* This researah was supported by National. Saienae Foundation Gr'ant Nwribe'l'
GU-2059.
,
GENERATION OF THE ADMISSIBLE BoUNDARY
OF A CoNVEX POLYTOPE
by
Richard Shachtman
Department of Statistias
University of North Carolina at Chapel. Bin
Institute of Statistics Mimeo Series No. 743
Feb~y
24, 1971
•
•
GENERATION OF THE ADMISSIBLE EoUNDARY OF A CoNvEx POLYTOPE*
Richard Shachtman
University of NorthCaro Una at Chap~Z_lIfZZ
Cnapet HtZZ~ No~th CaroZina 27514
ABSTRACT
The purpose of this paper is to develop a constructive algorithm for generation of the admissible boundary of a convex polytope, when the polytope is the
convex hull of a given (finite) set of points as well as to mention some related
results.
One application is to the sensitivity of decisions to changes in the
prior distribution for problems in statistical decision theory.
1.
INTRODUCTION
Problems in linear optimization are often reduced to examining the boundary,
or part of the boundary, of a convex polyhedral set,
P. In particular, prob-
lems in the related areas of linear programming, game theory, and statistical
decision theory require this sort of search.
half-spaces) defining
In some cases, the equations (or
P may be given and the objective is to determine one or
more "optimal" vertices (extreme points) of
A may be given and the convex hull of
A,
P. In other models, a set of points
C(A),
is of interest.
compact and has a finite number of extreme points, then
C(A)
If
C(A)
is a bounded (con-
vex) polyhedral set, or, as we will refer to such a set here, a polytope;
GrUnbaum [5, chp. 3].
is
see
The results here are directed predominantly towards this
second class of problems.
The main purpose of this paper is to develop a constructive algorithm for
generation of the so-called admissible boundary,
A(A),
of a polytope
* This l'esea1"ah was Suppol'ted by NationaZ Scnenae Foundation Gl'ant Numbel'
GU-2059.
2
P = C(A), as well as to mention some related results.
immediate appli-
An
cation follows if one is interested in maximizing a non-negative linear
func~,
-
tic:>nal f
on
P,
since it then suffices to considerA,(A);
Of course, if
f
is fixed, one searches for the point
is a maximum.
However, in many contexts
f
a
€
see Section 2.
A for which
f(a)
is an approximation to a "true"
linear functional, or is formulated from subjective estimates.
This is the case,
for "example, in applications of statistical decision theory, or to be more precise, when one must assess a prior distribution for certain unknown states of
1
nature in a decision analysis ; see Section 5.
Thus, constructing the admissible boundary should be useful in analyzing
problems involving non-negative linear functionals.
Here, these models are re-
duced to linear programs and the efficiency and flexibility of the simplex algorithm and of existing post-optimality techniques is brought to bear.
It is
thought that the admissible boundary approach should be useful in other contexts
where objective functions, or transformations of them, have non-negative coefficients or, in general, in studying the structure of convex polytopes; for example, an application may be made to integer programming problems such as those
discussed in Elmaghraby [3].
Background and notation is covered in Section 2.
In Sections 3 and 4, an
algorithm is indicated which is a constructive method for generating a certain
convex set containing A,
called the Bayes hull of
missible boundary as the polytope
from that of the Bayes hull of
working directly with
C(A).
A.
C(A).
A,
which has the same ad-
The description of
A(A)
is obtained
This turns out to be more efficient than
Horeover, for an "optimal" point, or set of points,
one may identify sets of adjacent extreme points.
The application to decision
analysis is described in Section 5, as well as an interpretation for the
1
Baakground and further 1'efe1'enaes for deaision anaZysis may be found in
H()U)ard [8" 9., 10J and Raiffa [lJJ.
3
associated linear program and its dual.
Results similar to those in Section 2
are known (e.g., see Blackwell and Girshick [2]); brief new proofs are supplied
to make the presentation-self-contained and easily accessible.
The algorithm is being programmed; initial efficiences. are indicated, but
extended comments on computational experience must be reported later.
2.
BACKGROlJ\ID AND NOTATION
Definition 2.1. We will denote the convex hull of a set
by
Suppose
(vl" ••• ,v,)',
J
is maximized over
= l, ... ,m.
j
mJ
Oefinition 2.2.
A linear functional
m
IR •
for all
x ~ 0,
linear.
Equivalently,
= I~aixi
and
X
€
f
ai ~ 0
i = l, ••• ,m
{x
defined
is non-negative if
f ~ 0
is non-negative on
for all
]Rm: x ~ O}.
€
f:IRm +IR
We will write
and all functionals will be
lR
m
m
Then
IR •
€
x
x
i
Definition 2.4. A point y
A(s)
= A(C(S»
b
>
For
'
O
x
is pure admissible.
0
= a'x
if
xi
~
A c: IR m,
Yi'
x
€
A
If
A is convex,
e: C(S)
and
Denote
A by M(A).
is said to be on the admissible boundary of
if it is contained in a supporting hyperplane of
such that
y
is mixed admissible, or, simply, admissible.
the set of admissible points of
a'x=b},
f(x)
A if it is not dominated by any other point in A.
A is finite, we say that such an
we say that such an
if and only i f
dominates
and the inequality is strict for some
is an admissible point of
f(x) ~ 0
= l, •.. ,m.
i
Definition 2.3. Let x and y
write
A linear functional
S if and only if it is maximized over the polytope
is contained in the upper orthant of JRm,
C(S)
is a
Moreover, we may and do assume, without 10s8 of generality, that .5
C(S).
If
x' •
is a set of points in m-space; each
(column) vector
on JRm
m
x e: :JR
by
and the transpose of
C(A)
in real m-space,
A
a
>
0
(i.e.,
a
i
>
0
for the admissible boundary of
C(S),
for
C(S).
i
H = {X€nf:
= l, ••• ,m).
We
This simplified
4
structure will suffice for our use.
For a slightly different setting and
alternative proof of the following, see Blackwell and Girshick [2].
-Corollary 2.5.-
Proof. y
a'y
= band
a'y
c
Z
a'x
~
M(S).
= b,
M(s).
H = {x: a'x=b} such that a
A(S) => there exists
€
such that
A(S)
=
A(S)
~
b
for all
x
Y and
zi > Yi
for'some
y
Conversely,
there exists
~
€
C(S).
If
i.
C(S)
r
R ~.
with
cp,
a'z
>
a'z
H with
b.
each y
Zrt > Yr'
z
r
> 0,
Let
say
z = latz(t),
~ y
= lat(zlt, .•• ,zme)'
y 4 M(s).
€
a
~
there exists
and
= (Yl""'Ym)'
y
0 < at < 1;
where
a. = 0
for
1
~
then
z
€
i
C(S)
€
Hence
a > 0, b > 0
Write
by
a'y.
>
Z
C(S)
where the inequality is strict for
r,
and
= y
r
r'
2
and b = a'y = Yr' This generates (at most m) hyperplanes H = {x: a'x=b},
r
r € R, and points z(t) = (zlt"",zme)' € C(S) with a'z(t) > Yr 2 or
~pr
define
M(S),
Then b
A(S) => for all
Z E
~
Y
b > 0,
0,
>
a
and
r
E
R.
Hence
0
Theorem 2.6. If f
~
:m. m, then max{f(x): xe:A(S)} = max{ f(x): XEC(S)}.
on
0
Proof. Let d and c be the maximum values f attains on A(S) and C(S),
respectively.
c = f(y)
(The values are attained since
for some
y
C(S).
€
Define
supporting hyperplane of
C(S).
necessarily equal to
with
f(w)
is compact.)
H = {x: f(x)=c} and note that it is a
H contains a point w (not
We will show that
w
€
Hence,
A(S).
Therefore since, obviously,
d
~
c,
= c = d.
If
such that
fine
y)
C(S)
f(x) = a'x
ai
=0
and
for
i
w = (wl" •• ,wm)'
(1)
we are finished.
>
€
K and
For
i4K,
a
i
>
Otherwise, let
K
c
{l, ••• ,m}
=H n
C(S).
set the linear program wi
= max
0
for
i
~
K.
Let
F
De-
by the following:
For each successive index
subject to
(2)
0,
d
xe:F
wi
i€~,
and w = max x '
j
j
= Yi
where
j
< 1.
y = (YI""'Ym)'.
xi'
5
Suppose there exists
Z E C(S)
such that
z
~
wand
Zj > w
j
f(z) = li4K aiz i ~ li4K aiw i = f(w) = c ~ f(z) => zEH => zeF
w =mttx x
- sequently,
j
~
j
zr
Contradiction.
3.
Definition 3.1.
= 1,
i
00.
Xi
= {XEIR m:
and
j
E K.
w :is admissIble.
Renee
j.
Con-
0--
HULL
THE BAYES
xi=O}
for some
is the
i
th
coordinate hyperplane,
,m.
We use the following matrix for the points in
v
(i,j) vI
v
where we have retained only those points in
we make the translation insuring that
S c
S:
n
ron
S which are pure admissible.
m
: x~O}
{X€ IR
Also,
and that at least one
entry in each row of the matrix is zero; i.e., there is at least one point of
in each
,\.
Definition 3.2. Compute ui = max{vij : j = l, ... ,n}, i = l, ... ,m. and define
the "corner" points c = (0, ••• ,0,ui,0, •.• ,0)', i = l, ... ,m.
i
i
= l, ••• ,m}.
Assume
u
i
> 0,
for all
i.
Definition 3.3. Fix i, i = l, ••• ,m, and observe that u
i
k
= k(i),
jections of
Pi
Pr
say
v
vk
= (vlk, ••• ,vmk)'.
on each coordinate hyperplane
k
= {Prk: ~i} and
= O. P = UmI Pi'
Note that
Define
Prk
= (PI""'Pm)
with
P may include some points in
Definition 3.4. Let the polytope
(convex) hull of
5,
or
B-hull.
B(5)
Pi
= vik
for some
to be the set of (point) pro-
X , r
r
~
i.
That is
Pi
= v ik
for
5
and/or
C.
= C(SuCuPu{O})
i ~ r
and
be called the Bayes
5
6
Corollary 3.5.
C = {c }.
i
points
Proof.
y
such that
Zj
k
€
a'y = b
€
C(S)
and
a'x
b
S;
for all
C,
Zj = (O, ••• ,O,ui,O, ... ,O)',
€
Z = Prl'
j
Io.ja'vh(j)
Thus
y
€
s;
J
€
B(S)
Y
= Io. j zj'
S;
v
vk '
r :f: i,
Zj
j
B(S}
~
J
for some
€
s.
For
i,
Zj
€
and
H is such that
y
€
A(B(s»; i. e. ,
b
0,
=1
La j
[or
= c..e.
i.e.,
u = v
i
ik
P,
€
Zj
°
S;
Zj
S;
vl
and
s.
€
x
€
and
vh(j)
€
S,
and
a'x
and
C(S}.
Z.
J
€
s; b
<
Laja'vk(j)
A(B(S)} c A(S).
S;
b.
for all
A(S} c A(B(S».
a'x
But
S;
b
for
Hence, it suffices to show
SuCUP.
If
Prl] for some l = l(j)
y
C(S).
€
¢ C(S), there exists j
[or
Contradiction.
y
Zj
r
S;
In either
and l] •
v
k
€
S,
As a result,
for some
Thus
y
€
A(S);
0
whenever we wish to maximize an
f
over
B(S)
S and perform any sensitivity anal-
Moreover, we will see that this construction will, in general, reduce
computational effort in generating
section.
~
i
There-
The last, combined with Theorem 2.6, tells us that we may construct
ysis.
For
for some
Pi'
with the inequality strict for at least one component.
= a'y = Iaja'zj
SUCuP.
€
Zj
where
H exists as above.
and
B(S)
€
a'y = b
case, we see from the first part of the proof that
k = k(j),
and
= 1
hence,
l=.e.(j);
°
b >
C(S).
La.J
0,
~
0..
€
a > 0,
S; b.
implies the same for
a
k
x
ISajz j + Icajvk(j} + Lpo.jv.e.(j) = Io.jVh(j)'
A(B(s)} =-> y
x
such that
€
Z.
Consequently,
X € B(S) •
y
o s;
for some
x = lo.jZ j
S;
then .x = Io.jzj'
hence,
H = {x: a'x=b},
and there exists
x
fore,
a'x
B(s),
= A(C(S»=A(s).
If
= k(j);
and
A(13(S»
A(s) => Y
€
through the corner
O
H n B(s) = C(C).
O
Then
-Theorem ··3.6.
H
Construct the (unique) hyperplane
A(S)
by use of the algorithm in the next
In particular, surfaces of the polytope
sible are not generated; some
C(S)
which are not admis-
{vj } which are pure admissible but not (mixed)
7
admissible (including some extreme points of
C(8»
do not appear in the iter-
ations of the algorithm and the number of constructions is reduced.
HO = {x: aO'x=b O}
The following guarantees that using the hyperplane
through the corner points
C
= {ci }
for the initial construction in the al-
gorithm does not exclude any points in the admissible boundary.
Theorem 3.7.
Proof. Let y
y
~
C.
Y
A(s)
= Lajcj'
before,
As
E
~
cj
and suppose that
a.
J
vk '
~
for some
= Ct,eu,e = a,ev,ek(,e)
component
y,e
z
Contradiction.
E
C(8).
La j = 1
0,
a 'y
0
bOo
<
j,
and for some
k
= k(j),
<
LCtjV,ek(j)'
y
and
or
Then
s;
y
y
say
Iajvk(j)
e:
C(Cu{O})
j
= l, o
=z
and
<
a,e
<
1.
with the
is dominated by the point
0
Corollary 3.8. All extreme points of B(8), except 0,
are in
{x: aO'x ~ bOlo
4.
THE ALGORITHM
The procedure which defines an algorithm for the construction of the po1ytope
8(8)
from the set
S consists of starting with the hyperplane
B(8)
searching for the extreme point (or points) of
distant" from
which is (are) "most
HO and generating hyperplanes interseating H n B(8)
O
and the point(s).
HO'
= C(C)
The procedure is repeated until, at the final iteration, the
hyperplanes defining
B(8)
have been generated.
hyperplanes defining the admissible boundary,
This collection includes those
A(8).
Definition 4.1. Let F be the intersection ofa polytope P with a supporting
hyperplane.
If a polytope
F has dimension k
e:
{O, ••• ,m},
it is a
k-face.
O-faces are vertices; 1-faces are edges; (m-1)-faces are called facets and
is determined by a finite number of (hyperplanes defining) facets; see
GrUnbaum [5, chp. 3].
P
8
Let
T denote
8 u CuP,
with points
Zj
T,
€
j
in some (finite) set
J.
uThecrrem-4.2~
H ... {x: a'x =b} be a nyperplane through m points in
Let
T and suppose there exists at least one point
6 = max{a'Zj: Zj
Write
T}.
€
an extreme point of
points with
In (2), if
Proof.
satisfying
zk'
E c S,
(1)
is in
ar
L
~o.
r
= 1'
r,
say
zk
{z:
r r
r
€
= s,
b.
>
is
and
~ K}
is the set of
B(8).
A(S).
Suppose
k.
=6
a'z
k
is a face of
zk is the unique point satisfying
j :/:
for some
C(E)
the face
C(E)
and
= {zk:
E
If
for all
~ 0,
= 6,
a'zk
a'z
or
B(8);
there is more than one such point, say
(2)
T such that
€
Then either
there exists exactly one point,
(1)
Z
is not an extreme point, then
a'z
then
a' Z .... 6,
Z
k
j
< a'z
= La r Zr ,
a subset of the extreme points of B(8),
R}
0<0.
Hence,
<1.
s
a'z k
= La r a'z r
k
<
and
La r a'z k = a'zk •
Contradiction.
6 = a'zk
(2)
> a'Zj
HI'
termine a hyperplane,
extreme points
= L~a r a'z r
as does
C(E).
Z
r
~ a'z
€
k
€
K,
j
~
K.
The points in
= 6.
k
Thus for all
That is,
E either de-
H, or lie in HI. In either
Observe that for all
8(8),
€
K,
y
€
a'zk
B(8),
~
a'zr
y
= Lar z r
for all
and
E lies in a supporting hyperplane of
B(8),
0
After establishing
hyperplane
k
which is parallel to
HI = {x: a'x = 6}.
case,
a'y
for all
T
=8
u CuP,
the procedure is to start with the
HO and determine a set E = {zk: k
€
K}
as in the preceeding
theorem.
(3.7 and 3.8 guarantee that this excludes no nontrivial extreme points
of
and, in particular, no admissible points.)
B(8)
E = {zO},
m-l
construct the
corner points and
If
K is a singleton,
m new hyperplanes determined by each collection of
z00
If
K is not a singleton, then form all new
9
hyperplanes determined by each collection of
in
E.
corner points and each point
For each hyperplane at this stage, and for subsequent stages, repeat
the above procedure.
(m+l)_l
m
m-l
=m
Each stage involves the construction of at most
new hyperplanes.
Theorem 4.3. The algorithm converges. At any stage of the algorithm, the
procedure locates and includes at least one new extreme point, or face, of
8(8).
The set of extreme points is finite.
some stage, for all hyperplanes
The algorithm terminates when, at
H = {x: a'x = b} there exist no points z
with
a'z > b.
{~i}'
include those hyperplanes which determine all facets of
€
T
These hyperplanes, in addition to the coordinate hyperplanes
8(S).
In a computer program for the procedure, one stores (the equation of) a
hyperplane containing
C(E)
if the cardinality of
K is
m or more.
Note
that such a listing may include hyperplanes determining k-faces with k
<
m - 1.
This is not necessarily a problem since the equations just provide redundant
information; e.g., this doesn't affect post-optimality analysis.
However, re-
moving these may decrease computational requirements and is useful in
fying extreme points.
identi~
If there are not too many such cases, the following
lemma suggests a technique for identifying the facets.
Lemma 4.4. A hyperplane H = {x: aix = b
if and only if the
>
O}
{Xj}~
is determined by
{Xj}~ are linearly independent.
Proof. The result is known for arbitrary b when the set in the sufficiency
is replaced by
m-1
{xj -xm}l '
{Xj}~
the sufficiency with
=> {xj-x }
m
xj
follows since
linearly independent.
= (xlj, ••• ,Xmj )'
(xl' ••• ,xm).
renumbering if necessary (see Hadley [7]).
Then
and
Xa
linearly independent
J
For the necessity, let
m
{aj}l scalars.
= O.
{x.}
Suppose that
Let
Hence,
L~ajxj = 0,
X be the matrix with columns
a # O.
Then
Thus,
det(X) = 0 and there
satisfies
Blx
j
=0
10
for all
j;
m
that is,
{x }
j
is contained in a hyperplane through the origin.
0
Contradiction.
- - - 1:t-fs evident that any - H ... {x: a 'x = b J w:1.ih-whTch we are concerned will
have
b > O.
An alternative technique for removing redundant constraints may be derived
from Rubin [15].
Beyond "mathematical" convergence, the algorithm is thotJght to be computationally efficient since at each stage, a surface point of the polytope is
located and, as mentioned in Section 3, constructions involving pure admissible
extreme points which are not admissible are eliminated.
by using
B(S)
instead of
This last is achieved
C(S).
If we denote the set of admissible extreme points of
C(S)
by
E,
then
the total number of stages, or steps, of the algorithm is bounded by the car-
E u P.
dinality of
planes is at most
At each step, the total number of constructions of hyperm.
Of course, in general, the number of iterations will be
related to the number of facets of
B(s).
The so-called upper bound conjecture
for the number of facets of a polytope, given the number of vertices, has recently been shown to be true by P. McMullen; see the announcement in Grllnbaum
[6, p.1183] and formulae in [5], [6] and [11].
m
Assume S is non-degenerate in IR
(not contained in a hyperplane).
Lemma 4.5.
(at least)
Proof.
is an extreme point of
m facets of
Write
determine
v
C(S)
F = H n C(S),
k
C(S).
if necessary).
k
Suppose
Let
y =
{X€r{=lFk : IIx-vll < d.
show that points in
to range in m-r
C(S)
containing
v.
k = l' ••• 9K,
r
v € Ok=l Fk
and
N (v)
e:
for the (distinct) facets which
v
~n{ II v-Hkll: k > r}
The dimension of
if and only if there exist
4 Uk>r Fk ,
> 0, o < e:
r
flk=lHk = m-r
may be selected which have
intervals, thus contradicting that
2:
r < m (renumbering,
< y
1;
m-r
v
and
N (v) =
e:
hence, one can
components free
is an extreme point.
11
Suppose
0 ~ v€F
= n~Fk = n~HknC(S).
Each
Fk
has dimension
m-1.
If
dim(FjnFk ) = m-1, j ~ k, then dim(HjnH ) = m-1 <=> the dimension of the sok
lution-space of the equatiorisde.firiing --Hj,-H
is m"'f => the equations are
k-
=>
redundant
= Hk =>
Hj
dim(FjoFk ) S m-Z.
= Fk •
Fj
By induction,
O-faces are vertices and
Corollary 4.6.
(i)
F
But the facets are distinct; hence,
dim(n~Fk) S m-r,
Ax S 1 where
Al
.
A = (A )'
1 = (1, ••• ,1)',
2
are the normals to hyperplanes defining the fa-
and the rows of the submatrix A
are the normals to hyper-
Z
B(8)
planes defining the facets of
in
B(s)
The final form of the constraint set defining
the rows of the submatrix A
1
A(S)
dim(F) = O.
and
= {v}. 0
may be written as the matrix equation
cets of
r S m,
which contain at least one (extreme) point
CuP.
(ii)
If we maximize
f
~
0
subject to
consistent, bounded and the solution is in
(iii)
Ax
S
1,
the linear program is
A(S).
We may list the admissible extreme points of
C(S)
[B(S)]
corre-
sponding to each facet defined in
A
[A].
1
When the algorithm has terminated, all hyperplanes will be of the form
H = {x: a'x
=b
> OJ;
those with at least one
those with
a.
~
=0
a > 0
correspond to facets of
A(S)
correspond to non-admissible facets of
For each hyperplane, there is a corresponding set of points of
S.
and
B(8).
Many
are already identified as extreme points (whenever an E-set is a singleton or
doubleton); the others may be tested and identified by using 4.5 (whenever a
point satisfies
m or more hyperplanes defining facets).
Hence, for an "opti-
mar' point, or set of points, one may identify sets of "adjacent" extreme
points; these sets are useful for post-optimality analysis.
Techniques discussed in Balinski [1] may be used to locate all extreme
points of a polytope when the set of defining equations,
In our model,
Ax
=b
Ax = b,
is given.
is to be generated and much of the effort in identifying
12
extreme points is expended in the generation; hence, it is thought that little
advantage is to be gained by using the techniques of [1].
However, computa-
tional --comparisons depend on the particular polytope ofiriterest . I n examples
for which the E-sets are always singletons or
doubletons~
all extreme points
are quickly located by our procedure on the first "pass" without recourse to
further checking.
Sa
APPLICATIONS
As mentioned, one application of the admissible boundary approach is to
decision analysis.
In the sequel, we rely on a description of the normal form
of analysis for statistical decision theory; see, for example, Raiffa [13,
chp. 6] and Raiffa and Sch1aifer [14, chp. 1].
For applications to decision
analysis, see Howard [8]2, [9].
We assume that part of a decision analysis model is described by a decision tree, namely the probabilistic (nature's) tree of the normal form.
(The
dual mode of analysis is by the extensive form via the chronological tree; both
modes lead to the same final decision.)
possible strategy,
s .,
J
j
= 1, ••. ,n,
Each path through the tree describes a
and has been assigned a certain utility
value given a particular state of nature,
6 ,
i
i
= 1, ••• ,m;
to denote the utility value for strategy
Sj
i.e., we write
when
is the
"true" state of nature.
To this point we have assumed only that the above values have been assigned
and, consequently, any analysis is independent of prior distribution assessments.
Hence, a major advantage of normal form analysis is that prior distribution input occurs at the final stage of the analysis thus reserving subjective judgements until the end, and dealing only with admissible strategies (as few strategies as possible).
In [8" p.39J reverse the labels for Figures 4 and 5; FiguT'e 4 is natUT'e's
tT'ee and FiguT'e 5 is the ahT'onologiaaZ deaision tree.
2
13
The vector
v
lecting strategy
= (v ., ••• ,v .)'
1J
mJ
j
s.,
represents
m possible values for se-
the vector of joint conditional values.
J
S ,,;; {vj:-j-;; 1, ... ,nr
If
represents the set of values -for all strategies, then
C(S) is the set of all mixed strategies and if we accept the basic axioms of
preference (utility) theory, our objective is to maximize the expected utility
ED = I~=lPiui
= p'u
prior distribution.
over the polytope
C(S),
where
p = (Pl, ••• ,Pm)'
As we have seen, it suffices to maximize
EU
is a
over
A(S).
It is often the case that the decision maker is uncertain about the precise values for his prior distribution on the unknown states of nature, or (perhaps to a lesser extent) about the utility values
tentative estimates for
p
and the
{v },
ij
{v }.
ij
If he establishes
we may solve the above linear pro-
gram and use post-optimality analysis to determine classes of probability distributions for which the decision remains the same.
varied by modifications on the
Av
1
=1
describing
(The
{v
A(S)
in Cor. 4.6.)
ij
} may be
Alternative approaches, where values are assumed to be fixed, are suggested in
Fishburn, Murphy and Isaacs [4] and Pierce and Folks [12].
In their
main approach, the decision maker determines a i1nearestlV probability distribution for which his decision changes and uses this as a guideline for the
sensitivity of his decision to his initial estimates for
p.
Comparative ef-
ficiencies of the procedures appear to be problem-dependent.
The linear program of 4.6 and its dual may be interpreted for decision
analysis.
.,
~n
y
Let
(P)
be
max piX, s.t. Alx
01, s.t. y i
Al ~'
P , Y
and A is
rxm where
~
00
cr(F),
strategies in
S.
and
x
1, x
~
0
and
have dimension
(D)
A(S).
Each such facet
be
mXl;
m is the number of states of nature and
ber of facets which determine
(unique) set,
p
~
y
r
is
rxl
is the num-
F is associated with a
of joint conditional values corresponding to certain (pure)
(We identify
sk
and
v = (vlk, ••• ,v )'.)
k
mk
14
For any point
x = (x , ••• ,x )'
1
m
given the state of nature
A(S),
€
xi
is the amount of utility value
x must be in at least one facet
0 ••
1
L~ = 1 and vk
thus
€
a(f).
f
of
Hence
is
the amount of value deriving from these strategies with proportions
each strategy
If
sk'
= y*'l = L~=lYj*'
where
a
for
k
0 •
given the state of nature
x* = (x *, ••• ,x *)'
m
1
A(S),
i
is an optimal basic solution to (P),
y* = (Y1*" •• 'Yn*)'
then
p'x*
is an optimal solution to
(D)o
By complementary slackness, if the jth constraint in
corresponding slack variable is positive), then
y.*
J
(P)
= 0;
is non-binding (the
thus the only
yj
*
which may be in the solution at a positive level are those associated with facets
f
j
whose hyperplanes are binding.
responding
Yj* = 0,
in other facets
fk
(If some
f.
J
is binding and the cor-
this just means that all the alternate optima in
with corresponding
Y * > 0.)
k
Yj*[L~=lYk*]-l is the proportion
Yj *
of value derived from
the strategies defining the facet
terpretation holds for
the constraint set in
Y* > 0
j
(P),
tation to be reasonable.
or Y * = 00
e.g.,
j
A1x
~
1,
are
Consequently, an interpre-
tation for the dual variables
a(f ),
j
is that
(P)
fjo
This in-
Note that a normalized form of
must be used for this interpre-
Parametric programming may be used to generate sets
of proportions of value derived from strategies given various prior distributions.
Modifications on the algorithm will handle the addition or change of
strategies.
The algorithm is being programmed and further investigation has the objective of resolving such conjectures as: the number of cases in which the algorithm must resort to identifying facets by independence checks is small in re1ation to the number of facets of
A(s).
15
AcKNOWLEDGMENT
The author is grateful to Professor Ronald A. Howard whose Decision
Analysis Seminar prompted this study.
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e
1.
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olsions~
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IsraeZ
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