38
ON SUFFICIENT CONDITIONS FOR OPTIMALITY
s.
Let
f,g 1 , ••• ,gm
defined on a domain
n
•
RoLemaz~
WaP8BClbJa
*)
be continuously differentiable real valued functions
of n-dimensional real space
lRn.
we consider the
following optimization problem.
f(x) + inf
(1)
x
gi(x) s 0,
Let
i.e.
x0
€
Q
gi (x 0 ) =
o
We assume that at
x0
all constraints
THEOREM 1 ( [91 l : Suppose that at the point
Vgi ~ aPe "lineaPLy independent.
conditions foX' optimaLity
\
>
o ,
i = 1,2, ..• ,m ~
are active,
aLL gradients of g.
x0
i.e. the'l'e a'l'e
~
If aU
gi
~
~
Suppose that at x 0 Kuhn-Tuake'l' neaessa'l'Y
V(f + ~ A.g.)
(2)
(1)
hoLd~
n .
€
then
~
Ixo =
x0
Ai
~
0
suah that
0 •
is a LocaL minimum of p'l'obLem
if and onLy if it is a LocaL minimum of the foUOUJing equaLity p:t'obLem
f(x) +inf
(3)
gi(x)
= 0.
The proof of Theorem 1 is elementary and uses only the implicit
functions theorem.
Theorem 1 gives a very useful algorithm for reducing a
problem of sufficient condition for problem (1) to well-known classical
*)
This work was
Victoria.
~artially
supported by Monash University, Clayton,
39
problem (3).
In this way we can obtain sufficient condition of op·timality
of order higher ·tha.Jl 2.
It is important that this alg·ori·thm needs only to
invert one fixed matrix determined by the gradients.
Now we shall presen·t a simple example
EXA~lPLE
1 ( [9]).
Let
g 1 (x,y,z)
-(x + y) +
g 2 (x,y,z)
-
y + z
4
f(x,y,z) = x + 2y- x
I·t is easy to check tha·t for
(0,0,0)
2
+ y
2
- z
2
Kuhn-Tucker conditions hold for
Using the theorem vJe can replace problem (l) by problem (3).
y = z
that
4
, x = z
2
(0,0,0)
- z
4
and
?
f = x + 2y - x- + y
2
=
2z 6 •
Thus
It implies
is a local minimum of problem (1).
Replacing
f
by
ax
a
we are able to prove that for
have local minimum, at
(0,0,0)
> 1
.
2
+
the corresponding problem does not
In both cases ·the corresponding
conditions are of the order higher than 2.
(6 in .the first case 4 in t:he
second one.)
follo~tJing
This basic theorem can be ex·tended in the
THEOREM 2
Suppose that at the point
([9]):
are linearly independent.
Suppose that at
necessary aondi.tions for optimaUty hold.
i
= 1,2, ••• ,p,
of the problem
problem
A.i
(1)
0
for
x0
i = p+l, •. .,m
way.
all gradients of gi
X
the Kuhn-Tucker
0
Suppose that
Then
3
X
0
A. >
~
o,
is a local minirrrum
if and only if it is a local minirrrum of the following
40
f(x) -+ inf
0
i = 1,,2, ••• ,p
gi (x) ,; 0
i = p+l, ••• ,m
g. (x)
(4)
~
There is a natural question.
.
The existing second order sufficient
conditions [3], [7] (historical discussion of the subject is well presented
in [4]) did not request linear independence of the gradients
V'gi
but
positiveness of the second differential on the set
m
(5)
n
i=l
T
T.
~
where
0}
if
A.
>
o
T. = {x: (V'g.,x),; 0}
if
A.
=
o .
T.
~
~
(6)
~
~
Observe that in fact the set
independent subset V'g.
~.
where
T
can be described by a linearly
span(V'g. )
Theorem 2 we can obtain the following
([12]):
Thus basing on
~.
J
THEOREM 3
~
J
k-time continuously
Let f,g 1 , ••• ,gm be
diffePentiable functions defined on a domain n c IRn. Suppose
aU constPaintsaPeactive, i.e.
thePe aPe
A1 , ... ,Am
~
gi(x 0 ) = o, i = 1,2, ••• ,m. Suppose that
o such that the diffePentials of the LagPangian
m
L(x)
~
f(x) +
Aigi(x)
i=l
aPe equal to zePo till the oPdeP k-1
foP all hE IRn, i.e.
i
1,2, ••• ,k-l
and that
for
whePe T is defined by (4).
Thus
h
E
T
x0 is a Local minimum of pPoblem (1).
41
For
k = 2
the result wa.s known much earlier [3],
[7]o
~:rhe n.a·tural ques"tion ·which arises is abou·t ext.ensions of the ·theoremS
spaces~
given above on Banach
[lOll:
THEOREM 4
Let
be ordered spaces..
function de fined on
Let
It can be done in ·the foll.cvNi.n.g way
z be
A
Let
Q
Bar~<_-cch
x
be a domain 'i..n
Let
F
Let
'
y2
be a real valued
n
be
: fo + Y1 , G2 : ll + Y2 , H :X -> Z
operatOl"S.
o
spaces,
We considex' the fo Zlowing problem
d-ifferentiabLe
0
F (l,) ~· inf
(x)
s
0
(x)
s
0
(7)
H(x)
Suppose that at the point
x0
0
aZZ constraints are active
0
H
Suppose that the differenl;iol
Y1 x
x
z .
'i7G 1 x I/G 2 x 'i7H
0
is a surjection of X onto
Suppose that there are li-near funct·iarzals
such that the differ•entiol of the Lagrangian
(8)
If
0 .
d(L(x) ,h)
is uniformly
positive~.i.e.
there is
c >
0
such that
(9)
for aU
Y
E
Y1 , Y ~ 0 , then
x0
is a local solution of problem (6)
if and only if it is a local solution of the following equality problem
42
F(x)
inf
+
G1 (x)
G2 (x)
0
0
:S
H(x)
¢1
The hypothesis that
,
~2
W
,
0
are linear is not essential;
it is
enough that they are odd.
Theorem 4 can be generalized to Lipschitz functions in the followir1g
way.
THEOREM 5 ([lOll:
Let
x,
Y1 , Y2 ,
H : X +
those
operato1•s~
z .
z • n be as in Theorem 4.
Let
We shalL not assume conUnui-/;y of
but we assume that the muLti-function
z}
is 7.-ocaUy Hausdorf
continuous~
there is a neighbourhood Q1
each neighbourhood Q of
i.e.
of x 0 , Q1 c Q and a neighbourhood
in the space
(Gl (xo) ,
a:nd a constant
xy 2 xz
x0
W of
K
>
0
such that
,'z) n
where
d
arbitrary
denotes the Hausdorf dista:nce of spaces and
no~
in
x Y2 x
z , . coinciding on
such that
;:: 0 '
;:: 0
and
~~ 1
i.e. II (y,O,Ol 11 1
Y1
If ther'e are linear conti..nuous j'unctionaZs¢ 1
denotes an
II 11 1
E
* , ~2
Yk
E
Y*
2 ,
·1:s uniformly positive (i.e.
and the function
:satisfies the Lipschitz condition with constant
i'L,
then
(9)
WE
Z*
hotds)
43
solution of the problem (6)if and only if it a local solution of
problem
(7) provided
(lOl
< 1 .
~~c
Theorem 5 gives Theorem 4 via Ljusternik theorem [5].
r
warranting
o-ther conditions
is locally Lipschitz, can be found in the papers [1],
[2],
[8].
Theorem 5 presented above can also be used ·to obtain results of
sufficient conditions for Pareto minimization.
Let X
THEOREM 6 ([13]) :
·that the spaces
yl • y2
let
•
F
•
Gl ~ G2
•
>
~
p
y2
z
3
p
3
a1'e ordered.
be Bcrnach spaces.
Suppose
Let u be a domain in
X
and
be continuously differentiable operators
H
,
u
H :
+
z .
We are looking for a local
Pareto m·inimum of the following problem
F(x) + inf
(ll)
G1 (x)
:<:;
0
G2 (x)
$
0
H(x)
0
Suppose that
(i) there are continuous linear functionals
a
E P
*
y
E
such that
where
\IF , \/G 1
at the point
\/G 2 , VH
x0
are the differentials of F , G1 , G2 , H taken
(this is caUed a necessary condition of optimality
of the Kuhn-Tucker type) •
(ii)
the functionals
a , A. 1 , >.. 2
are positive, and
a1 ,
>. 1 are
44
posltive
that for
i.e., there are positive constant
p :::: 0 ,
(::Lii)
0
?
the constrm:nts are ooiJh;e at
"" 0
F
(iv)
y
l
X
0
Q
1:s a surjection on
P
0
H
f
and.
is a
, 17H)
on
XH
(v)
the space
ker 'i7F and the halfsub.space
ker 'i7Gl n ker 'i7H n {x : 'i7G 2 (x)
have a positive gap
d
such
O}
, i.e.;
d
max(inf{llx -yll, x
:S:
E
L1 , y
E
L2 ,
llxll
1}, inf{llx- yll,
J(
E
> 0 .
Then
x0
is a local Pareto rninimum of problem
(11).
In the theorem presented above condition (v) is very restric:·tive.
Unfortunately simple examples [14] show ·that
~chis
condition is essen·tiaL
REFERENCES
[ l]
S. Dolecki,
'Semicontinui ty in constrained optimizai:ion' , Con"tloo l
a:nd Cybernetics 1. (1978) Ia, No. 2, 5-16, Ib, No. 3, 17··26.
[2]
S. Dolecki and S. Rolewicz,
'Exact penalties for local minima", SIAM
Jou:mal of Cont1ool and Optim. 17 (1979), 596-606.
[3]
M.R. Hes·teness, 'An indirect sufficiency proof for the problem of
Bolza in nonparametric form', Trans. Amer. Math. Soe. 62 (194 7),
509-535.
45
[ 4]
F. Lempio and J. Zowe, 'Higher order or:Ytimali ty conditions •
(p:r,eprint of
rr::,
,_ .J.u
Is.,A .. Ljus·ternik !'
Russian)
[6]
l~~ath.
3,
Institute, Univ. Bayreuth).
conditional es·trerna o£ func·tionals
~On
!Vlatem, Sao
u ,
(in
(1934). 390-401.
H .. r.riaurer and ;]" .Zo~J·Je 1
]Pi:rst and second order necessary and
sufficient op·timalit:y conditions for infinite-dimensional
Mal; h. PY•ogr'wmni.ng 16
pr::>g-:camming problems' ,
[7]
E.J •.McShane,
'Sufficient condit:ions fm:' a weak re1a·tive minili1mn in
'che problem of Bolza', T1nans. IJJ71er.
[8]
S. Role>·licz,
S. Rolewicz,
52 (1942), 344-379.
Se1~.
Optimizat·ion
Math.
(1989) , 3-lL
'On sufficient conditions of optimality in ma·th,smatica.1
prog:ra.rnming', Oper.
[10]
Math. Soc.
'On intersections of mu1tifunc'cions',
OpeY.•forschung und s·,';atisU.k.,
[9]
(1979) , 98-110.
Res. Ver'f . .1.!.'-!.. (1981),
149-152.
S. Rolewicz, 'On sufficient condition of optimality for Lipschitzian
functions" , F1.ooc.
Conf. Game :rheoi'Y an£1 Mcrthemat·ical Economy, ed.
0. Moeschl:Ln and D. Pallaschke, North-Holland Publishing Co. 1981,
[ll]
S. Role>vicz,
'On sufficient condition of vector optimization' ,
OpeY'.
Res. Verf. (Methods of Operation Research) _'!;!_ (1982), 151--157.
[12]
s.
Rolewicz,
order' ,
[ 13]
'On sufficient condi·tions of op-timality of second
Ann. Po Z. Math.
S. Rolewicz,
(1983)' 297-300.
'On Pano:·to optimization in Banach spaces' ,
S'tud. Ma-i;h.
(·to appear).
[14]
s.
Rolewicz,
'Remarks on s·wfficient conditions of optimality of
vector optliaization',
Math" Operationsforschung and Statistik,
8e1"ies Optimization (in prin·t) .