-:,: N- 1:1 0 ? ?S :
~
: :
:·::
0-::X:i0::f
·i
··._.I
u:-ff
: 0054: u
:
;':-u:8E
::0
:.-;:·
·-·;:·
: aSil S: D : 00
A: 2 ED
~:
pape
k::i
'--wor
::i;-
:
fi:; : S r000
-:~]
]
I·-·:
-·.
·I:
;
'1
''
· :;:
.- ;·
.t
r
· ·-:·
:···I--·
·
i..i'r
i
;···
·. :,·-,
·-.;·.,:---.
·
:·
-·
:-·
·
:···-.·
-----.-:,
· :.
·-:·
·.---: ·· ·
:I
..
i-.
,;·
·"·
-i-···
..
·- I:;··-·i
-·.··
i
i;.
;·'
;;
:·
r.
i;
·:.
i
I-· --:i;·-:· `·· :
;-··i.
..
:- ·
:?
i
..-· -;..
-r
·
-
1-
:
1;
ISTITU UTE
5-OF T ECHNOLOG Y
MAS.C:T;
'·
THE STRUCTURE OF ADMISSIBLE POINTS
WITH RESPECT TO CONE DOMINANCE+
by
Gabriel R. Bitran*
and.
Thomas L. Magnanti*t
OR 074-78
April 1978
+ Several results from this paper were presented in less general
form at the National ORSA/TIMS meeting, Chicago, May 1975. This
paper is a revision of CORE Discussion Paper #7716, June 1977.
* Sloan School of Management, M.I.T.
t Research partially conducted as a fellow at the Center for Operations Research and Econometrics (CORE), Universite Catholique de
Louvain, and supported, in part, by the U.S. Army Research Office
contract DAAG29-76-C-0064 and by the U.S. Office of Naval Research
contract N00014-75-C-0556.
·L
Abstract
We study the set of admissible (pareto-optimal) points of a closed
convex set X when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions of X,
we
(i) extend a representation theorem of Arrow, Barankin and Blackwell
by showing that all admissible points are either limit points of certain
"strictly admissible" alternatives or translations of such limit points
by rays in the closure of the preference cone, and
(ii) show that the set of strictly admissible points is connected,
as is the full set of admissible points.
Relaxing the convexity assumption imposed upon X, we also consider
local properties of admissible points in terms of Kuhn-Tucker type
characterizations. We specify necessary and sufficient conditions for
an element of X to be a Kuhn-Tucker point, conditions which, in addition,
provide local characterizations of strictly admissible points.
I
I.
INTRODUCTION
Formal approaches
to decision making almost always pre-
sume that an underlying preference relation governs choices
from available alternatives. Rich theories now go far toward
either characterizing or computing solutions that are in some
sense
"optimal".
Mathematical programming techniques predom-
nate when preferences can be embodi-edoin a real valued
utility function (Debreu [1],
ditions. See also Bowen
[2] discusses appropriate con-
[3] and Arrow and Hahn [4, page
Multi-attribute utility theory (Keeney and Raiffa
[5])
106]).
pro-
vides one means for considering multi-objective situations
which involve several, possibly conflicting, criteria. In
summarizing methods for studying multi-objective decision
making, Mac Crimmon [6] has classified approaches as weighting methods,
including statistical analysis;
ination techniques;
sequential elim-
mathematical programming procedures;
and special proximity methods.
Although these theories have made impressive contributions to decision making, they have yet to resolve a number
of issues that are fundamental
to both descriptive and
o
prescriptive theory. For arbitrary preference relations,
still
little is known, and possibly can be said, about such
an essential concept as admissible alternatives, also called
- 2 -
nondominated, efficient, or pareto-optimal alternatives. Even
when a convex cone
,
the set of points
x+
{x +p : p
P}
specifying those alternatives preferred to x, describes
preferences, admissible points have not been characterized
completely. Is the set of admissible points connected ? Are
there representation theorems which characterize admissible
points ? What are local
haracterizations of admissible
points ? Can the notion of admissible points be exploited
within the context of solving mathematical programs ?
In this paper, we consider several of these issues. In
Section 2, we
introduce notation and concepts to be used
throughout the paper. The next section considers global characterizations of
admissible points, including existence. We
show that the sets of admissible and strictly admissible
points are both connected when
is convex,
nonzero
(ii)
p E F}
the set
(i)
F
{p
is nonempty,
(iii)
+
the preference cone
E R
n
s
+
· p > 0 for all
the set of available
alternatives X is convex and closed, and
of
:p
P
(iv)
some element
makes an obtuse angle with every direction of reces-
sion of X. In this section, we also present a representation
theorem which partially characterizes admissible points. This
result
says that "all
limit points of
admissible points can be expressed as
'strictly' admissible alternatives or as a
translation of such limit points by certain rays in the
- 3 -
closure of
the preference
characterization of
approximations.
cone
Tucker characterizations
Our analysis
possible
extensions and
These
the properties
restrictions
set of
y.
Later
Geoffrion
y,
studied
economic
Simon
is
several
preferred
to y,
of papers
properties and
and Titus
results
[20]
[9],
Kuhn
[11]).
or suggested
Smale
and
[13],
and
to
[16],
[17]
x
y,
[14],
problems. More
of
recently
by cones as here,
this paper.
[21] have
the
that
computational aspects
related
Wan
and/or
that
is defined by
[12],
vector maximization
literature,
linear
most notably
polyhedral
considered preferences defined
including
in
that we consider here under various
in a series
certain nonlinear
[15]
x is
statistics
(Koopmans
either proved
on the problem structure,
x
is an outgrowth
[8]),
economic planning
available alternatives
preference
Yu
in
The final
convex analysis
(Gale, Kuhn and Tucker
fundamental contributions
many of
x
and
linear
applications.
This approach
Arrow, Barankin and Blackwell
[10]),
of
local
to the usual Kuhn-
1950 in mathematical
and nonlinear programming
and Tucker
are related
is based upon results of
of work conducted around
[71,
in terms
of nonlinear programming.
and mathematical programming.
(Wald
Section 4 considers
admissible points
These results
section discusses
".
and
[18],
In
Rand
studied local
the
[19],
properties
-----
------------111
- 4 -
of
admissible points
ogy. A number
from the viewpoint
of other
papers
of differential
(Charnes and
topol-
[22], Ecker
Cooper
.,
and Kouada
[23],
[24], Evans
Geoffrion,
Dyer and Feinberg
and Yu and
Zeleny
mining and
investigating admissible
have
[30])
[25],
and Steuer
Philip
[27],
treated
Gal
[28], Sachtman
points,
primarily for
'
Applications
include
ical
equilibrium
the concept of
[31],
and many others
dominance
maximizing utility vectors
[34],
as
cone
risk-return trade-offs
risk sharing and
suggested
in
are varied
planning [4],[11], mathemat-
in economic
[41,% [32],
[33],
selection
of
efficiency
statistics
29],
algorithms for deter-
linear problems.
and
[26],
[36]
in exchange
in portfolio
group decisions
and
the
collection
[35],
[37].
a
2.
PRELIMINARIES
Throughout
nontrivial,
n-vector x
cone
,
say
i.e.,
FP
{0},
is preferred
denoted
x
We
our discussion, we
x -y,
y + P --
that a point
{y+
y
to
cone
let
in
R
an n-vector
when
x
y
P
be
. We
y
a nonempty and
say
that an
with respect
to
the
and
p: p eP}.
is admissible
for
the cone
P
over a
- 5 -
given set
to
y;
X
when
that
yE X
A(X)
denote
x
contains no
points preferred
) = {y}.
the
Frequently,
Then
X
is,
x n (y+
Let
and
y
set of all
the cone
when
x
y
P
and
admissible points
is
y
in
X.
he nonnegative orthant
x *
y,
and
R n
admissibility is
commonly called Pareto-optimality.
As a useful variant of
this
P
example,
the preference
cone
Pk= {0} U (...,Xkyk+l
In this
'n)
"
case the preference
is
given by
E Rn: all Xi >
and (
relation compares only
'
the
O
k).
first
k
components of any vector.
The preference
problem of
of
k
zEZ
zE Z,
is
arises
"optimizing" a vector
called
efficient
satisfying
If
x
k
from the vector maximization
f(z)
=
(fl(z),f2(z),...,
real valued criteria over a subset
least one
xEX
cone
f(z)
in
this problem if
f(z)
with
of
R
n-k
there
the inequality
.
is
A point
no
strict
point
in at
component.
X =
{x = (y,z) E R
n
: z E Z
is admissible with respect
= (y,i),
Z
fk(z))
y
tion problem.
= f ()
By
and
this
z
is
to
and
y
the cone
efficient
f(z)},
Pk if
then a point
and only
if
in the vector maximiza-
construction, we have
expressed,
and
can
6 -
-
defined by
-
z
study, the preference order
f(z), in terms of a cone preference
f(z)
f(Z),
f(z)
Pk in an enlarged
space.
that both closed and nonclosed
These examples suggest
preference cones might be studied profitably since both arise
in practice.
Many properties of admissible points depend upon the
separation of the sets
P
is a nonzero n-vector
x
p
p
p ·y
P)
(y+
since
convex
are
X
and
by a hyperplane.
X
and
n X
p
such
that
*(y
+
for all
p)
=
{y}.
xE X
That
and
we may reexpress
p E F,
p
a nonzero vector
of
(2.1)
there
is,
p E P.
(2.1)
p
p > 0
Because the right-most inequality can be restated as
for all
For any
such a separation is possible whenever
y,
admissible point
y+ P
by saying that
contained in the positive polar
there is
cone
P
defined by
P
P
+
+
E
n
+
with the property that
p
y
0
for all
p E '}
solves the optimization problem
max {p . x : xE X}.
The positive polar cone is closed and convex without any
assumption on
P.
(2.2)
-
There
for
is
a point
a partial
y
to be
P
If
this
or
that
set
converse to
Let
P
defined by
{p
E R
is
is
: p
only at
p
e P
if
y
y
>
0
for
: p
In this
for
.+
E P)
x -y
implies
that
solve
(2.2).
y does not
sible
points
2.1
admissible
if
max
{p4
some
p
p
: An
it
e Ps
p
= 0}
that
the
any strict
Ps
factis
· (x -y)
and
> 0
defines
intersects
converse
support
a
simple
con-
x
y
(i e., x *
that
these special
admis-
definition
y
:
A(X)
is
strictly
the maximization problem
.x: xE X}
E PF.
Any other admissible
point
is
said
to be
nonstrict.
We
let
in
X.
AS(X)
denote
the
set of
y,
and, therefore,
We distinguish
admissible point
solves
(any
supported
+
p
in the following
Definition
for
that
p E P}.
is strictly
x
admissible. This
sequence of observing
set of strict
terminology,
(2.2)
require any
all nonzero
P
that
{xE R
solves
is
necessary condition
the
a strictly supported cone
then
,
denote
S
nonempty we say
the origin).
states othat
1PF
.p
a supporting hyperplane
P
this
admissible, which does not
convexity assumptions.
supports of
7 -
strictly admissible
points
----------·I---
C·-U-cl-rr-r
Ulc
Y
II-LIIIL-
WIIIU.-
- 8 -
Most of our subsequent results require that preference
cones be strictly supported. When
this condition is equivalent
containing no lines.
P
is closed and convex,
to it being pointed, that is
In an appendix, we extend this charac-
terization in order to interpret the
strictly supported con-
dition directly in terms of the underlying preference cone
whenever
P
is convex. The following characterizatiofi .isoa
consequence of this development.
Proposition 2.1.
Let
be a convex cone in
R n . P is supP+
ported strictly if and only if the relative interior of
is contained in
P.
S
As an example, if
or
P1 > 0
=P
{p E R
and
\ {0},
3
: p1
P2 > 0},
P = {p
then
(pIP P2
= cl
P
R 3 : (!'P 2'P
3E)
and the relative interior of
> 0, P 2 > 0
and
=
p3 = 0 ,
:
n{p
+
3
3 )=(,O,O)
=
0},
is the set
+
a strict subset of
Remark 2.1
Throughout
the remainder of this paper we frequently
apply elementary results of convex analysis without reference.
We also
translate many properties of polar cones usually
formulated in terms of the (negative) polar of any set
denoted by
S
- {y ERn :y
x g O
for all
ments concerning the positive polar
S+
xE S}
S,
into state-
of S. Standard texts
- 9-
in convex analysis (Rockafellar
[38], Stoer and Witzgal
[39])
discuss those results that we use.
In addition to the notation introduced earlier in this
section, we let
RC(X)
set,
ri(S),
let
cl(S),
denote the recession cone of a convex
and
-S
{x :- xE S}
denote
closure, relative interior, and negative of any set
the
S,
let
S\T denote the set 'theoretic difference of two sets
and
T.
We adopt Rockafellar's
the origin in
RC(X),
[38] terminology by
and
S
including
but by defining direction of recessions
as only the nonzero elements of this cone. Finally, we use
the Euclidean norm to define the open and closed unit balls
in
3.
Rn
GLOBAL CHARACTERIZATIONS
When the set of available alternatives
closed and the preference
cone
strictly, the admissible set
erties
: it is connected
support to
P
X
is convex and
is convex and supported
A(X)
has
important global
(see Theorem 3.4)
prop-
if some strict
P makes an obtuse angle with every direction of
recession of X, and
admissible points
it can be characterized in terms of strictly
(see Theorem 3.1)
of X belongs to the closure of
P.
if no direction of recession
-
Before
sider the
3.1
establishing
-
10
these properties,
existence of admissible
we briefly
con-
points.
Existence
Yu
is
[15] has
compact
fices)
and
or- if
P
or X.
conditions
the
interior of
he problem
tion for some
of
observed that
p
He
EP .
not
FP+
max {p
is nonempty
is nonempty
x : xE X}
if either
(
that
A(X)
$
S
may be
X
suf-
has a unique
Neither condition requires
also notes
are
A(X)
solf-
convexity
empty when these
satisfied.
o
-The following
of
admissible
ily bounded,
propositions characterize
points
sets
supported
Proposition
3.1.
cone and let
X
and only if
both
P
:
x E X,
and
If
If
any
be a strictly
a nonempty closed
the origin
is
and the recession
Proof
yEX
so no
but not
point
necessarare defined
supported closed convex
convex set.
the only element
cone of
x
X
) RC(X) = {0},
Then
contained
A(X)
in
X.
O E P n RC(X) theno> x + y
y
P
convex,
closed convex cones.
Let
be
closed and
of alternatives when preferences
by strictly
if
for
the existence
)
(x +
X
for
any
is admissible.
then
(y +
)
([38], Thm. 8.4). Consequently,
X
is
for
compact
any
p
s
for
E P
S
-
there
is
max {p
sible
an optimal
solution
11
-
z
to
· x :x E (y + P) n X}. The
in
(y+
admissible
P)
in
X
X,
with
for
z + p E (y + P) n X
is
point
respect
if
to
To obtain a dual version
of
Proposition 3.2.
X
P.
(II)
Consequently,
P
if and only if
FP
set
this
convex
whenever
is a polyhedron.
any
p
0E
if
and
in
and
is any
closed, then
and alternative
P nRC(X)
Then exactly
X
is nonempty
0.
Moreover,
(I)
o
valid :
of admissible points
(II),
Alternatives
set.
0.
g
alternative
Proof:
supported closed
{0}
convex cone, not necessarily
X
then
proposition, we note
be a closed
n RC(X)*
(b)
admis-
z.
strictly
rnRC(X)*
the
z + pE X,
P be a
nRC(X) *
'P
strictly
It also must be
and
one of the following alternatives is
(I)
is
the alternative.
(a) Let
convex cone
and let
0
z
to
0 E P
p
preferred
the followiig theorem of
the optimization problem
(II)
(II)
ps E
s
AS(X) *
n
RC(X
~
implies that
cannot
s
strictly supported
both be
)
must
implies
AS(X)
valid
satisfy
since
-
+
p Pp
Ps
>
Since
O
0
Ps*P
and
0.
RC(X)
that is,
real number
and all
and is
p
of
F+
s
S
that
a closed
shows
Since
0
for all
so
p
+
that one
Since,
P+
+.
is
2
if
alternative
(II)
u
(I)
P
++
thus
= P
and. (II)
y
RC(X)
s
inequality
inequality
> 0
u. p
and
is
uE P n RC(X)
always
satisfied.
is nonempty
is not satisfied,
A(X) *
~
if
if
is valid.
the final assertions
of
the proposition,
con-
the optimization problem
max
x:
p
p+E P+ .
w
s
xE X}
z E A
If
(3.1)
(X),
we can select
this problem. Therefore
implying
Closure
2
and
the previous proposition, A(X)
and only
solves
and a
the recession cone
right most
E FP
Consequently,
(I)
-where
n
5
alternative
sider
for all
The left most
since
P
for all
if
prove
O.
closed. The
and only
To
uE R
a limit point of
of alternaotives
by
they can be
origin is
, 8 =
.
=
u-ps
= RC(X),
convex set
>
P s nRC(X)
u-y < B
the
uE RC(X)
s
+
that
is a nonzero vector
in. RC(X)
u- p+
that
there
satisfying
contained
implies
So suppose
are nonempty convex sets,
S
separated;
-
+
and
PF
12
Ps E RC(X)
that
of
P
and
Ps Y
that
is not required for
Every point
of
sition 2.1,
ri (PF)
P
+
is
a limit
C
P
C P
s
p
s
so that
O0 for all
P+
S
this
point of
RC(X)
z
y E RC(X)
* 0.
conclusion.
P
+
,
since,
by Propo-
.
-
_
__.I_
I_
_
_I
-
Ps E F
If
y E RC(X)
for all
that
n RC (X)
there
is
13
-
and X
is polyhedral,
then
implying, by linear programming
a solution to
problem
(3.1).
p
.y
0
theory,
This solution
is
strictly admissible.
i
The following examples
is
closed
verses
is
to
required for
part
(b)
of
the
show
that
the hypothesis
these propositions
last proposition
and
that
P
that con-
are not
possible.
O
Example
and
3. 1.
P
let
Let
=
X =
{(x
(xi ,x )E)R
2
I
,x
2
:x 1
0
IR2 :x x
>
1<
and
x2
2 )
< 0
and
P
x2
, x
>
0) U
I
>
O,
X2
>
(O,O)).}
I
:X)
r-
I
X
III
X 1
Figure 3.1
IP nRC(X) =
+
Then
p
IP
P
(-
,0)E
is not
RC(X) *
valid.
part
{(}
of
the
S
closed
or
In addition,
(b)
and No Admissible
Points
+
=
S
cone
{01
PF
nRC(X)
and
A(X) =
and
neither of
nRC(X)
= 0
the converse
last proposition
to
.
The
preference
the conditions
for
A(X)
to be
empty is
the first assertion in
is violated.
I__ 1____ ______
O}
I
-
Example 3.2.
Let
those points
in the
be
= {(0,A,0) :
RC(X)
Then
space
12>
and
3
)E R
is the line
{(X,X 2 ,3)
0
xl- x 2
E
3
> x
= O0 x
:x 3
A(1,1,0),
A(X) = {(x,x,x
2
3)
the sub-
is
3
{( ,XX
i.e.,
x I < x 2 +1}.
and
2
for
0 and
~ °
n A3 >>1 }.
2
XE R; RC(X)
=
except
the x2-axis,
) 3 :
2X
°
2 3
A1 ~> 0,
~
: x1 +x 2 = 0}; P=
A 3 > 01 and
3
xl
plane not on
R3
in
the positive orthant
> 0} u {(XI
X = {(xlx2x
Let
14 -
3
3)
:X,
E R: x 3 = 0 and
x3
Xl--=X2}X2
I
1
0
2.
x
1
X1
7
III
J
.
121
..
a
"I
X3
Figure 3.2
Proposition 3.2 Requires
P
this
instance,
or
(II)
of Proposition 3.2
that A(X)
when X
closed and
is not
In
is valid.
$ 0 does not necessarily
is polyhedral, unless
P
to
be
Closed-
neither alternative
The
example also
(I)
shows
imply alternative II even
is closed.
15 -
-
3.2
Representation
[8] have
Arrow, Barankin and Blackwell
= R
points whenever
all preference
Example
X,
as
with vertex
illustrated
of strictly admissible
The
is convex and compact.
this property is
x3 =0}. Let
x3
and
(X
A2 )
0}
in Figure 3.3,
not valid
for
= P2
2
P
{}
U{(
which "ignores"
1
,
,
2
2
points
on the
points
is
the
the
interior of
entire
the arc KS,
arc, but
line segment
the
3) :
AS(X)
X2
X1
Figure
3.3
A(X)
consists
admissible points
xa
cl AS(X)
O
2>
± (0,0,1).
the closure of
KV
2
0
33 )
in
: x2 + X2
1
2
{x E
the direction
strictly admissible points
set of
cone
be a truncated
and circular base
V = (0,1)
and
include
that
every
cones.
and
The
and X
that
3.3
Let
R
n
example shows
following
point
is a limit
admissible point
shown
of
those
these
also
-
Observe
those
that
points
in
the direction
closure of
next
X
admissible
(0,0,1),
result,
which
points
more general
X.
cone
P2
that
shows
upon
following result
of
but not
example
the arc KS by
in place
P2
to
the
itself.
Our
hypothesis,
extends
the
representation by permitting
and by
coones
their
are
this observation characterizes
under very general
preference
We build
in this
a direction which blongs
Barankin and Blackwell
Arrow,
points
translations
which are
the preference
admissible
of
the
16 -
relaxing compactness
clever proof
techniques
using
the von Neumann minimax
of
the
theorem.
0
Lemma
3.1.
Let
Rn satisfying
(ii)
RC(C) n D
u * v
that for some
u
:
The
Rockafellar
exists
if
lemma
E C
u
[38,
is a
of
saddlepoint on
and
· v
u
f (v)
u
By
.v
for
for
the
C.
u E C and
al
a theorem due
for
their
vE ri(D)
recession over their domain
in
vED.
to
theorem a saddlepoint
this
uv
recession over
-u-v
C x D
E D
v
specialization of
the functions
gv(u)
{0}.
Theorem 37.6].
common direction of
functions
=
0}
has a
u.v
Proof
nonempty closed convex sets in
b
C n RC(D) =
the function
sense
D
and
the conditions
(i)
Then
C
o
u E ri(C)
domain
have no
A direction
--
D
have no
and
the
common direction
of recession
~---- -~- -----
y
-
for
f
(v)
is a nonzero
Therefore,
the
recession
u .y
-< 0
all
the
functions
for
of recessioh
impossible
if
to
3.1.
strictly
is
x
to
inequalities
C
=
have no
-y.
condition
X
{0}.
belongs
to
common
satisfies
{0}.
Similarly,
common direction
v < 0
for
be a closed convex set,
-
no
0.
all
vE D
yERC(C)
are
is
(ii).
and suppose
Then any point
=x
u.y
let
be a
o
that
xE A(X)
can be written as
p
AS(X)
cl
and either
p = 0
or
p
belongs
cl P\F.
:
Proof
We
first establish preliminary results
the proof. Let
3
RC(D)
vE ri(D)
supported convex cone,
x
where
is,
yE RC(D)
satisfy simultaneously whenever
Let
RC(X) =
cl P
if no nonzero
that
satisfying
for. uri(C) have
U
for
the
nonzero, which
Theorem
f (v)
uE C;
gv(u)
-
element of. RC(D)
functions
direction of
17
B
be
the
closed
unit ball
in
to
Rn
be used
and
in
let 3
Several choices are possible. We may, in fact, choose polyhedral cones
for the S.. Let B
denote the open unit ball in
j=1,2,..., let F.
be a finite set of points in
with the property that
hull
H
of
[ri(P ) u{0}]n B °
The convex
is polyhedral and, so then, is the cone
S. that it generates [38, Cor.19.7.1]. Any
in F.
Q
and for each
-balls about these points cover Q.
{0}UF UF2 u...UF
of a simplex in
Rn
xEQ
belongs to the interior
Q whose dimension equals that of
Q. For some j,
points
are close enough to the vertices of this simplex so that xEH..
ThereforeU
{H.
: j > 1}3 = Q and the union of the
Sj's
is ri(+ )U
3~~~~~~~~~
{0}.
18 -
-
cl
+
P
=
1~+
+
)
= ri
ri(
cl P = P
whose union is
Sj CS j+)
(i.e.,
cones
S..
j
j>1
n RC(X) = {O}
integer J, S
+
(B nS
finite
T.
sets
x
let
RC(Z) = RC(X).
where
conditions
which we
there
(i)
E X,
zj
t
E T
< z - t j
3.1
lemma
j
< z
t > z .t
zj . s>
valid as-well.
S
0
> 0
and
.t
to
for
Tj
and
) = T.
reduce
and
i s compact,
T.
Since
let
i Is convex and
C = Z,
to
RC(Z)
for
each
T
j
=
{(0}
J
satisfying
for all
Z
Z
Therefore,
zj E Z
and
X
in
Then
with
above.
the
z -= 0 belongs
The definitions of
are
x E X}.
established previously.
are points
x
+
as
of
(ii)
and
convex
=S..
for some
is defined
z-t
Since
4
T.
We apply Lemma
J
are nonempty,
any admissible point
be
: z =x -x
Z = {z E R
J
O
contrary to hypothesis.
for j =1,2,...,
S. n B
In addition,
and compact.
Now,
-
so is RC(X)1
is closed,
X
intersection property [since
The
j
j > J. Otherwise,
j which implies by the
for all
(B ncl P) n (B nRC(X)) * {0},
that
Then
some positive
for
that
Note
) U (0}.
ri(P
whenever
(Bn RC(X)) * {0}
)
convex and increasing
be nonempty, closed,
Sj for j =1,2,...,
z E Z,
t e T
(3.2)
and
all
imply
for all
.
tET
that
sE S
j
the
inequalities
(3.3)
-
is
4
ri (
) u{O}, the inequalities
p + E ri(P +),p
quently,
j}
if
for all j sufficiently
z -x
-x
is a limit point of
p
because
z
cl P.
Therefore,
x
gives
=x
x
z
Ecl A
j
= x
(X).
-x
any
limit
satisfies
0
But
x
tj
Consequently,
[x
-
x
solves
>
x]
max {tJ-x :
and
x ji
A
s (
X)
the strictly
To
{zj
point
because
a
as
(x
= 0
the sequence
or
z
the
---^--
{z
.
E cl P \
j }
j
This
theorem
[x -x ]0
if
with
x E X.
for all
is
a
limit point
of
{xj}
we must
shotw that
Suppose ,to
Euclidean norms
increases.
x
XJ
As we ha,ve
the
sequence
the contrary,
of
the
z
that
must grow
noted previously,
1--111.1_
-------II
P
-x0)
gives
limit point.
j
to
tj EP + . Thus
contains
wi tho out bound
belongs
and
E X}
the proof,
the
t he sequence
then
,
any
Conse-
this condition,, since,
admissible points
Then
z
z
(3.2)
complete
it doies not.
large.
()
conclusion of
the
t
x
j=1,2,...
A(X).
fulfills
, expression
*
for
that for
0
x
with either
- z
representation
If
chosen from
was
Eri
*
++
=P
x
p
O0 for all
Sj
imply
> o
then
ri (F
(3.3)
·z
++
to
the increasing sets
the union of
Beca tuse
-
19
*---1---
.I-VIC--UUCr--_
.-
- 20 -
for each
j
++
p
e ri (P+),
sufficiently
+
'
>
0
cl P
contrary
to hypothesis,
0
and hence
and
p
+
ri (
e
0 for all
our assumption
to
the
). Therefore
([38],
y ERC(X) = RC(Z)
that
is
untenable.
P
is
closed,
or
P
RC(X)
limit point
contains no
Zi
.
·
p
any limit point y
for all
and
)=
=
j+
then for
y
P
yE ri (P
z'} jl
z
large. But
Zi
jl
sequence
p
the
Thm.8.2),
sequence
U
O
When the preference
P
RC(X) * {0}
cone
A(X)
M
=
and
fills the hypothesis
of the
characterization simplifies
Corollary
3.1.
Let
X
A
is a
Example
3.4.
Example
3.3.
ful-
to
o
:
Then
P
be
every admis-
limit point of strictly admissible points.
to Example 3.3
shows
the
that
might not be valid when
{0}.
Y
be
Then if
P
Let
the half-line
from
admissible points
We
which
be a closed convex set and let
slight modification
n RC(X)
{0}
the
representation of Theorem 3.1
cl
=
theorem. In either case,
a strictly supported closed convex cone.
sible point
either
should
V
is
the cone generated by
= P2, the
passing
through
K.
The
set of
set of
in
V+
Y
is
strictly
empty, however.
emphasize that
_
admissible
X- V
----
the previous
I'-~~~~~~~~~~-~~~-^1~~~~~'-- "I-----------
results do not
1~~~~~~~~~~~~~~~~~-I~~~~~~~-I
-' ----
--
~~~~~~~~~~~~
-
an example
show by
Blackwell
that points expressed as
shows
Exampl.e
As
3.5
in Example
P= {(0,X,0) :
Let
X
the
be
:
X = {(XlX2,0)
in
,Every point
is
X
2
in
R
3
following example
let
as in Theorem 3.4
- p
E clA (X)
P
3)
is
admissible.
be defined by
XII
0
2
0 and
A 3 > O}.
given by
x>
on the line
x
x
X
1 X2'X
> 0,
admissible and
strictly
3.2,
0u
triangle
x =
even when
admissible,
need not be
a limit point of strictly
that
admissible. The
points need not be
admissible
-
completely. Arrow, Barankin
admissible points
characterize
and
21
and xl +x
2
segment
2
<
1}.
x
satisfying
every point
x
in
X
+ x2 = 1
can be written
as
x = x
for
some
-p
sible. The
xI =
3.3
x2
E
x
E cl P \P;
and
admissible points
and x
+x
2
=
are
but
not every point
the points
in
X
is
admis-
on the
lines
1.
Connectedness
When
parametric
X
is a polyhedron
analysis
and
P
is a polyhedral
in linear programming
shows
that
cone,
A(X)
is
- 22 -
connected
(see, for example, Koopmans
[30] for related results).
[11], or Yu and Zeleny
To study the connectedness of
A(X)
in a more general setting, we also use a parametrization of a
mathematical programming objective function.
We first set some additional notation. Let
p(l)
n
and
+ (l-8)p(O).
et
be any points in
p(6) =
p(l)
solutions to
R
for each
X(6)
0
e
p(O)
and
1, let
denote the set of optimal
the optimization problem
v(e) = sup {p(6).x
: xEX}
(3.4)
0
for a given set
X,
generic element of
Berge
not necessarily convex; x(o) denotes a
X(8).
[40] discusses several properties of parametric
optimization problems which encompass results for
this prob-
lem. See also Hildenbrand and Kirman [41], whose introductory
description of parametric analysis is formulated
to include
the following result.
Let X be an arbitrary closed subset of E n and assume
Lemma 3.2.
that the solution set
compact for all
X(8)
O0 6
to problem (3.4) is nonempty and
1. Then the point to set mapping
8 - X(e) is upper semi-continuous; that is, if
G is an open set in
real number
!e -e'
I
6.
R n containing
6 >0 such that
X(e),
0 < 8 <1 and
then there is a
X(') C G whenever
0 <8'
I and
23 -
-
Using this
lemma,
we
first
consider the
set of
strictly
admissible points.
Theorem 3.2.
Let
be a strictly supported convex cone, let
X be a given subset of
Rn, and suppose that the solution set
to the optimization problem
max {p
compact and connected for ech
.x :x
X}
is nonempty,
p+ E P+. Then
A (X)
is
connected.
O
Proof
that
G
: Suppose
to
the
contrary that
there are disjoint open sets
is,
As(X) ° 0
x(O) EG
5, H n A S(X)
s
A
*
and
0
A
implies
and
that
p(l) E P
so
that
for
p(8) E P
= 0
X(O) C G
+
(-e)p(O).
(3.5)
and
I,
e' =
above,
Since
the
is nonempty
x(8)
strictly
p(O) E P
solves
as
set of optimal
and connected,
contained
in H.
solutions
each X(8)
In particular,
X(I) C H.
sup
e :0 <
By the previous lemma,
either
The definition of
(3.5)
as
contained in G or
and
Let
with
E X}
is defined
X(8) to problem
is either
H
.,
max {p(( 3).x :
p(I)
and
connected
there are vectors
5
p(O) =
G
is not
(X) C GU Ho. Let
(X). and x(l) EH nAs(X).
admissible points
where
AS(X)
G
or
H.
But
tion. For
if
X(8') C G,
<I1 and X(e) CG for
e'> O. Now
either
then
X(')
assumption
X(e) C G
O < e <
}.
must be contained in
leads
for
all
to
every
a contradic8
e
+ 6
- 24 -
and some 6 > 0
and if
by Lemma 3.2, contrary to the definition of 8';
X(8') C H, then by Lemma 3.2 again, X(e) C H
8 > 8'-6
and some
Therefore,
6 >0 contrary to the definition of
our assumption that
is untenable and
for every
AS(X)
'.
is not connected
U
the theorem has been proven.
The following version of this
theorem is probably more
useful in applications.
Theorem 3.3.
Let
P
be a strictly supported convex cone, let
X be a closed convex set, and suppose that
Then
AS(X)
Note
some
+om
+
,
sup {P
necessarily
+ n RC (X)
S
the condition on.+
p+
<0
*
cone of X. That is,
lem
n RC(X)
S
+
.
S
is connected.
that
Ps E
- P
every element y in
the solution set
· x :x EX}
all,
for
states that for
S
to
the recession
the optimization prob-
must be bounded for some, but not
strict
supports
ps
to
P.
This
weakening
of
the hypothesis of Theorem 3.2 is offset by the stronger assumption that X is convex.
With slight modifications, the proof of Theorem 3.2 proves
Theorem 3.3 as well. Without loss of generality, we may select
x(1)
in the proof as
- p(l)E RC(X)
of Lemma 3.2.
the solution to
. Then we
invoke
max {p(l).x :xE X}
the following result
(Observe that each solution set
---
--
instead
X(e), 0
I-
-
-
where
e
----------------------
I,
_.---rr---rrrrr*pC
- 25 -
to problem (3.5)
is convex and hence connected.)
Lemma 3.3. Let X be a closed convex set in in.
tion set
X(8)
to the problem
= sup {([p(l)
v(e)
is nonempty for
e)p(O)]
+ (1-
to set mapping
strict
- X(e)
lemma
When X
is
of -this
X}
is
to
theorem
I
<
Then
and the point
is upper semi-continuous.
proved
in Appendix B.
compact, RC(X) =
support
xx:
e = 0 and nonempty and compact for e = 1 .
X(8) is nonempty and compact for all
This
Assume that the soZu-
RC(X);
{0}
and every
point i n
consequently the hypothesis -
is valid whenever
P
is
strictly
is
.
s
a
nRC(X)
S
$
supported.
Therefore, we have
Corollary 3.2.
Let
be a strictly supported convex cone and
P
let X be convex and compact.
addition, P
Proof
is closed or
: As we have
connected. Whenever
is connected. By
connected
set
since
Then
P\{0}
just noted,
is open,
P\{0} is open, P+
is
is connected. If, in
then
A(X)
Theorem 3.3 shows
Corollary 3.1,
it
AS(X)
contained
= P
whenever P
in
and
is
the closure
A(X) and
contains
The next
two examples show that
is connected.
that AS(X)
so
A(X) = A(X)
closed
of
is
the
A(X)
is
connected
A (X).
the "strictly
___I____II
supported"
_ ______II__
__
- 26 -
condition imposed upon P in this corollary is indispensible,
- P
as is the condition
Example
s
n RC(X)4
* 0 of Theorem 3.3.
s
3.6. Let X be the closed unit ball in
P = {(0,X)E
2
:X
R}.
12
and le-t
R
is closed and convex, but P+ = 0.
Then P
·.
Since
S
the only admissible points are
(-1,0) and
(1,0),
A(X)
is not connected.
~~x3Let S = f(x 1 ,x 2 ,x 3 ) E FR :x 1 =0, X 3 g O and ~~~3
Example 3.7.
3 :
3- I < x 2
0} and let T be the halfline {(1,0,)
0 O}.
Define X, which is closed ( [38], Cor. 9.8. 1 ) , as the
.bao
:egative
negative combinations
be generated by nontrivial non-
P
hull of S amd T and let
p
of
convex
OS
p(0)
= (-1,0,0) and pII) = (1,-1,0).
Since the third component of every element of
PF
is zero, a
point in X is strictly admissible if and only if it belongs
to
y+ T
for some point y that is strictly admissible
set obtained by projecting X onto the
The solutions
-
x 2 axis.
to problem (3.5) are
S
X(e)
x
in the
=
if
=0
l
T U
T
T
(O
I , 0)
if 8=1.
x2
th '%2
I
fx
/
I
4
Alternatives X
.l
I
3
(0,-1)
Al
---3
Projection on xl-x
2 Plane
Figure 3.4
-
In this
case,
RC(X)
=
S
+
-
RC(X)
S
=
S
and
To show that
27 -
{(Xix
2,x 3
A (X) = S U T
A(X)
is
let
requires
£(y)
additional
£(y) i
and
<
},
is not connected.
connected without assuming
the underlying preference cone
is open,
3
) ER
P
is
closed
argument.
denote the
For
or
that
P\ {O}
that
n
yE R,
any vector
subspace generated
by
y,
and its orthogonal subspace.
Lemma 3.4.
Let
Suppose that
max {p
P and
p
belongs to
-x : xE X}.
alent
(ii)
(iii)
and that
y
solves
Then the following conditions are
equiv-
y E A(X)
yEA(X) where
: Suppose
tion of
= X
X
y is admissible in
y,
p = 0;
p
that
sequently, y
p can be
violated and
Corollary
that
z = y+
(y+
is,
p EC(p+ )
to
y
p E X
+ p
p
if and
3.3.
X
z
is not
chosen
(y +(p
) )
£(p )
)
with respe ct to
)
PI.(p
p
P
.
:
(i)
Proof
n
X be a cone and arbitrary set in
for
only
if
y.
p E P.
p
p
and
z-E (y +.C(p
+
respect
to
admissible with
be nonzero,
some
But
By defini-
since
)
p
)X.
p
Con-
r,
that
if and
only
if
condition
(iii)
is violated.
Let p+ E P+ \P+
and let
y
0,
condition
is,
(ii) is
U
solve the problem
--
-
--
-r*-rr*rr-n
S
- 28 -
max {p.
p
x: xE X}.
contained in
Proof
: Apply
p EP fn
(p
Then either
belongs
to
lemma,
the
+
Corollary
3.4.
Let
for aZl
z E X
problem
max {(p
Proof
A(X)
p
let
:x EX}
that
for some
any point
U
.
+
.
S
A
X :
X = {x
to
-x
> p
+
y
z
to the
A(X).
(X) C A(X)
and,
by Lemma 3.3,
U
to A(X).
the representation
for proving
p
Then any solution
belongs
to
y beloongs
and
noting
boundary of
E P
S
belongs
These results
connected,
y + p E X
or
+
E P,
let
.x
y
X,
ingredients
p
and
: Since
A(X) =
A(X)
the boundary of P.
the previous
)
y
that
without requiring
the set
P
theorem 3.1
of admissible
to be
provide
points
closed or P\ {0}
is
to
be
open.
Theorem 3.4.
Let
P
be a
let X be a
closed convex
Then
is connected.
A(X)
Proof
: We use
k =1, A(X) is
the
theorem
sion less
Note,
strictly supported convex cone and
set satisfying
-
S
n RC(X)+
induction on the dimension k of
an interval and
is valid for
than k
first,
and
all
hence
closed
that X has
that no
connected,
S
X.
0.
Whenever
so assume
that
convex sets with dimen-
dimension k.
generality
is
lost by assuming that
- 29
X has full
that
dimension.
the origin belongs
and only
if it
containing X.
belongs
to
the
if and
orthogonal
+
pL
ps
esis of
- P
the theorem
of
+
solutions
nected,
and
(ii)
admissible points
to
some
this
cl
A
in X
the set
(i)
+
RC(X)s
has
O and
>
n L).
A(X).
EP
if
Ps
E P
s
+
and
pi
- (P
L , p
+4
=
y
PL
,
n L
S
p EP n
y
L
L)s
<
is
0O and
0.
the hypoth-
we may assume
Then y
. Let
problem.
that
X
If
and
(i)
,
is connected.
A(X)
(ii).
will
solves
X denote
(X) nA(X) ni
(
in X
Since
A(X)
the
set of
X
is con-
then by Theorem 3.3
together with
y EA(X)
was
the
chosen
be connected as'well.
establish
Since
the
X CX,
+
C RC(X)
that x
in X with
Consequently,
in L and
p
Consequently, we will
conditions
(i.e.,
+
+
is valid
Q
if
+
p
PL E L
that
strictly admissible points
arbitrarily,
implies
+
L
E RC(X)s
Rn
linear subspace
s
where
shows
p
= PL
.x: xE X} for
set of
in
is admissible
any
necessary,
Rk.
optimal
the
it
P
of L,
+
*p
ps
and
+
Let y be any element of
p
the smallest
+
PL
f[RC(X)rn
-Q
then
is an element
max
is connected
admissibility
if
+
=
subspace
is
Q
in L,
only
+
as
+
Thus
Then X
if
+
+
yERC(X),
X.
Moreover, expressing
L.
E RC(X)s
nonempty as
to
The definition of
Pn
+
suppose, by translation
is connected
A(X)
respect to
-s
For
-
theorem by verifying
RC(X) C RC(X)
and
+
. Therefore
dimension less
than k,
- P
the
+
RC(X)
*
and,
inductive hypothesis
___1_1
_II__
__
_1
_
since
implies
·_
X
o
- 30 -
that
A(X)
is
condition (i)
connected.
is
+
p
condition
let
x
S
Corollary 3.4,
x
x
p Ecl P\ P.
+
0
max (p
4.
= x
,
cl Pn RC(X)
would
-p
=
{0}
satisfy
+
p
+
x
is
+
. x
x
0
EA(X).
Since
(any
q
.x: xX}
0
p
+
x
4
q
and
Thus
x
0
p >0) and
can be repre-
either
_
-
+
-P + n RC(X)
s
s
p
pp
+
p
x
or
4
and
4
+
-
As
q
Ecl A (X) and
= p
* p < ps
-Ps
. x: xE X}.
(ii)
where
Therefore
= p
x
tion
max{p
+
0
the representation theorem 3.1 applies.
sented as
and
O
By
.
p E cl P rn RC(X)
PS
solve
+
RC(X)
s
s
and
(ii),
+
EF
S
+
P5
-
3.4, A(X)m X =A(X)
satisfied.
To establish
for some
By Lemma
*x
a result,
,and
x
so x
A(X)
E X
and
cl A(X)
x
solves
and condi-
satisfied.
i
LOCAL CHARACTERIZATIONS
Studying properties
of an underlying set by applying
convex analysis to approximations
ring and fruitful
adopt
this
is defined
the
not necessarily
x
to form an
assuming
that
convex.
By approximating
approximation
that
the
set has been a recurIn this section we
set of alternatives X
intersection of a convex set
to X,
ity in X via the approximation. We
hypotheses,
the
theme in optimization.
viewpoint,
as
of
C with a set D,
D at a given point
we investigate
admissibil-
show, with appropriate
strict admissibility
in X
is equivalent
to
- 31
-
admissibility in the approximation. We also establish a
Kuhn-Tucker theory in the setting of cone dominance, which
the Kuhn-Tucker theory of nonlinear
when specialized,becomes
programming.
4.1
General Setting
at
If,
x
if
L(x)
- x
-
-{(x
in addition, D C L(x0 ),
0
D at x . A support to
a canonical approximation to
L(x )
Let us call a set
D
x)
: x E L(x )} is a closed cone.
we say that
at x
0
is
L(x° )
a polyhedron. In this case,
L(x )
In many applications the set
is a support to
iO
said to be finite if it
D
> 0, i =l,...,m}.
two canonical approximations to
D
predominate
in the optimization literature. When each function
Vh.(x 0 ),
with gradient
x
O
D at x
is defined by a system
of nonlinear inequalities, D = {x :hi(x)
differentiable at
is
is the intersection of a
finite number of half-spaces, each supporting
In this case,
D
hi(x)
is
then
1
L(xo ) = {xE Rn :Vh i(x ° ) (x
and when each function
s. at x
(i.e.,
)
h.(x)
1
s
+
for
o
o
)
all
for all
x
xE
R
(4.1)
n)o
),
inequality
then
{xE Rn: si(x-x 0 ) > 0 for all i with hi(x 0 ) =0}.
11
=0}
is concave with supergradient
satisfies the "supergradient"
s(1
11
=
o) > 0 for all i with hi(x
h.(x)
h (x ) + s. (x -x )
L(x)
-x
(4.2)
__L____1__^
_-·-QCUIIIL.---^-_L-I -----·--- . _._-·----·-·-·----·-e -
--- CDIII----I··IIIU----r-----
- 32 When the functions h i(x) are both differentiable and concave, the
(4.1) and (4.2) coincide.
finite supports
As a preview of the results to be presented in this section,
2
1
2
< 1, x 1
+ x2
> 0,
x2
0
R2
Let C = Rn, X = D = {x
we begin with the following example.
2
= R.
and let
is
(0,1)
Then, x
admissible for the cone I over X, but not over the linear approxi(4.1) to X at x
mation
0
while'x
1
1/2) is strictly admissible
= (1//2-,
over X as well as over its linear approximation (4.1) at x .
formalize these observations.
remainder of this section we will
Our first result relates admissibility
L(x )
in
to strict
admissibility in D. We will apply some simple, but useful,
observations concerning admissible points
in cones.
Q
Lemma 4.1.
P and
Let
suppose that
(ii)
and (iii)
Proof
x
x
0
E A(x
: Conclusions
A(x 0 + Y) = ~
with part
A(x
x ° EAS(x 0 + Y)
of definitions. If
and
Y be closed convex cones in Rn and
0
+ Y)
0
+ Y)
=
g,
if and only if P+ n y*+
+ Y)
x
whenever
(i) and
x
A(x
or
(ii)
,
E
,
EA(x0 + Y)
are immediate consequences
0 A (xo + Y),
then
P
YY
=
by
the definition
(ii)
by Proposition 3.2. This observation coupled
U
(i) establishes conclusion (iii).
In the next
two propositions we assume
X = C nD
that
C= R n
in
of X.
P be a strictly supported closed con-
Proposition 4.1.
Let
vex cone and let
x E X C R.
x0
x
P is supported strictly. Then for any
(i) either
Then
is admissible in some support
In the
x E A (X) if and only if
L(x0 )
to X at x.
33
\
Proof
: Whenever
x
is
str-i-ctly
admissible,
+
problem
L(x
max
) =
If x
in
.x
S
since
for
p
<
some
·x
S
}
support
x
X C
L(x ),
that
implies
x
is
= V =
(0,1,1)
examp-les.
Then
and
x
or X even though
L(x ° ) = Y
is
the
R
2
and
support
x
is
L(x
O
3.4
show that
the
is
_
it
is
X
and
Y
X
to X at
it
is
Although
closedness
in both of
as
R
,
+
admissible
L(x°)
in this
+
let
would be
a support
is
in these
these
0
x
+Y
sets and
that whenever x
is
in every support
L(x
X be
support.
}
2
square in
) : x
1}
x
is
a
= (1,1),
the support
chosen in
of
°)
the unit
In fact,
to X
0
for which x
of
are worth noting.
{x = ((Xl,
. X <o 4+
*x 1p
• p · x
Ps E P+
L(x ).
admissi-
in either
the strictly admissible point
o r -Rn
in
at x.
this point. For example,
.
to X at x
be defined
admissible
a support to
= R
)
Proposition. Let
the proposition depends upon knowledge
O+
and x
the definition of strict
conclusion does not state
{x
at x
is not strictly admissible
not admissible
) =
set
AS (X).
let
strictly admissible
to X at
X
L(x
Certain features of Proposition 4.1
First,
The
strictly admissible
necessary in the previous lemma and
0
the
+
E P
p
supports
in some
previous lemma, x
Examples 3.3 and
x
solves
L(x ).
is admissible
then by the
bility
+
.x: x E X
Rn
{x
admissible
But
{p
it
the proof of
a strict
support
+
maximizes
Ps
x
More useful
over X.
that depends only upon local
----
·-----
--
--
----
I
information
f---
--
-----------
----- L1111--
·-
_
-
0
at x.,
and
34 -
such as the supports specified in expressions (4.1)
(4:2).
Our next results delineate a wide class of prob-
lems where such supports are possible.
Proposition 4.2.
Let
P
be a strictly supported closed con-
vex cone and let X be a polyhedron. Then every admissible
point
x
point
x E X is strictly admissible in the support
X is strictly admissible. Moreover, any admissible
defined by (4.1) with
Proof
: Let
h(x)
D= X.
a x - b.
1
1.
for i
1,2,...,m
affine functions defining X and suppose that
L(x )
that
admissible
denote linear-
x E A(X)
and
is defined by (4.1).. We first note that x
in
L(x ),
"~xO>
n(
L(x°
)
e8
0, satisfies
(x
L(x ° )
+
for otherwise some
o
). But then
ai y
a
i
b.
z
z
x
b. and y -- x
0
+
is
belongs to
0
(z-x), where
a i ·x o = b..
for all indices i with
1
Choosing
a
.x
e
small enough, a .y > b.
> b.
as well. Therefore, y E X, y
contradicting
Since x
x
i with
and
yE x
+
,
x E A(X).
is admissible in
closed convex cone, x E A
quently
for every
L(x )
(L(x )),
and
L(x ) - x
is a
by Lemma 4.1, and conse-
x E A(X).
Previously, Evans and Steuer
A(X) = AS(X)
[25] have shown that
when
, as well as X, is polyhedral.
_1_1_
·___
____
_
I___
__
___
___
-
35
-
We next consider instances when X
x0
is
X = Cn D,
admissible in
strictly
is
If
nonpolyhedral.
then
it solves
the
optimization problem
Max
p .(x
xECCnD
s
for some
p
+
+
P
S
L(x)
°
) -x
=
is
{yE Rn:
to
X >
0
objective function in
ED
0
a support at x
°)
-x
removes '
]
with
example, we
(4.1),
O0 for all
I
L(x ) - x
is a
u E]Rm : u = X-7h(x)
=
A-h(x0 )
= 0}
h(x )
i with
=
.
In this
case, the
the
linear approximation to
(4.3).
Recalling the usual
x
u E [L(x
defined by
(4.4)
Lagrangian function of
with respect
L(x ),
(4.4)
hi(xo)y
Lemma,
some vector
for this
0
Replacing D by
L(x )
and, by Farkas'
call
to the cone
terminology of nonlinear programming
x
P
a Kuhn-Tucker point
and
support
L(x )
in
C
X =
to D at x
D
,
and
Max
xEC
for
(4.3)
+ u) · (x -x ).
(p
Note that when
if
)
constraints and gives
from the
Sup
xEC
for
0
"dualizing" with respect
and
L(x
.
S
-x
(p
+ u) (x - x)
= 0
some "Kuhn-Tucker" multipliers
for
Ps E
+
5
and
0 multipliers
0"Kuhn-Tucker"
uE [L(x )
----- -----·^--- --------- ------I*---C-·---------
-
x
'-CICI*··-CTCI-
- 36 -
The following proposition characterizes
such Kuhn-Tucker
points
x0
.Proposition 4.3.
to the cone
is a Kuhn-Tucker point in X with respect
L(x °)
and support
to D at
if and only if
x
the following two conditions are satisfied:
(i) x0 is strictly admissible in
x
solves
v = max
(ii)
{p
· (x - x)
+
for some
p
for this
p
S
where
: x EC
L(x
)}
+
E P
s
and
,
:u E [L(x ° ) -x 0 ]
v = min {v(u)
v(u) = sup {(p
+ u) (x -x 0 )
: x EC}.
Proof : The validity of conditions (i) and (ii)
O - v
for
some
= max
u E [L(x ) -x
versely, if x
u E [L(x ° ) -xO]
,
+ u) (x - x)
S
so x
p
: x E C}
is a Kuhn-Tucker point. Con-
and u, then
, u-(x- x )
p+
> 0
and
(x -x
x ECn L(x ).
v < v(u) = 0. Since
v = v(u) = 0
{(p4
implies that
is a Kuhn-Tucker point with associated Kuhn-
Tucker multipliers
whenever
that is,
C nL(x°);
x
v(u) = 0.
consequently
) (x -x
These
) <
inequalities
ECn L(x ),
Since
v(u)
imply that
v > 0. Therefore,
satisfies conditions (i) and
(ii).
_Llli______L__ypyspI
_ _ __(_ __
U
- 37 -
ri(C) nL(x° ) $ ~,
([38],
tion
the duality condition
28.2.2). In particular,
Cor.
(ii)
is polyhedral, as in
L(x )
Whenever
if
or
(4.1)
(ii)
(4.2),
and
is fulfilled
C = Rn
then condi-
and we may sharpen proposition
becomes superfluous
4.1 by specifying necessary conditions for strict admissibilof easily computed supports.
ity in terms
let
P be a strictly supported convex cone,
Let
Corollary 4.1.
x ERn : h.(x) > 0, i= 1,2,...,m},
X = D =
and assume
that each constraint-function h. is differentiable at a
strictly admissible point x0 solving
max {(p
Ps
where
.
qualification4,
.x
: x
Then,
x
(4.5)
X}
if problem (4.5)
satisfies any constraint
is strictly admissible
support
in the
L(x
°)
defined by. (4.1).
Corollary 4.2.
let
X = C
D
Let
P be a strictly supported convex cone and
where C is a convex set and
D = {xe
n : hi(x)
0,
i = 1,2,...,m} is defined by concave functions. Then, if X
satisfies the Slater condition
for some
x E C, x
hi(x ) > 0
for
i
1,2,...,m
is strictly admissible in the support
defined by (4.2).
Conditions, like linear independence of the vectors Vhi(x )
for indices i with hi(x o ) = 0, that ensure that'x ° satisfies
the Kuhn-Tucker conditions of nonlinear programming for
problem (4.5).
·_·._
___ ____ ____
____
I_
- 38 -
4.2 The Vector Maximization Problem
To illustrate the previous results in-a somewhat more
concrete setting, let us consider the vector optimization
problem introduced in Section 2 with a criterion-function
f(z)
=
(fl(z),
= {(y,z)
Let X
is defined by Z
2,...,m}.
let h(z)
hn(z))
Let
denote
... ,fk(z))
and
: z E Z and
E R
e = {(y,z)
the
: y
n
z E
=
,
R
: hi(z)
z E C}.
subvector of h(z)
=
s
{((,y)
k
E
x
If each of the functions
entiable,
ing to
to
P
Note,
=
: X > 0 and
n-k
f.(z)
and
As we
in this
0}.
is differ-
h.(z)
correspond-
then the linear approximation to X at x
(4.1) becomes
y < vn
f(z)
L(xO) -x 0 = (yz)E
Since
...,
= (f(z), z0 ) is admissible in X
with respect to the preference cone ! = Ike
that
= 0.
D,
is efficient in the vector maximization
problem if and only if x
case,
h 2 (z),
(hl(z),
Z
i = 1,
0,
>
For any z
restricted to those components with hi(z)
noted in Section 2, z0
that
and suppose
y < f(z)}
C n D where D
n-k
alternatives Z c g
a set of
ny admissible point
must
admissible
satisfy
in
L(x 0 )
y
(y
= Vf(z)
,z )
z
and Vh.(z0 )
z
e
L(x )
in
x
=
(y
z
0.
with respect
,z )
is
strictly
e whenever it solves
v = max{ X Vf(z° ) · (z-z
0)
: zEC and V h(z).(z-z 0 ) > 0}
---
----
·--
-----
· 1-
(4.6)
I---L-·-----·l---i-C11I-
- 39 -
for some positive k-vector X.
As we have noted previously, Farkas' Lemma implies
the polar to the polyhedral cone
[L(x)-x]
=
(u,u E
:u
L(x ) - x
=
is given by
X and u
for some
Therefore,
0o
(y 0 ,z ) is a Kuhn-Tucker point
that
AVf(z
+jVh
O and
X
(z
Q0}.
if it solves
max {[XVf(z 0°) +~ Vhz)] · (z- z) + (a
yERk
X)
(y-y
)}
zE C
for ,sofe positive k-vector a. The value ofothis optimization
problem is +
point, or z
unless
a =X. Thus,
(f(z ),)z 0 )
is a Kuhn-Tucker
is a Kuhn-Tucker point in the vector maximization
problem, whenever
max {[XVf(z ) +
zEC
Vh(z)]-(z -z)} = 0
(4.7)
for some positive k-vector X.
Note that for
z
to be a Kuhn-Tucker point to
the vector
maximization problem requires a choice of positive weights X
for the vector criterion function
f(z)
so that
z
is a Kuhn-
Tucker point to the nonlinear program
max {(Xf(z) : z'E C
Proposition 4.3 shows
and
h(z) > 0}.
(4.8)
that necessary and sufficient conditions
____
______CI____IC_____jr_
I-LIII·-r-·ry--lrru--·IL-Ur·-ULIUWUI
o
z
be a Kuhn-Tucker
to
first
order
(ii)
the
linear
within the
objective
weights
so that
unaltered.
That
(i)
are
of
constraints
as well
a duality
of
the
solves
z,
and
choice of
the problem
are
the
incorporated
can be
remains
associated with
that
guaranteeing
certain solutions
o
(4.8) at
to
to
Kuhn-Tucker points
conditixon
(ii)
z
(i)
function by an appropriate
imation inherits
as
that
(4.6)
the optimal value
is,
a regularity
point
approximation (4.6)
inequality
-CUlill-P·IL-I--'-·L
40 -
-
for
C--i-
approx-
a linear
from a nonlinear program,
condition guaranteeing dualization
linear approximation problem.
o
When
the vector maximization problem becomes
k= 1
to value
normalized
ditions.
for
the
the
with
positive
(see,
seem
[43]
McCormick
the duality
dition
first stated
related
of
results
this
required.
not
[44]
Magnanti
the context
in
presents
to have
condition is
unpublished report,
42]).
example, Mangasarian
the
[46]
and
C= Rn
In a section of an
introduced
context
Fiacco
fact when
the
nonlinear programming.
in
of the
of nonlinear
conditions
qualification
for
con-
duality conditions
regularity and
characterizing Kuhn-Tucker points subsume all
programming
scalar X
the usual Kuhn-Tucker
to
1, reduces
Consequently,
numerous constraint
and
(4.8),
condition
linear program and
a non-
duality
Halkin
con-
[45]
of nonlinear program-
derived
first order necessary
ming. More
recently, Robinson
conditions
for a general vector optimization program in infinite
dimensions by
studying
multivalued maps.
preference
orderings
in the
image of
certain
-
41
-
When specialized to the vector maximization problem,
Corollary 4.1
be strictly admissible
x
that
=
z
(i.e.,
solves problem (4.8) or,
is strictly admissible in X)
(f(z ),z )
is admissible
(f(z),z )
dition is equivalent to z
0
Vf(z ).
L(x ).
in
being efficient
ZLL M {z E ]n-k
-k: Vh (z o )-(z-z o ) > O}
vector criterion
to
then a necessary condition for z
dition is fulfilled,
equivalently
and the regularity con-
C =
that if
shows
is
The last con-
in
to the
with respect
z. Therefore efficiency
in the
linear approximation to the vector optimization problem is
necessary for strict admissibility in the problem itself.
Remark 4.1.
These results are related to the notion of
proper efficiency introduced by Geoffrion [12]
Kuhn and Tucker
[10]).
0
By definition, z
(see also
a
is a proper effi-
cient point in the vector maximization problem if there is a
M > 0 with the property that for every
scalar
fi(z)
index i satisfying
fi(z)
fj(z )
-
the inequality
> f.(zo)
1
z E Z and each
1
f (z)
-
f(z)
is valid for some index j such that
f.(z) < f(z°).
As
Geoffrion shows, whenever Z is a convex set and each funcf(z)
tion
of is
of z
for
j =1,2,...,k
equivalent
to
z
is equivalent to z
is concave, proper efficiency
soling
problem
(4.8
solving problem (4.8)
for
some
for some
___
XO.
X>0.
___
__
- 42 -
As we
this
last
is equivalent
condition
) ,z ) being strictly admissible in X. If
(f(z
then,
if
noted,
have
our comment made
restate
z
is
proble m
and
z
EZ
the
, we
is
fulfilled,
efficient point in
Z
L
then
with
(4.9)
o
the
to
respect
also note
We might
condition
(4.9)
crite rion
vector
that
f(.)
if
that
implies
Vf(z
and h(.)
in the vector maximization problem. To
lem
fies
(4.6)
the
for
f our
because
conditions
then
concave,
this
establish
solves
z
. Therefore
(4.7)
hypothesis,
concavity
are
that
k-vector
some positive
Kuhn-Tucker
implies
(4.9)
condition
that
we note
)-z.
a proper efficient point
is
z
may,
vector maximization
the
regularity condition
is a proper
n
C= R
to this remark as :
just prior
strictly admissible in
to
with
z
CER
solves
fact,
prob-
0
z
satis-
and,
the
Lagrangean
maximization
max
zE
Since,
by
{Xf (z) +
definition, iih*(z
problem equals
Xf(z ),
optimization problem
proper
h (z)}.
n -k
efficient
SHere we use the
programming.
point
and
) = 0 · the
since
z
E Z,
max {Xf(z) : h(z) >
in
the
to
this
solve
the
optimal value
it must
0}
and hence
be a
vec tor maximization problem.
standard weak duality argument
of
nonlinear
- 43
We should point-out,
point
let
z
terion
h
(z)
in
stance,
)
2
O
2 -
and
the regularity
not admissible
that a proper efficient
condition (4.9).
As
an example,
the vector maximization problem with cri-
f(z) =
=
though,
need not satisfy
= (0,0)
z
-
in
=
2
and
2
h 2 (z)
constraints
2-
z
condition fails
ZL =
2
(
0
In
since
: Z2
this
in-
the origin
is
0}
O
In a recent paper Borwein
topological
spaces,
[47]
several results
local characterizations
presents,
for
intimately
of efficient points.
more general
related to our
He defines a point
0
z
E Z to be'proper efficient with respect
cone S if
z
({y
f(z),
: y =
is efficient and the tangent cone
z
E
Z
-
S)
In
implies this
notion of proper efficiency
the absence of convexity,
while under convexity assumptions
([47],
convexity,
theorem 2).
This
equating L(x ) - x
to the set
= f(z ) intersects
at y
the origin.
valent
to a closed convex
strict efficiency
([47],
theorem 1)
these definitions
last result shows
with the
S only at
are equi-
that, with
tangent cone fulfills
the support condition guaranteed by proposition 4.1.
5.
DISCUSSION
In the previous sections we have studied
perties of admissible points with respect
to
structural proa convex cone.
Our results provide global characterizations of
admissible
points
- 44 -
in terms of strictly admissible points and local characterizations in terms of linear approximations.
We have also shown,
with appropriate hypotheses imposed upon the problem structure,
that the sets of admissible and strictly admissible points are
both connected.
In this section, we briefly discuss a few
potential extensions and applications.
First, we might comment.on the frequently evoked assumption that the underlying preference cone is strictly supported.
According
to Proposition A.1, this assumption rules out "grass
is greener" preferences in which each of two alternatives is
preferred to
More generally, it does not permit
the other.
and y>
situations in which xy
converging to y.
points y
x for j = 1,2,...
for some
As an example, lexicographic
orderings define preference cones that are not strictly
supported.
Whenever underlying preferences are described by a closed
convex indifference cone
y>
x),
the set
=
2
belongs
I
then
(i.e., y
p
E
if and only if
This cone is strictly supported for if
the lineality space of cl
0 = p -
x +
: 0> x} describes a strictly sup-
x
ported preference cone.
p E
l
p + P
or
0OX p,
E cl
, i.e., -p E cl
a contradiction.
We
should emphasize, however, that even though this construction
provides strictly supported cones, our development does not
presume the existence of any "weak" preference relation>..
-
45 -
There are several ways in which our results might be
extended.
Replacing
the preference cone P by a convex set C
or, more generally, by a family of convex sets Cx, C x denoting the set of points preferred to x, would add possibilities
for broader applications.
Another line of investigation would
be to retain our hypothesis and to see what additional assumptions might lead to stronger conclusions.
and Hahn
4]
show that if I = {x
For example, Arrow
R: x = 0 or all xi > O}
then the following restrictions on the
(convex) set of alterna-
tives X
(i)
(ii)
(iii)
0 belongs
X nR
to the interior of X;
is compact;
free disposal, i.e., x-y
whenever x
imply that A(X) n
simplex.
and
X for any y E
n
E X
n is homeomorphic to an (n-l)-dimensional
This substantial strengthening of connectedness
is
possible, with similar hypothesis, for other convex cones as
well.
In general, it may be that the set A(X) is homeomorphic
to a union of simplices with some special structure.
In a paper not yet available to us, Naccache [48]
initiated investigations
of another nature.
has
He studies sta-
bility of the set of admissible points with respect to perturbations in X and
of A(X),
.
He has also studied connectedness
but with assumptions that may be related to free
disposal.
___
_
__
__
_
_
I_
_I
_____
- 46 -
There are a number of ways in which the structural properties discussed in this paper and these extensions aid deConsider, for example, the vector optimization
cision making.
In practice, it is convenient to generate strictly
problem.
admissible points by solving
k
max { Z
J.l
Xfj(z) : z
E
(5.1)
Z
(5.1)
with positive weights Xj associated with the criterion funcBy varying the weights, decision makers can generate
tions.
all strictly admissible points, or by choosing a sequence of
positive weights appropriately they might move toward some
admissible point that is "best" with respect to some auxiliary
criterion (see, for example,
Considering, as before,
[27]).
the vector maximization poblem in terms of the set
X = {(y,z)
=
k'
E
n
: y
we see from the
admissible point
(y
(y
and z
Z} and
the preference
representation theorem
= f(z ),
that
cone
every
z ) is a limit point of s8lutions
= f(z ), z ) to (5.1) or a translation of such limit
points by vectors
k
R
f(z)
the image f(.)
(O,z)
cl
k\
k.
That is, "in the space
of efficient points is contained in the clo-
sure of the image of the proper efficient points"
(see Geoffrion
6
[12])
.
In this context,
that the solutions to
the representation theorem 3.1 shows
(5.1) delineate all potential values
6
n RC(X) = {O} of
Some assumption such as the hypothesis cl
For example, let
Theorem 3.1 is required for this statement.
V and Y be defined as in example 3.4, let Z = V + Y and let
fl(z) =
1 and f 2 (z) = z 2 .
The efficient set is the halfline
Every
from V passing through the point K (see Figure 3.1).
efficient point is non-proper, though, so that the statement
is not valid in this instance.
-
47
-
of the criterion function when evaluated at efficient points;
connectedness
of the admissible points shows that to move from
any (proper) efficient point to another, we can restrict ourselves to local movements among (proper) efficient points only,
such as local changes in the coefficients Xj of (5.1).
We would expect similar benefits from the structural
properties of admissible points
in general, especially when
solving for strictly admissible points is attractive computationally.
One applications of admissibility that might be explored
profitably concerns the optimization of monotonic
functions
h(x), where, say, h(x) is strictly increasing in each component x. of the vector x.
In this instance, any optimal solu-
tion to the problem
max {h(x) : x
X}
n
E IR+
is admissible for X with respect to
This observation
suggests that optimization algorithms might restrict their
search to the admissible points A(X),
especially once any
algorithm first identifies a point in this set.
Do mathematical programming algorithms have this property?
The answer, at least in terms
of the simplex method is no.
the example,
max
subject to
2xI +
x2
4x 1 + 7x 2 > 18
x1 +
x1
2x
2
0O,x
2
<
5
0
__
__
_
_
_II
_I__
In
- 48 -
the
admissible
a =
(1;2)
and
the
the
segment joining
solution
c =
b =
same phenomenon. In
simplex method
iteration.
enhanced
Starting
[361,
of
the admis-
once the
it
an admissible point,
admissible point at
Our understanding
from
we have never ob-
these examples,
first-encountered
always generated an
(5,0).
points
In a number of experi-
(4.5,0).
conducted on larger problems
served this
the
the simplex method moves off
a,
to the point
sible set
line
optimal
the extreme point
ments
is
set
the
each successive
simplex method might be
if we would explain this behavior.
0
6.
APPENDIX
A
The following,
ize the
Recall
of
rather intuitive,
propositions character-
strictly supported condition for a convex cone.
that
lineality space L of any cone C
the
lines contained
Proposition A.1.
in C,
Let
L
i.e., L = C
be
closure of the convex cone
the
is
the
(-C).
lineality space for the
is
. Then-
supported strictly
if and only if
p nL =
{0}.
Proof
: Let
denote
the orthogonally complementary
to
Then
L.
L
cl
P
=
L G
(cl P r L i
set
)
is a direct
-- -
-I----1---
subspace
sum representa-
-
------
CII---·L
- 49 -
tion, and
Now
p
for all
= L1 n (cl PF
(cl P)
I+
E L
s
whenever
p
S
L)+
L
E P
$
p nL. Consequently,
p
.
+
+
C
(cl P)
-
+
and
p S *p =
P nL = (01
To establish the converse, note that since
is convex, closed and pointed,
has full dimension
([38],
to the interior of
nonzero
L
and
all nonzero
P =
and
p
y ·p > 0
must satisfy
for
some
y
p
for all
y = YL
L ).
If
PLE L
and
> 0.
(YL + y
that
pEP
of
and F
P n L =
that is,
F
is
is a strict
I
y
L =
p >
{0},
E (cl P
p
{0}
with
implies
0
for
then
n
),
that
supported strictly.
any point contained in both
L +)
(cl P
+ yI
=
some nonzero
Therefore
p E P;
)p
The proof of this proposition shows
FPL = {01,
y p > 0 for all
I
we note
,
p E(cl
PL + P
y
(cl F n L )
.L
E L
y
Cor.14.6.1). Any point y belonging
nL ). Expressing y as
I
YL E
(cl , n L )
its positive polar
i.
p E (cl
cl P n L
support to
that whenever
L
and the interior
P. We next establish
the converse to this statement.
Proposition A.2.
Let L be the lineality space for the
closure of the convex cone
P. Then
ri(P
+)
C P+
if and only
-5
if
Ln
p 4 C ri(P
SL
L
{0}.
=
)
Moreover,
and,
if
cl Pn
consequently,
ri (
L
= P n L
) =P
,
then
if and only if
=
F =
(0}.
---·--------- ^-
-· -------.-
____
__
-
Proof
50 -
cl P = L 0 (cl F n L )
: Let
+
tation. Then
(cl P)+
=
intersects
L
ri (
(cl
sum represenJ.
and, since L
L )
has a nonempty interior I which
L
)
+
= L
+
is a subspace and (cl P nL)
be a direct
I.
L = {(0
By the remark preceding the proposition,
implies
that
F
P+
and
S
ri(P
LP
+
) C
+.
Conversely,
P
to the relative boundary of
EL
pi EPnwhich we scale
satisfies
{pJ }j>I
+
) C P+
then
L
= P
and let y belong
. Then there are vectors
converging to y satisfying
sequence
ri(F
= {0} bythe previous proposition.
Finally, suppose that. cl PFL
yi
if
pi
yj
< 0
for some
to unit norm. Any limit point p of the
belongs
to
cl
y- p < 0. Consequently, y
L=
n L
+
F + and
S
c F
P
+
S-
and
ri(++
C ri(P ).
When combined, these propositions establish Proposition 2.1
of the text. Note
Proposition 2.1
shows
that
that the illustration following
ri(PF)
P
s
is possible.
--- \II~~-----
--------
·--
I'II-~-"
-
7.
APPENDIX
51
-
B
We prove
for
lemma required
the continuity
Theorem 3.3,
namely
Lemma
set
the solution
to
v(c)
is
to set
: Let
y <
any
y be
0.
X(6)
e -
mapping
all
Consequently,
+
(I -)p(O)]
on the
xEX}
= 1.
Then
Since X(O)
p(O)
y <
0
is
and
the
is
the case
< e < 6}
X(e)
[,
proof
is nonempty and
if,
at
1]
-
for
=
For
compact.
is upper
0 < e
I1.
show that
this
0.
some
= 0,
X(e)
for any
we must
at
is bounded.
semi-continuous
0
the mapping
interval
to complete
indeed
U {X() :0
a
:
and the point
of X.
compact,
y <
mapping is upper semi-continuous
,
x
O <8 < 1
all
recession direction
implying that
semi-continuous
is not upper
]
Therefore
0 <8e
This
)p(O)
upper semi-continuous.
is
According to Lemma 3.2,
S6
(1-
is nonempty and
[ep(I)
for
that
Assume
.
and nonempty and compact for
= 0
nonempty and X(1)
p(l)
+
nonempty and compact for
X(e) is
R
the problem
sup { [ep()
=
nonempty for
Proof
closed convex set in
Let X be a
3.3.
if
6 >0, the
set
the mapping
then there
-
X(e)
is an open
1. Ir
I·--Lr-r-n·aLI
--·--ih-
- 52 -
set G
containing X(O)
e8 >0
numbers
approaching
p(e j ) ·x
any
and points
limit point
x
0.
> p(e)
(such a
x
E X(6j)\ G for
some real
Since
-x
for all xE X
limit point
eventually
lie
in the bounded set
satisfies
x
G
S
)
exists
to
the
since
the x j
sequence
{xi}j>
and
1
I
ao
x
p(0)
But
then
x
>
E X(O)\ G,
al xEX.
for
p(0).x
contradicting
Therefore to establish
the
X(O) C G.
theorem we only need
0
to
show
c
that
S
i-s bounded
for
some
suppose, by translation
6 >0. For fiotational
if necessary, that
simplicity
O E X(O). Then by
definition
p(O)
for
any
x(e)
x(9) E X(e),
[ep(I)
implying,
Now
(1),
(I
X()
are
-
if
,0
S
0 E X,
x(3) >
>- 0.
is unbounded
Since
Since
(1)
that
and points x(0)
approach + A.
{x(eJ)
0 < e < I.
+ (1 -e)p(O)]
from
p
p(O). x(O) = 0
(2)
for every
E X(
0 <6 < 1,
then
there
) whose Euclidean norms
0 EX, any limit
point y
to
X.
the sequence
/ Xj}j>l is a direction of recession of X. But the
inequality
(2)
impli-es
that
p(l).y > 0,
contradicting
the
-
hypothesis
that
bounded for some
X(1)
6> 0
53 -
is bounded. Consequently, S
and the proof is complete.
must be
U
Acknowledgements
We are indebted to Arjang Assad for several helpful discussions and to Andrzej Wieczorek for his careful reading of
an earlier version of
this paper.
_.~~~~~~~~~~~~~~~Il
-
54 -
REFERENCES
1.
Debreu, G., "Representation of a Preference Ordering by
a Numerical Function", Decision Processes,
R.M. Thrall, C.H. Coombs and R.L. Davis (Eds.),
Wiley, New York, pp. 159-166, 1954.
2.
Debreu, G., Theory of Value, John Wiley and
York, 1959.
3.
Bowen, R., "A New Proof of a Theorem in Utility Theory",
International Economic Review, Vol.9, p. 374, 1968.
4.
Arrow, K.J. and Hahn, F.H., General Competitive Equilibrium, Holden-Day, San Francisco, 1971.
5.
Keeney, R. and Raiffa, H., Decisions with Multiple Objectives : Preferences and Value Trade-offs, Wiley,
New York, 1976.
6.
MacCrimmon, K.R., "An Overview of Multiple Objective
Decision Making", pp. 18-44 in reference 37.
7.
Wald, A., Statistical Decision Functions, Wiley, New York,
1950.
8.
Arrow, K.J., Barankin, E.W., and Blackwell, D., "Admissible Points of Convex Sets", Contributions to the
Theory of Games, Vol. II, H.W. Kuhn and A.W. Tucker
(Eds.), Annals of Mathematics Studies, No. 28,
Princeton University Press, Princeton, New Jersey,
1953.
9.
Gale, D., Kuhn, H.W., and Tucker, A.W., "Linear Programming and the Theory of Games", Activity Analysis
of Production and Allocation, T.C. Koopmans (Ed.),
Wiley, New York, 1951.
Sons, New
10.
Kuhn, H.W. and Tucker, A.W., "Nonlinear Programming",
Proceedings of the Second Berkeley Symposium on
Mathematical Statistics and Probability, University
of California Press, Berkeley, California,
pp. 481-492, 1950.
11.
Koopmans, T.C., Three Essays on the State of Economic
Science, McGraw-Hill, New York, 1957.
- 55 -
12.
Geoffrion, A.M., "Proper Efficiency and the Theory of
Vector Maximization", Journal of Mathematical
Analysis and Applications, Vol. 22, pp. 618-630,
1968.
13.
Geoffrion, A.M., "Strictly Concave Parametric Programming,
Parts I and II", Management Science, Vol. 13,
No. 3 and 5, pp. 244-253 and 359-370, 1967.
14.
Geoffrion, A.M., "Solving Bi-Criterion Mathematical
Programs", Operations Research, Vol. 15, pp.
1967.
t5s.
Yu, P.L.,
"Cone Convexity,
Cone Extreme Points,
39-54,
and Non-
dominated Solutions in Decision Problems with
Multiobjectives", Journal of Optimization Theory
and Applications, Vol. 14, pp. 319-377, 1974.
16.
Smale,
S., "Global Analysis and Economics III : Pareto
Optima and Price Equilibria", Journal o f Mathematical Economics, Vol. 1, pp. 107-117, 1974.
17.
Smale,
S., "Global Analysis and Economics V : Pareto
Theory with Constraints"
Journa
of Mathematical
Economics, Vol. 1, pp. 2 13-221, 1974.
18.
Smale,
S., "Global Analysis and Economics VII : Geometric
Analysis of Pareto Optima and Price Equilibria
Under Classical Hypothesis", Journal of Mathematical Economics, Vol. 3, pp. 1-14, 1976.
a
19.
Rand,
D., "Thesholds in Pareto
matical Economics, Vol.
20.
Simon,
21.
Wan, Y.H.,
Sets", Journal of Mathe3, pp. 139-154, 1976.
C.P. and Titus, C., "Characterization of Optima
in Smooth Pareto Economic Systems", Journal of
Mathematical Economics, Vol. 2, pp. 297-330, 1975.
"On Local Pareto Optimum", Journal of Mathe-
maticaZ Economics, Vol. 2, pp. 35-42,
1975.
22.
Charnes, A. and Cooper, W.W., Management Models and
Industrial Applications of Linear Programming,
Vol. 1, Ch. 9, Wiley, 1961.
23.
Ecker,
a
J.G. and Kouada, I.A., "Finding Efficient Points
for Linear Multiple Objective Programs", Mathematical Programming, Vol. 8, pp. 375-377, 1975.
-'-11F-
- 56
-
J.G. and Kouada, I.A., "Generating Maximal Efficient Faces for Multiple Objective Linear
Programs", Discussion Paper 7617, CORE, UniversitE
Catholique de Louvain, Heverlee, Belgium, July
1976.
24.
Ecker,
25.
Evans, J.P. and Steuer, R.E., "A Revised Simplex Method
for Linear Multiple Objective Programs",
Mathematical Programming, Vol. 5, pp. 54-72, 1973.
26.
Gal, T., "A General Method for Determining the Set of All
Efficient Solutions to a Linear Vector Maximum
Problem", Report 76/12, Institut fr Wirtschaftswissenschaften, Aachen, Germany, September 1976.
27.
Geoffrion, A.M., Dyer, J.S., and Feinberg, A., "An
Interactive Approach for Multi-Criterion Optimization with an Application to the Operation of an
Academic Department", Management Science, Vol. 19,
pp. 357-368, 1972.
28.
Philip,
29.
Schachtman, R., "Generation of the Admissible Boundary
of a Convex Polytope", Operations Research,
Vol. 22, pp. 151-159, 1974.
30.
Yu, P.L. and Zeleny, M., "The Set of All Nondominated
Solutions in Linear Cases and a Multicriteria
Simplex Method", Journal of Mathematical Analysis
and Applications, Vol. 49, pp. 430-468, 1975.
31.
Ferguson, T.S., Mathematical
New York, 1967.
32.
Whittle, P., Optimization Under Constraints, Ch.
John Wiley and Sons, New York, 1971.
33.
Markowitz, H., "The Optimization of Quadratic Functions
Subject to Linear Constraints", Naval Research
Logistics Quarterly, Vol. 3, No. 1 and 2,
pp. 111-133, 1956.
34.
Markowitz, H., Portfolio Selection : Efficient
fication of Investments, John Wiley and
New York, 1962.
J., "Algorithms for the Vector Maximization
Problem", Mathematical Programming, Vol. 2,
pp. 207-229, 1972.
Statistics, Academic Press,
10,
DiversiSons,
-
57
-
35.
Raiffa, H., Decision Analysis : Introductory Lectures on
Choice Under Uncertainty,'Addison-Wesley, Reading,
Mass., 1970.
36.
Bitran,
37.
Cochrane, J.L. and Zeleny, M. (Eds.), Multiple Criteria
Decision Making, University of South Carolina
Press, Columbus, South Carolina, 1973.
-38.
G.R., "Admissible Points and Vector Optimization :
A Unified Approach", Operations Research Center,
MIT, Ph.D. Thesis, February 1975.
Rockafellar, R.T., Convex Analysis, Princeton University
Press, Princeton,oNew Jersey, 1970.
39.
Stoer, J. and Witzgall, C.,
in Finite Dimensions
1970.
40.
Berge, C., Topological Spaces, MacMillan,
41.
Hildenbrand, W. and Kirman, A.P., Introduction to Equilibrium Analysis, North-Holland, Amsterdam, 1976.
42.
Mangasarian, O.L., Nonlinear. Programming, McGraw-Hill,
New York, 1969.
43.
Fiacco, A.V. and McCormick, G.P., Nonlinear Programming
Sequential Unconstrained Minimization Techniques,
Wiley, New York, 1968.
44.
Magnanti, T.L., "A Linear Approximation Approach to
Duality in Nonlinear Programming", Working Paper
OR016-73, Operations Research Center, MIT, 1973.
45.
Halkin,
46.
Robinson, S.M., "First Order Conditions for General
Nonlinear Optimization", SIAM Journal on Applied
Mathematics, Vol. 30, pp. 597-607, 1976.
Convexity and Optimization
I, Springer-Verlag, Berlin,
New York,
1963.
H., "Nonlinear Nonconvex Programming in an
Infinite Dimensional Space", Mathematical Theory
of Control, A.V. Balakrishnan and L.W. Neustadt
(Eds.), Academic Press, New York, 1967.
- 58 -
47.
Borwein, J., "Proper Efficient Points for Maximizations
with Respect to Cones", SIAM Journal on Control and
Optimization, Vol. 15, pp. 57-63, 1977.
48.
Naccache, P., "Stability in Multicriteria Optimization",
Department of Mathematics, University of California
at Berkeley, Ph.D. Thesis Abstract, Berkeley,
California, January 1977.
·
r