LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAR 7
^^Kjjga»^ilMKiQ2ijj»228UiSiU
1973
THE FUNDAMENTAL CONTINUITY THEORY OF
OPTIMIZATION ON A COMPACT SPACE I
By
Murat R. Sertel
November 1972
629-72
Abstract
Optimizers on a compact feasible region are abstractly
specified and, as set-valued mappings, are studied for sufficient
conditions yielding them (as well as certain associated maps and
certain restrictions of all these) continuous, using function space
methods.
In particular, the study concerns the continuity of the
set of optimal solutions as a function of the three arguments:
(i) objective
function used, (ii) an incentive (or "penalty/reward")
function imposed, and (iii) an abstract "parameter".
An interpretation of the mathematical apparatus is suggested and a brief gametheoretic illustration given.
:
Thanks are due Miss Janet Allen for kindly (and meticulously) typing
what follows.
RECEIVED
JAN 18 1973
M.
I.
T.
LIBRARIES
Notation
Let S and T be topological spaces.
:
tinuous mappings of
W
c
T
,
the set
{
f
T^
e
I
f (B)
C
will denote the power set of
empty closed subsets of
{A
e
C(S)
I
AC
Given B
.
W} will be denoted by
(B
,
W)
.
C
S
and
(P (S)
t(S) will denote the set of non-
and
S
Given B
S.
S
c:
S
,
^B)
will denote the set
Throughout, R will be a non-empty set equipped
B}.
^
with a total order
The set of all con-
into T will be denoted by T
S
and a (Hausdorff) topology which is at least as
fine as the order topology.
Aim
:
Given non-empty topological spaces X = X
is compact with X
^
X„ and Y, of which X
(compact and) Hausdorff, in this paper we seek
sufficient conditions for the continuity of (a) the optimizers
R^""^ X
ex':
Y^
x^ ^
X
^(X^)
defined by the specification
a'
(u,g,x
^
)
= {x
1112
e
u(x ,x ,g(x ,x
X
I
12 ))
>:
Sup u(- ,x ,g(- ,x ))}
2
2
^1
(ueR^""^, geY^, x^eX2)
[a
specification which is quickly seen to yield a' as being into C(X )],
or (b) of its "associate"
a:
R^""^ X
determined by a(u, g) (x
"'(u,g) =
)
Y^ - (P(Xj
= oi'(u,
g,
^
x ),
or (c) of the restrictions
«'|{(u,g)}xX^=-^(^> ^'-'
G37018
^° ^("'g)>
^^'
-2-
2
'
X
"
"u
Motivation
:
^
~"
"|{u}xY^-
^
^^1^
of
to {u}
ex
Y^.
Our motivation derives from a desire to understand in
above) is continuous,
what sense optimization on a compact space (X
a
X
question we consider to be perhaps the fundamental topological
question of the theory of constrained optimization.
Interpretation
The functions u
:
or objective functions
R
e
XxY
may be called utility functions
perhaps the former being preferred so as to
,
These are understood to be
emphasize their merely ordinal nature.
representations of complete (i.e., "total" or "decisive") transitive
relations on X
x
y.
Concerning the existence of such continuous rep-
resentations we refer the reader to [Debreu, 1954] and [Sertel, 1971b].
For the functions g
e
Y
we prefer the term Incentive function
[Sertel, 1971a] to the more restrictive terms "payoff function", "penalty/
reward function", etc.
(although we use the term "payoff function" in a
game-theoretic illustration below).
For R we refer back to our Notation
of the familiar feasible region
.
.
As to X
e
Y, so that externally instituted rules g
optimal behavior, i.e.,
a'(u,
g, x
it plays the role
The role of Y is that the utility
functions u are in general sensitive, not only to x
y
,
).
X
e
Y
e
X, but also to
may serve to influence
may be considered as an abstract
"parameter space" in which a point is exogenously determined, subject to
which optimization is to be carried out.
A game-theoretic context may serve to illustrate.
For simplicity,
3-
consider a non-cooperative two-person game in
which X^ and X^ are the
strategy spaces of players
1
and 2, respectively.
Then g specifies the
"payoff" to player 1, for whom u serves
as generic utility function,
and a^ determines the player
(1) cc-mpletely: for mathematical
purposes,
we are interested essentially in
cx^Cg);
how this
x^
p
^
(x^
)
,
and would study
("reaction function") varies as
g is varied, given the "tastes"
or "preferences" as represented by u.
Here we also investigate how,
from the viewpoint of continuity, " the whole map a,
as well as its
associate a' and the restrictions listed
under
Method/Procedure
above, behaves.
The nature of the investigation makes it natural
to
:
proceed in the reverse order
<(c)
tinuity of the maps listed above.
topology on
(c)
(b)
,
,
in studying the con-
(a)>
We always take the unner semi-finite
C(X^) and equip the function sp pcp..
with their respective compact-on e n topolo giP.-.
and a^ (Theorem I).
.
R^"""^,
Y^ and
C(X
First we study
a'
^
)
(u,g)
Then we turn to a'^ in the Corollary to Theorem
I.
Finally, we study a and a
Topological Refresher
:
'
in Theorem II, taking Y locally compact.
Let
S
and T be topological spaces.
The
compact -0£en topology on the function space
T^ is the one generated
by {(B, W)
B
I
'^^^
{
<W>
1
c
S
is compact and W
c
W c T is open} as sub-base.
is said to be upper
S
^
C(T)
C(T) is the one generated by
A polnt-to-set map
ex:
S
^
semi-continnnn.^ iff it is continuous when
given the upper semi-finite topology.
ex:
T is open} as sub-base.
"PPer semi-finite topolopy on
^
C
T.
C (T)
is
It is easily seen that
is upper semi-continuous iff
is closed for every closed D
C (T)
the set {seS|
[Michael, 1951].
a(s)
H
D
?^
0}
-4-
Proposition
1
Let X, Y and Z be topological spaces, of which X is
:
/yxY")
locally compact, and let u
= g
co
Then w
is continuous.
u
Giving Y
Proof:
X
and Z
topologies, let (B, W)
such that
00
(g)
Z
Define a map w
.
X
e
Y
X
X
^ Z
X
by
and x
e
CZ X
be a subbaslc open set, and take any g
Denote G
C
and H = u"l (W)
= a)-l((B, W))
G
and, by continuity of u, H
b e
B,
(b,
(b,
g(b)); furthermore, using continuity of g, we may assume g(A,
e
X
x
Y is open.
H and there exists an open box nbd A^ x
V,
and, since X is locally compact, we may actually assume that
relatively compact (i.e., its closure
and A^
X
V,
C
H.
Now
{A,
b e
|
affording a finite subcover
{A,
G
CI
Y
is a
1=1,
|
1
..., m}.
to
e
A
H of
)
C
Is
C
V,
(compact) B,
Defining
A
X
Thus,
.
1
basic open set with g
is open and proving that
C
is compact) with g(A^)
B} is an open cover of
1=1
we see that A C Y
A,
Y
e
Hence, for every
g e
g(b))
X.
their respective compact-open
(B, W).
e
Y
:
and g (x) = u(x, g(x)) for every g
writing
(g)
e
C
is continuous.
G
,
showing that
V,
;
-5-
Lema_l:
S
Y:
^
compact.
Proof;
S
Let S and T be compact spaces,
of which S is Hauadorff.
cm
be a map whose graph
Let D
C
T be closed.
P is
S\
{
(s
t)
,
e
S
x
t|
t
e
Let
y(s)} Is
Then D Is compact, since T is
so.
is compact, S x D is also compact.
e
=
Then y is upper semi- continuous.
that its projection P into
(s
T
Y(s)
closed.
n
S
Thus
is compact.
,
F
O
(S
x
d)
Since
is compact, so
But P is precisely the set
D ^ 0}; and, being compact in
the Hausdorff space S,
This proves that y is upper
semi-continuous.
#
Proposition
2
Let X = X
:
Hausdorff; and, for every v
aCv)Cx2) = {x^
so that V
H- a(v)
be a non-empty compact space with X
X
x
R
e
Xj
e
define a
,
(v)
x^)
v(x^-,
>
'X
determines a map a on R
.
X^
:
^ (X
-*
Sup v(., x^)},
Then
X
(1)
^
^
C(X^)
maps R^ into
a
[i.e., for each v
C (X
semi-continuously into
Proof
(ad
:
Let v
(1)):
R
e
by
)
) ]
R^, a (v) maps X^ upper
X
,
and
;
For every x
.
e
e
X^
x
{x
is compact,
}
so the continuous v attains a supremum on a non-empty closed subset,
specifically on aCvXx^)
{x
x
=
{
(x
,
X
£
)
x|
{x }, whereby a (v) (x
x
(X
)
.
X
e
a (v) (x
) }
to show that) the function v:
X
->-
,
)
range is Hausdorff, their graphs
T
= {(x,
},
respectively, are closed.
,
r)
e
X
X
r|
r =
^(x
)
r)
e
X
x
r|
continuity of v and compactness of X, v(X) is compact.
set K = (X^
X
r
)
n
r
CX
X v(X)
is compact.
into X is precisely the graph r, so
is proved.
->
= Sup v(*,
X^
Since v and v are continuous
(x
C(Xj),
X
C(X^)
(it is straightforward
R determined by v(x
is well-defined and continuous.
{
:
is compact, and this we now do.
From the continuity of v and compactness of X
r_ =
e
)
To prove that a (v)
using Lemma 1, it suffices to show that the
is upper semi-continuous,
r
C
into
and we see that a(v) maps X
graph
c X
}
T
x
)
and their
r = v(x)} and
Also, by
Thus, the closed
Now the projection of K
is compact and, by Lemma 1,
(1)
(ad
P
Q
X
Q => v(p)
c R
|>
union
(
Agreeing on the notation
(2)):
<
Q
,
(- -,
((P,
n
t))
:
+ -))))
(t,
(Q,
U
<W>
such that a(v)
for every b
iust the
is
((P,
(-
-, i))
C
m
and there is no
and any non-empty open W
X
n
C
t
e
R
v
c
X^
R
X
Claim (established in the next paragraph):
there is a relatively compact open nbd V(b) of
\ W, is a nbd of
<|
V(b)
x
w*^,
V(b)
In that case, {V(b)
v.
)
|
i =
|
1,
K(b)
x
b
e
|>,
b
and a
where
B} is an open cover
..., m}
,
and it follows
_
O V(b.)
1=1""
x
<|
<W>
c: (B,
P,
is a subbasic open set, and take any
)
of B affording a finite subcover {V(b
a(G)
<|
e
q)
(p,
|
^2
(X
<W»,
(B,
e
B,
c
C C
)
compact set K(b) c W such that
that G =
|>
\J
(
^={(s, s)f;RxR|s.<s
Take any compact non-empty B
so that (B,
W*^ = X
R
e
(s.,s)eQ
+ «)))), where
(s,
= {v
>
<| P, Q
teR
(Q,
|
v(q)} for subsets P, Q «^X, first we note that
<
is open when P and Q are compact
U
P
|
V(b.)
w'^,
K(b )|> is a
x
nbd of v such that
I
showing that a is continuous.
),
To complete the proof,
we turn to establish our claim above.
For each b
(since W
B, v
e
compact)
is
attains a supremum s(b) =
s^(b)
=
Sup
v on {blxw'^,
Sup v on {b} x w, and
{b}xW
Now, for all b e B,
{blxw""
s^(b)
s(b) holds, and there are just two possibilities:
<
element t(b) such that
element.
If
<
s.(b)
t(b)
<
s(b), and
holds, define T (b) =
(i)
(- -,
As these are both open sets and v continuous,
v(b, w(b))
£
T
(b),
V^(b)
X
-
u(b)
C
V
-1
MT
R owns an
R owns no such
t(b)) and T*(b) = (t(b), + -).
taking any w(b)
W with
e
using (local) compactness of X, we may find a relatively
compact open box nbd V (b)
-
(ii)
(i)
*
(b)).
x
u(b) of
As W
c
(b
,
w(b)) such that
(the closure)
is compact, we may also find a
relatively compact open nbd V (b) of
Thus, writing V(b) = V^ (b)
V(b)
X
K(b)
and T (b) =
|>
is a nbd of v.
(£(b)
+<=°)
,
This completes the proof.
b
such that V (b)
V^Cb) and K(b) = U(b),
If
(ii) holds,
<|
w'^
C
V(b)
x
x
define T^(b) =
v"-' (T
(b)).
W^,
(-
<»,7(b))
and find V(b) and K(b) as in the case of (i).
FUNDAMENTAL CONTINUITY THEOREM
I
Let X = X
;
and Y be non-empty
X
x
topological spaces, of which X is compact with X
u
e
(XxY)
R
.
each g
e
Y"^,
\(&)
Consider the function
a (unique) map
(x^) = {x^
e
a
Cg)
u(Xj
X^
I
,
^
a
u
on Y
X
Hausdorff, and let
which determines, for
defined on X
x^, g(x^
,
x^))
by
uC
>^
Sup
e
Y^, a (g) maps X^
,
x^, g(-
,
x^))},
Then
(1)
a^ maps Y^ into
C2)
a
C(X
)
^
[i.e., for each g
upper semi- continuously into
Proof
C(X
and
)];
is cont]
Let g
(ad (1)):
:
x.
e
By continuity of u, the function
g^ = "u^^^ defined in Proposition 1 is continuous, i.e., g
and (1) is proved.
(ad
in g
,
(2)):
Proposition 2(2) shows that a
and from Proposition 1 we see that g
Thus, as the composition
a
^
'
as to be shown.
u
= a o w
:
u
Y
X
This completes the proof.
e
R
.
"X
Thus, Proposition 2(1) applies, showing that a (g) = a(g
= a(g
(g)
)
is
C(X
d?(X
)
1
,
a
u
)
continuous
is continuous in g
^2
->
)
e
(g
c
Y
is continuous,
)
•10-
COROLLARY
In Theorem I,
;
the map a'
Y
:
x
x
a'^(B, x^) = a^(g)Cx^)
is continuous
Proof
:
argument
(g e Y^,
e
e
X^)
(in both arguments together )
see that
From Theorem I, we already
^
(g
x^
Y^ and x
e
X
)
separately.
a'
u
But X
is continuous in each
is compact, hence
locally compact, in which case the result is a well-known consequence
of the compact-open topology (on
C(X
)
).
Lemma
2
Let X, Y and Z be non-empty topological spaces of which X and
;
Y are locally compact.
Z^''^ X
y:
Y^
(u e Z^'''^, g
->
e
6
Y^, X
X).
e
is continuous;
(2)
y,
too, is continuous,
C
Z
x)
C
X and V
C
be any open nbd of y'(u, g, x)
Y such that u(U
we may assume u(U
assume that g(U)
C
x
nbd of (u, g, x) with
(ad
(2)):
C
v)
V
g,
in the domain of p',
so continuity of u yields an open box nbd U
U
and
Then
Take any (u, g,
(ad (1)):
and let W
.X
xy^xX->-Z
and
y'
;
Z
z^ by y'Cu, g, x) = u(x, g(x)) and y(u, g)(.) = y'(u,
(1)
Proof
XxY
Define the maps v':
c
y
W;
V.
'
(N)
x
v)
c W.
Then (x, g(x))
.
x
e
u"-^
(W)
As X and Y are locally compact,
and, since g is continuous, we may also
Now we have N = (U
c W, showing that
x
y'
V, W)
x
(u, V)
x
u a
is continuous.
This follows directly from (1) and the well-known fact
that the compact-open topology on Z
X
,
V of (x, g(x)) with
is always splitting.
7
•12-
FUNDAMENTAL CONTINUITY THEOREM II;
compact space with X
compact space.
6
reference to Theorem I) by a'
g e Y^,
,
x
x
be a non-^empty
Hausdorff , and let Y be a non-empty locally
Consider the map
"associate" map a on R
(u e R^''
Let X = X^
x^
e
g, x
(u,
Y
x
)
XxY
=
x
Y
X
X
x
defined (with
a^CgXx^), and consider the
determined by a(u, g) = a
Then
X^).
on R
a'
is into
a'
C
(X^
)
(g)
and a is into
CCX^)
and both maps
ca/'
K^''-'.^.
are continuous,
Proof:
In Lemma 2(2), set Z = R.
Also, using Proposition
clearly, a = a
CiX^).
o
Since X
topology on
2
,
R^""^ x
y:
a:
R
R^ ^
Then y:
x
Y^ ^ R^ is continuous.
^0
X
C(Xj)
->
Y^ ^ C
(X^
)
^
^
is
continuous.
is continuous and a'
is compact, hence locally compact,
\
C(X^)
completes the proof.
is conjoining,
7
Now,
is into
the compact-open
so a', too, is continuous.
This
'^
.,
•13-
Historical Note
:
An earlier study of the continuity of the set of
optimal solutions is [Dantzig, Folkman and Shapiro, 1967].
It works
with feasible regions lying in a metric space and with objective
functions coinciding with real-valued incentive functions, and takes
the "parameter space" X„ as singleton, investigating the continuity
of the set of optimal solutions as a function of
used and (ii) feasible region.
(i)
objective function
Thus, neither does it contain, nor is it
contained in, the present study.
In a paper following the present and [Sertel, 1971a], we intend
to study the continuity of the set of optimal solutions as a function
of the feasible region as well.
(The present study extends a corrected
version of the author's "The Fundamental Continuity Theory of Optimization on a Compact Hausdorff Space", Alfred P. Sloan School of Management,
Working Paper 620-72, October, 1972.)
14-
FOOTNOTES
0.
Our notation R is intended to be mnemonic of the real line, i.e.,
the set of all real numbers taken with the usual (Euclidean)
topology, since objective functions in the literature are almost
always specified as real-valued.
(N.B
The usual topology on the
set of reals coincides, of course, with the order topology, using
the natural order of the reals).
Thus, the reader may wish to
interpret R, throughout this paper, as the real line; mathematically,
(s)he is quite free to do so.
.
1.
A terminological warning may be in order.
For, according to a
terminology not used here, a mapping into a space of non-empty
closed sets is called "continuous" iff it is continuous when the
range space is given its "finite" topology (see [Michael, 1951]),
a topology finer than the upper semi-finite.
When we are given
a map of a topological space into a space of non-empty closed sets
and the topology on the range is understood (e.g., to be the upper
iff
semi-finite) we feel free to speak of the map as continuous
it is so w.r.t. the topology on the domain and that on the
,
range!
*
2.
Note that a function Y
S ->
(f(T) which is singleton-valued
(i.e., such that, for each s e S, ¥*(s) is a one-element set)
is upper semi-continuous iff the function ^
S -> T, defined by
1'*(s) = {'f(s)}
(s e S), is continuous.
:
:
3.
4.
Denoting (R x pJs ^ by < by
and by (s_, + °°) we mean {re
,
(- «,
r|
s^
<
s) we mean {r e r|
r}.
r < s}
Strictly speaking, as made* clear in announcing our Aim, a is
..
As {u} x Y^ is homeomorphic to
on {u} xY^ and a (g) = a'
(u,g)'
Y
5.
,
our informality in treating a
See, e.g., Theorem 3.1 (2) on pp.
as a
Y'' should cause no
map on Y^
261 of [Dugundji,
1966]
Notice that, if Y is the real line, it is locally compact. The
relevance of this is that our theorem may be applied in the study
of optimization subject to real-valued (e.g., monetary) inceative
e.g., the case of
functions, such as in the usual economic case:
taxes or subsidies, the case of wages or salaries, and the general
case of a price system.
See
§
10 of Ch.
XII in [Dugundji, 1966].
REFERENCES
[Dantzig, Folkman and Shapiro, 1967],
J. Folkman and N, Shapiro, "On the Continuity
Dantzig, G. B.
of the Minimum Set of a Continuous Function", Journal of
Mathematical Analysis and Applications 17 1967, pp. 519-548.
,
.
2.
.
[Debreu, 1954].
"Representation of a Preference Ordering by a Numerical
Debreu, G.
C.H. Coombs and R.L. Davis (eds
Function," Ch. XI in Thrall, R.M.
(New York, 1954).
Decision Processes
,
,
.
,
3.
[Dugundji, 1966].
Dugundji, J., Topology
U.
,
(Allyn & Bacon, Boston, 1966).
[Michael, 1951].
Michael, E.
"Topologies on Spaces of Subsets", Transactions of the
American Mathematical Society, 71 1951, pp. 152-182.
,
.
5.
[Sertel, 1971a].
Sertel, M.R.
"Elements of Equilibrium Methods for Social Analysis,"
unpublished Ph.D. dissertation, M.I.T., January, 1971.
,
6.
[Sertel, 1971b].
Sertel, M.R., "Order, Topology and Preference," Alfred P. Sloan
School of Management, M.I.T., Working Paper 565-71, October, 1971,
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