x
t:
Then b1 (5.1i). x ~ C and
L
S...
•
£1.+1
x*<oo>
<
cos<t - L ~
2) •
~in
L
~
80
2 •
'!'hUB
. +~....
Jt*(x) < sin L': 2 ·I'x".
Vr1tiq x· ~ + X*(x).X , we have
2
<5.7)
x*(x) < sin L : 2(11~11 +
'x*(x)1 ·ll x2 11)·
It X*(x) ~ O. then f(x) ~ 0 since P(~). (Pop) (x) ~ 0 tor all
x c X.
It X*(x) > 0, then fran (5.7) ve obtain
'I
sin L + 2
X*(x) <
1 -
II x211
. It tollowa:f'i"al (5.6) that
II~II
• IIp(x)11
< £L'
sin L: 2
·1 I xl"
'. Clt+l·II~II·
P(xl ) · (Pop)(x) ~ClL+l·II~11 it
15
then tor x
S , x
€
t
~
a, we, have
ll~ 1.1 ~
lip II· J Ix II .i '£r. t
and hence
x*(x) ~ CYl·'I~II .i P(~). (pop)(x).
Thus
Since
at
-
r{x) < 0 which completes the proof' of' the lemma.
t{e)
= 0,
it is clear that
t
hu a local constrained max1111\D1
xo ' thus completing the proof 01' theorem 2.
It X is a Banach space such that the t"unction
ditterentiable except at
P, and hence the function
eYerphere.
x. a,
x + Ilxll
is
e.g. a Hilbert space, then the tunction
1', can be mod1tied so as
In this case the point
the class of ob3ective functions.
X
to be ditterentiable
o need not be specified to detine
The 'ob3ective functions will be the
SJII&l1er class 01' tunctions which are difterentiable in an open set containing . S.
see
For a discussion of when
th~
norm tlmction is differentiable,
[1).
6. SUpp1ement&1')' results.
~ ~ ~
In this section we will clarity and prove
scae 01' the statements' concerning tangent cones made ee.r1ier.
First w
shall prove propertl iv 01' tangent cones.
PROPOSITIOlf 6.1.
_
and
It
B
!!..! convex
subset
.2!..!. Banach
space
Z
c B, then T{B,ze) - T (B,z ).
v
0
To prove this proposition we shall emplOl the tollowing resultf'rca
I
o
i8.!. sequence 01' elements .!a!. Banach space X ~­
n
verging, weakly to x E X, !!:!!a.!!!!!. sequence 01' convex combinations9l. i!!!.
. [1]:
{X }
n
11:
{x }
converges!2. x.
Proof !!!. proposition
g.
From property 11 of tugent cones it is
evident that we need o~ shoy that
Tv (B'llo ) ~ T(B,IO).
16
.Let
Z
£
Tw(B, zO). . Then there exist sequences
{zn ~ e: B and
U n } e: R*' such
z II ... 0 and ).nn
(z - z ) ... £ ~.
. that II zn- o
0
For each It > 0 let lilt be chosen such that tor all n ~ N~,
lizn - z0 II < 1k .
(6.1)
Denote by
Nt
{zk)
k
{A }
and
n
the sequences
n
{z}
n
and
U} with the first
n
terms removed and set
k( k
)
x-k
n = An zn - z0 •
k
For each fixed It, x ...
that each
i:
n
A
Z
.weakly.
Thus there is a sequence
is a convex combination ot elements in {x:}
k
such
{xn}
and as
Iii: - ill ... o.
(6.2)
lt
Since the ink are convex combinations ot the elements x .).k( zk - z ) t
n
n
n
0
. it can be shown that each it can be written as
n
Ak . ~k{Ak
x
z
n
n n
where the
~
z0 )
zkn
i nk and
are convex combinations ot the elements in {A It}
n
and {zlt} respectively. Thus the 'zk are in B.
n
n
From (6.2), we have that tor each k there 1s an ~ such that
II i~
Thus as k .... , x
At
(Ak
A
~
since each
• >.
~
z
-
ill
<
~
•
- z ) ... z in the norm topolo&V.
A
~
Moreover,
0
z~ is a convex combination ot elements
from (6.1) that
:It
in {In}' it tollows
It then tollows from the
z
definition ot T{B,z) that
£ T(B,z)
which -proves the proposition.
o
0
,
The following example ~hOYs that in general T (B,z ) to R(B,z) for
woo
B an arbitrary subset
EXAMPLE 6.1.
Let
ot a Banach space Z.
Z. 1
2
and set
1
0
•
e
and
2
1
B • {x c I. : x • i<e1 + ek)' k a positive integer} u
tel.
Here
elt
17
denotes the
II~
Thus
80
k
- ell •
th
unit vector.
as
0
k ....
1
l
~d that
1
=keel
+ e ),
k
e) •
k(~ -
we see that .
e 1 weakly as
k.-.
It is easily verified that T(B,O). R(B,e) • {el,
e « Tw(B,e).
that Tw(B,8)
co
Taking .~
R(B,e).
This example mq be JIOdified to give an example
for which T (B,e) • Z and R(B,e). {el.
w
Examples tor which R(B,zo)
l
~ be e&8i~ constructed
Tw(B,zo)
in finite-dimensional spaces where T (B,z ) • T(B,z ).
woo
That equality does not hold in general in property v ot tangent
cones tollows trom the above example, property 11i ot polar cones, and
the tollowing proposition.
PROPOSITION 6.2.
T~(B,zo)· T' (B,zo)
:!!. ~ only !t
Ty(B,zo ) -c R(B.z
., 0 ).
SUppose T (B,s ) c R(B,z). Then
w
0 0
R'(B,z ) c T'(B,z ) c T'(B,z). But R'CB,z). T'(B,s }, ao
o-w
00
0
0
Proot.
T~(B,zo) • T'(B,Zo)'
Converse'-,
suppose Tw(B,z0
)¢ R(B,z).
"-J
0
Z t:
Tow(B,zo)
z~
such that
R(B,zo'.
Since
Then there is a
R(B,zo)
1s a closed convex
cone the separation theorem of section 4 can be applied.
a
z* c Z*
such that
z·(z)
for all
. But
z'.
z« RCB,zo)'
t
Thus there is
T'(B,z)
y.
0
Therefore
since
> 0 ~ z·(z)
z.« R' (B,zo)
z.(z) > O.
Thus
and hence
z·, T' (B,zo) •
Tt(B,z) ~ T'(D,zo)'
w
0
.
....
{'
18
References
[lJ
M. M. ~, Normed Linear Spaces, Springer-Verlag, Berlin, 1958.
[2)
N. Duntord, J. T. Schwartz, Linear 'Operators, Part I. Interecience
Publishers, New York,
[3]'
1964.
P•..J. Gould, J. W. Tolle, A necessary and sutficient qualitication
. for constrained optimization, SIAM J. Appl. Math. ,to appear.
[4]
M. GUignard, Generalized Kuhn-Tucker conditions for mathematioaJ.
programmiZlR problems in a Banach space, SIAM J .Control,
•
7 (1969), pp.232-241.
[5]
L. Hurwicz, Programming in linear spaces, Studies in Linear and
Nonlinear Programming, K. J. Arroy, L. Hurwicz, H. Uzava, ada.,
Stantord University Press, Stanford, 1958.
[6]
H. Kuhn, A. Tucker, Nonlinear programming, Proceedings of the
Second Berkeley Symposium, on 'Mathematical Statistics and
Probability, J. Neyman, ed., University ot California Presa,
. Berkeley, 1951, pp. 481-492.
[7]
o.
Mangasar1an, S. Fromovitz, The Fritz John necess&ry' optimality
conditions in the presence of equality and inequality constraints,
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. Russ~ll, The Kuhn-Tucker conditions in Banach space with an
application to control theory, J. Math. Anal. Appl.,
15 (1966),
pp. 200-212.
[10]
P. P. Varaira. Nonlineai' progrUllldng and optimal control, ERL
Tech. Memo. M-129 t University of CAlifornia, Berkel.,., 1965.
19
[11]
P. P. Varaiya, Nonlinear programming in Banach space, SIAM J.
Appl. Math'
t
15 (1967) t pp. 284-293.
.
..
"
==.==.. . I...........
DQCUNM ,ceNt_QAT•••• D
... ~~~~~~~~~~~~~~"'::·::"'==al~••
~
·~.~M~tM~1~.~""~.
~.~_~.~~~"~~f'~lIta~.'~fl_~.~
__JJ1.
T . .cu.. ~\' Q-.....14C.TlO..
Department of Statistics
University of North Carolina
... GROUP
Optimality Conditions and Constraint Qualifications in Banach Space
.
(.)
F. J. Gould and Jon W. Tolle
I'"
......
t ....NOOQI4-67-A-0321-003
....,...
\
~.
(NR047-095)
Mimeo Series
e.
e'
The distribution of this report is unlimited.
.....t......
·~.,.
Operations Research Program (Code 434)
Office of Naval Research
Washi,n ton, D.C. 20360
I
•
In this paper necessary optimality conditions for non-linear programs in Banach
spaces and constraint qualifications for their
applicabi~ity
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.
,
assumption is
kn~
made~
If a certain convexity
then this optimality condition is shown to reduce to the well-
extension of the Kuhn-Tucker conditions to Banach spaces.
constraint qualification is weaker than those previously given.
OD .':-:'••14 3
In this case the
•
.-
-0_
14.
tee" . .
Kuhn-Tucker conditions
. Optimality conditions
Constraint qualifications
,
..... "
Lno • •
-.&.&
w't
llI0"'C
.
.
'-iNK C
_?
"Oio.l!
.,.
.
Nonlinear programmillg
Banach spaces
.
.
...
.
.
.
•
.
... .
,
,
~
t
I
~
!
•
I
i
l