1
HarvaPd University. AwOl£)'s work was sponsored in paPt by the Office of
Navat Research under ~t number N00014-87A-0298-0019 (NR04r-004).
2
University of North CaroUna at Chapet Hitt. The work of Goutd and H07JJe
was sponsored in paPt by the Office of Navat Research under grant number
.l'IOQO__l {-67..A..0321..00G.a .iNRQ47.-098 J.
AGENERAl SADDLE PoINT RESULT FOR lmSTRAINED OPTIMIZATION.
t
by
K.J. Arrow l , F.J. Gould2 , S.M. Howe 2
DepaPtment of Statistics
University of North CaroUna at ChapeZ Hitt
Institute of Statistics Mimeo Series No. 774
September, 1971
.--....,.......--.
..
AGENERAL
SADDLE POINT RESULT FOR CONSTRAINED QPTH1lZATION
by
K. J. Arrow 1 , F. J. Gould 2 , S. M. Howe 2
I,
INTRODUCTION
In the context of nonlinear programming theory, the
existenc~
of a saddle
point of the Lagrangian function is known to be heavily dependent upon convexity properties of the underlying problem.
In particular, for a concave pro-
gram satisfying the Slater condition,.with a solution x*,
such that
x*, u*
there is a
is a saddle poine of the Lagrangian function.
u*
In 1956, mo-
tivated by game theoretical and economic implications, Arrow and Hurwicz
demonstrated thot the concavity assumptions could be relaxed via a Qodified
Lagrangian approach [1].
The results in this 1956 paper were
terms of a specific modified Lagrangian formulation.
pres.~ntecl
in
In 1958, in a discussion
of gradient methods, Arrow and Solow presented additional saddle point
in terms of a different modified Lagrangian function [2].
Another
r~sults
sp~cific
result along the same lines was presented by Gould and Howe in 1971 (4J.
In this work. we both generalize and simplify the above presentations.
Conditions will be given under which, for a nonconcave (as well as concave)
program, a quite general function
1
P will possess a saddle point corresponding
llarvard Univcpsity. Arr'O/J)'s 7POJ:>k 7.las sponsorcd in part by the Office of
unaeJ:> grant: n~70er N00014-87A-02B8-0019 (NR047-004).
Nava~ Rese(~ch
•
University of NOJ:>th Carolina at Chape~ HiZZ. The W01"K.. of GouZd and HOlJe
2
was sponsored in Pa:r't by the Office of Naval RSS8a:t'ch under g;rtant nwnber
NOOO-14-67-A-0321-0003 (NR04?-098).!__
2
to the prugram solution.
Specific
r~a1i~ations
of the
P
function will be the
modified Lagrangian expressions discussed by the above mentioned authors.
II, REDUCTION TO A~ U~CONSTRAINED PROBLEM
We" formulate the nonlinear program with both equality and inequality
constraints:
(P)
max
n
x e: R
f(x),
subject to
gi(X)
S
0,
i - l, ••• ,ro
h j (x) - 0,
j'" 1, ••• , p •
Througbout the paper it is assumed that all functions in (P) are twice
differentiable.
e
Corresponding to the inequality constraints (the correspondence will be
clear from the
P
A(~ ,T"l,a):
function formulation) let
RXR+XR+ -)- R
be
a
multiplier function having second partial derivatives with respect to the first
argument l , where we employ the notation
the following properties on
.(i)
for any
a >
(ii)
for any
(1
(i1i)
Note:
o2A/c~2
= All'
Impose
A
°
> 0
for each fixed
oA/o~ - Al'
Al (O,n,OI)
• n
for every
n
~
A1 (;,0,0I)
='
0
for every
t
< 0
n ::- 0
(ii) implies thlit, for any
All (O,o,a)
a > 0,
~
<
-loCO
0,
Corresponding to the equality constraints let
as
a
-+
0
110.
All 0;,0,01) ... O.
4l(t,Tl,a): RxRxR
+
-+
R
be
a multiplier function having second partial derivatives with respect to the
first argument, such that
(v)
(vi)
k
1 R
+
for any
a > 0
for each fixed
~l(O,n,a).
Tl
£
n
as a
R
denotes the nonnegative orthant of
-
for every
rf.
-+
0 € R
00.
3
Examples of the
Ita
functions are
A
I (l+cx),
loll:
n(E;+l)
M2:
(n/ ex) ex.p (a~)
r- n//~a,
l..•
2+nt.
~
.; ;!; -
1/201
~ > -
1/20.
an even_ integer
t s - n/2e
M4:
t > - n/2a.
An
example of the
function is
ep
M5:
Now define the modified Lagrangian function
P: RnxRmxRPXR
+
-l-
R as
follows:
If the multiplier functions are chosen as in Ml or M5 one obtains, respectively,
the P
£~~ctions
studied by Arrow and llurwicz [lJ and by Arrow and Solow [2].
By the choice M2 one obtains the P function studied by Gould and Howe [4].
The following result, sometimes referred to as Finsler's Lemma, will be
used (see Debreu [3], or the appendix to this paper):
lemma: Let" Q be a real nxn matrix and let L be a real mxn matriX.
Suppose
T
z Qz < 0
ficiently large,
for every
T
T
z'[Q-aL L]z
z =f 0 such that Lz· O.
=f
O.
tive definite on the null space of L then for
a
T
<
0
for all
z
I
Q-aL L is negative definite on the whole space.
Then, for all ex
suf-
That is, if Q is negasufficiently large
4
Let
L(X,ll,l/J)
= f{x)-<j.l,g(x»-<1/I,h(x», which
is the usual Lagrangian
function.
'l'hc firt'icrcsult dfthlspiipcris -the fol1owio h o
Theorem 1:
ditions for
Suppose
x*
(X*,ll*,ljJ*)
satisfy the second order sufficiency con-
to be an isolated local solution to
complementarity holds.
That is, we
(a)
VL(X*,ll*,ljJ*)
(b)
8 (X*) :;; 0,
~
ass~~e
i .., l" .. t>,m
= 0,
0,
and suppose strict
• 0
i
h. (x*)
J
(F)
j
(c)
ll.*
J.
(d)
yTV2L(x*J~*'W*)Y
= 1, ... ,p
ll.*
> 0 if and only if gl.'(x*)
J.
<
0
for every nonzero
= 0,
y·Vh 1 (x*). 0,
= l, ••• ,m
such that
y
.,..
and
i
j =
1, ••• ,p,
where
I
,.J
denotes the active inequality constraints.
I
= {i:
tion
gi(x*)~O).
Then, if
P(x,lJ*,1/I*,a): Rn
maximum at
x*.
Proof: For any a
> 0,
+
a
That is,
is sufficiently large, the func-
R has an unconstrained isolated local
m
VP(X*,ll*,ljJ*,a)
•
Vf(x*) -
l
1=1
- r ~l
j=J.
a
V2P(x*,lJ*.ljJ.*,a)
Vf(x*) -
A1 (Si(x*) 'lJi*,a)Vsi(x*)
(h j (x*)
''''j ,a)Vh j (x*)
IIlJ i*Vgi (x*)
-
I
ljJ.*Vh (x*)
j=1 J
j
5
::
V2 L (x*_, \J~( ,Jj!~)---l Al r(gi (x)n, J.l i *,CL)Yg (x*) ',;I
T
T
.L
Hr (x~\-)
- r ¢l,(h.(x*),~.*,a)Vh.(x*)Vh.T(x*).
')
j=l
Defining
JT
=
4
J
J
J
J
[VSi(x*), •• ,id, Vhl(x*), •• ,Vhp(x*)l,
T ...
Y v .. L(x*, u*) 1/J*)Y < 0 for
sufficiently large.
eVel"Y
Now if
a
nonzero
note that
such that
y
Jy
>=
0,
(d).
by
By
is sufficiently large we can ohtain
All(gi(X*)'~i*.a) > k
each
id
911 (hj (x*),ljij*,a) ;.. k
each
j::
l, •••• p
whence
...
+ t [k-A 11 (g"i (x*) ' \J i*, a)]V~.(x*)vg.T(x*)
V1
1
t"
+
which, for all
a
! [k-¢ll (h. (x*),1/Jj*,a)]Vh.(X*)Vh
j=l
J
J
T
j
(x*>
sufficiently large, is negative definite, since large
a
implies that each of the dyadic terms has a negative coefficient and is hence
negative semidefinite.
o
II L ALoCP-L SADDLE
\
POlf\;'T RESULT
To obtain the IDain result, it is necessary to impose the further assumptions
(iv)
if
over
a >
0,
n
~e (-co,O
~
0 then
A(~,n,~)
is monotonically nondecreasing
J•
It should be noted that condition (iv) is satisfied by the examples Ml thru M4.
6
If the muJ.tlplier fun~tior;s sati.sfy properties (i)-(vi), then
Theorem 2.:
'under the cond.i.tion:1 of Theorem 1, if
for every
in som~ neighborhood
x
N
a
is sufficiently large,
of
x*
nnd every point
(lJ,;~)CR+IDxRP,
where
m
f(x) +
r
+
Proof:
L
i=1
[A(O~~.,a) - A(gi(x)'~1,a)]
J.
~(hj(x)'~j,a)J.
~
[$(O,$.,a) J
j=l
From Theorem 1, it follows immediately that
p (x*, lJ''', Wi, ,\.\)
x
for evt!!.ry
in
h.(x*) :: 0,
J
SOI~,e nt:~ighborhood
J
f(x*) +
== 1, ••• ,po
r
1\
~*i
141,
= O.
=0
A(O'~i,a}-A(8i(X*)'~i,a) ~
0
for
J,.1.
i(l.
for any
-P(x*,~*,w*)a) ~ P(x*JjJ,~,a)
-
Also, since
x*.
(A(O,ui,a) - ~(8.(X*),~ .• a») •
. '~I
Integration of (ii? from
A(O'~*i,a)-A(gi(x*)'~*i'~)
of
we have
.1.1>
For
P(X,jJ*,lji*,a):S
for all
~i?
gi(x*)
to
0
shows that
From the monotonicity property (iv),
O.
It follows immediately that
m p
(~,~)€R+ xR •
o
APPENDIX
Lemma: If
T
Z
Qz < 0
is negative definite for
for eve-ry
z "1 0
such that
Lz "" 0
then
T
Q-o:L L
a !sufficiently large.
i
Proof:
s = S.l.l.
m
1
If
= miny
matrix
Let
Q.
y€S,
Sol
denote the nullspnce of
Y ~ 0,
[yTLTLy : YES, IIyl
Hence,
= I jLy! 12 > O.
let M· i IQ!I,
then yTLTLy
l=lJ, and
L.
Thel<e are three cases to consider.
S.1.
= {z:
Lz==O},
Let
the sup norm of the
7
Case (i):
is true for all
a
0
~
n
J.
Suppose
S ... R.
T
-L L is
since
Q is negative definite and the result
Then
neg~tive
semidefinite.
n
Case- (ii):
veR ,
v .; 0,
T
T
v [Q-aL L]v
T
<
V .; O.
T·
v Qv - av L Lv
III
Case (iii):
zTQz
T T
Suppose
v Qv
...
-
{OJ ~ S~ ~ an.
T
[v
a~lvl"l
T
L L
J IIvl1 2
11~1~
Then for every
zeS
.I.
t
Z
1: 0,
n
Hence. let max [zTQz : z€s~,l Izi I-IJ • -m2 < o. Suppose vcR,
z
.
.I.
Consequently t
Then we can write v· y+z for some y£S, zeS.
O.
T
T
T
T T
T
•
z Qz + Y ·Qz + z Qy + y"Qy - ay L Ly
~
- 1n211z112
- -
+
2Mllyll IlzlI
a~lIyl12 - ~[IIZIf - ~
+
Mllyl12 -
Ilyl
r
+
am111yll2
~
- lIy 11 2
(-am1 +~ +H) - m2 [ IIzll- -!; lIyll
Ityt 12 + HI JyJ12
J:o
8
D"'F"":'~~'CE'"
l\I:: Cru:n
,:)
[1]
K.J. Arrow and L. Hurwicz,
Reduc.tion of constrained maxima to saddleThird EcikeleySympodul1l on Hathema-tTcal
Statistics and Probability, ed. J. Neyman (University of
California Press, Berkeley, 1956).
POillt problems,
in:
[2J
K.J. Arrow and R.M. Solow, Gradient methods for constrained maxima,
with weakened assumptions,
in: Studies in Linear and Nonlinear Programming, ed. K. Arrow, L. Hurwicz, H. Uzawa
(Stanford University Press, St&\ford, 1958).
[3]
G. Debreu t Definite and semidefinite quadratic forms,
Econometrica XX, (1952) 295-300.
[4]
F.J. Gould and S.M. Howe, "A new result on interpreting Lagrange multipliers as dual variables", rnstit·ute of Statistics Mimeo
Series No. 738, Dept. of Statistics, University of North Carolina
(Chapel Hill), January, 1971.
/