Programming Under Uncertainty: The Solution Set
Author(s): Roger Wets
Reviewed work(s):
Source: SIAM Journal on Applied Mathematics, Vol. 14, No. 5 (Sep., 1966), pp. 1143-1151
Published by: Society for Industrial and Applied Mathematics
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J. SIAM APPL. MATH.
Vol. 14,No. 5, September,1966
Printedin U.S.A.
PROGRAMMING UNDER UNCERTAINTY: THE SOLUTION SET*
ROGER WETSt
Abstract.In a previouspaper [81,we describedand characterizedthe equivalent
convexprogramof a two-stagelinear programunderuncertainty.We proved that
thesolutionset of a linearprogramunderuncertainty
is convexand derivedexplicit
expressionsforthisset forsomeparticularcases. The mainresultof thispaper is to
showthatthesolutionset is notonlyconvexbut also polyhedral.
It is also shownthat
the equivalent convex programof a multi-stageprogramming
under uncertainty
problemis of the form:Minimizea convexfunctionsubject to linearconstraints.
1. Introduction.The standardformof the problemto be consideredin
thispaper is:
(1)
Minimizez(x)
=
cx + Et {min qy} subject to
Ax = b,
Tx+Wy-i, =
x > O,
y
on
F
O,
whereA is a m X n matrix,T is m X n, W is m X ii, and t is a random
vectordefinedon a probabilityspace (Z, i, F).
We will assume that problem (1) is solvable. One interpretsit as follows: The decision-maker
must select the activitylevels forx, say x = &,
he thenobservesthe randomevent t = i, and he is finallyallowedto take
- T77 and qy is minimum.
a correctiveaction y, such that y > 0, Wy =
This correctiveactiony can be thoughtof as a recoursethe decision-maker
possessesto "fix-up"the discrepanciesbetweenhis firstdecisionand the
observedvalue of the randomvariable. This recoursedecisiony is taken
when no uncertaintiesare leftin the problem.
All quantitiesconsideredherebelongto the reals,denotedby St. Vectors
will belong to finite-dimensional
spaces Tn and whetherthey are to be
regardedas row vectorsor columnvectorswill always be clear fromthe
contextin which they appear. No special provisionhas been made for
transposingvectors.
We assume that (Z, 9:, F) is the probabilityspace induced in k% F
determinesa Lebesgue-Stieltjesmeasureand 9Fis the completionforF of
the Borel algebrain WYm.
g, the set of all possibleoutcomesof the random
variables,is assumed convex. If not, we replace it by its convex hull and
fillup 9: with the appropriatesets of measure zero. We use the notation
* Received by the editorsSeptember1, 1965.
t MathematicsResearch Laboratory,Boeing ScientificResearch Laboratories,
Seattle,Washington.
1143
1144
ROGER
WETS
t to denotea randomvectorof dimensionm, as well as the specificvalues
assumed by this random variable, i.e., points of -. No confusionshould
arise fromthis abuse of notation.
The marginalprobabilityspace for i = 1,..., m will be denoted by
(si, Wi, Fi) where Zi is a subset of the real line. If they exist,let ai
and j3ibe respectivelythe greatestlowerbound and least upper bound of
In the firstpart of this paper,we characterizethe solutionset of problem (1); in the second part,we generalizeour resultsto multi-stageproblems.
2. The solutionset. A solutionof (1) is a decisionto be made (here and
now), thus a selectionof activitylevels forthe vectorx. The value to be
assigned to the vector y can be determinedby solvingthe deterministic
linearprogram:
(2)
Minimizeqy subject to
Wy
=
t-Tx,
y >o
i.e., afterx is selectedand t is observed.
In this paper,we are only interestedin some of the propertiesof a solution, i.e., the decisionvariable x. Nevertheless,the recourseproblem (2)
affectsour selectionof x in two ways. For each selectionof a vectorx, we
must take into account the expectedcosts of the recoursessuch an x may
generate.But also, we need to limit our selectionof a vectorx to those
forwhichthereexistsa feasiblerecourse,i.e., problem(2) is feasible.This
latterrestrictionand the conditionsAx = b, x > 0 determinethe set of
feasiblesolutionsof problem(1).
A vectorx is a feasiblesolutionto (1) ifit satisfiesthe first
DEFINITION.
stage constraintsand if problem(2) is feasibleforall t in t.
We do not call such a solutiona permanentlyfeasiblesolution.We rejected these termssince theyled to certainconfusions,see [6] and [8].
2.1. Definition and notation. A set C is convex if xl, x2 C C implies that [xl, x2] C C. C is a conewith vertexzero ifx C C impliesthat
Xx C C for all X > 0. C is a convexcone if xl, x2 C C impliesthat
X1+ X2 C. (a) is a ray,i.e.,(a) = {zIz = Xa forXin [0, + oo) and a E Tn}.
The rays (a) and (b) are distinctifb E (a) or a E (b). C* is the polarcone
of CifC* = {yIyx > 0, Vx GC). Thepolarconeofa ray (a) isahalfsee [4].
space that we denote by (a)*. For furtherrefereiice,
The theoryof positive linear dependencewas developed by Chandler
A set of rays { (a'), (a2), ... I
Davis [3].We reviewsome of the definitions.
PROGRAMMING UNDER UNCERTAINTY
1145
spans positively
a cone C if (b) E C impliesthat b =
1j Xja j forsome
selection of n rays a'j and some Xi > 0, j = 1, * , n. A set of
rays { (a), *
is positively
independent
if none of the a' is a positivecombinationof the others.Otherwise,the set is positivelydependent.A set of
vectors{al, a2,
determinesa frameforthe cone C if {(a'), (a2),
are positivelyindependentand span positivelythe cone C.
C is a convexpolyhedralconeif it is the sum of a finitenumberof rays,
C = {(b) I (b) = ,
(a') }. Equivalently C is a convexpolyhedral
cone
if it is the intersectionof a finitenumberof half-spaceswhose supporting
hyperplanespass throughthe origin.A set K is a convexpolyhedron
if it is
the intersectionof a finitenumberof half-spaces.A bounded convexpolyhedronP is a polytope.It is easy to verifythe followinglemmas.
LEMMA 1 [5]. Let C be a convexpolyhedral
cone,thenC* is also a convex
polyhedral
cone.
LEMMA 2. If C is a convexpolyhedral
cone,thenevery
frameofC is finite.
LEMMA 3. Every convexpolyhedron
K can be obtainedas the sum of a
polytopeP and a convexpolyhedral
coneC.
2.2. The polar matrix.Let A be a m X n matrix,and let C be the cone
spannedpositivelyby the columnsof A, i.e.,
C=
{yIy=Ax,x'O},
then C is a convex polyhedralcone. Moreover,a subset of A determines
a frameforC. The same cone C can also be definedas the intersectionof
a finitenumberof half-spaces.Moreover,there exists a minimalset of
hyperplaneswhichsupportC. This conceptled to the followingdefiniition.
DEFINITION. A* is the polarmatrixof A if
ly I Y = Ax,x >
0} = C = {y I A*y > 0)
and the matrixA* has mininmal
row cardinality.It is easy to see that
if A has fullrank,i.e., rank A = m, then the matrixA* has m nonzero
columns.If A * has m rows, then C is a simplicialcone; and if A * has
one row,then C is a half-space.If C = NMZ,then no hyperplanesupports
C, i.e., the numberof rows of A* is zero. An algebraiccharacterizationof
the facesof convexpolyhedralsis givenin [9].
2.3. Fixed constraints.Let
K1,
{x I Ax=b,
x > O}.
Since K1 does not involve any constraintsinvolvinglater stages condiwe say that the
tions,and sincethe constraintsof K1 are well-determined,
set K1 is the set representing
the fixedconstraints.
PROPOSITION 1. K1 is a convexpolyhedron.
If m = 0, thenK1 = T+n.
ROGER WETS
1146
2.4. Induced constraints.Let
{x I V CEX, 3y _ O
K2=
such that Wy =-Tx}.
the constraintsinducedon x by the
We say that K2 is the set representing
followingcondition:No matter what value is assumed by the random
variable i, thereexistsa feasiblerecoursey. If we define
K2t= {xIWy=
forsome y_O},
-Tx
then
K2= nK2,.
we can
In [8] it was shownthat K2 is convex.Withoutloss of generality,
assume that the matrixW has fulldimension(mn-);
otherwisethereexists
an equivalent systemof linear equations to the system Tx + Wy = t
with at least one equation of the form:Tix + 0 y =- , where0 is a row
vector of dimensionin. If tj is a constant,we can add that equation to
the systemof equationsAx = b. If {i is not a constant,then problem(1)
is not solvable and is thuswithoutinterest.
In orderto be able to appeal to some intuitivegeometricconcepts,we
definethe sets:
L ={
I Vt E :, 3y > O
such that Wy= X-I}
and
LE
x Ix
Wy
forsome y _ 0}.
We also get
L = n L.
x, x e Ld and K2 = {x I Tx = x, x E LI.
so is K2t. Similarly,if L
2. If Lt is a convexpolyhedron
so is K2 .
is a convexpolyhderon
Proof.If Lt is a convexpolyhedron,then by the definitionof a convex
polyhedronthereexistsa matrix,say V, and a vectord such that x E Lt
x VTx ? d}.
ifand onlyif x E {x I Vx ? d}. Then, by Lemma 4, K2t {x
SimilarlyforL and K2 .
Thus, in order to show that K2 is a convex polyhedron,it sufficesto
show that L is a convex polyhedron.For each i, Lt is the translateby t
of the cone spanned positivelyby the columnsof the matrix-W. Thus,
L is the resultof the intersectionof "parallel" cones, each one being the
translateof the same cone and having forvertexthe point t.
Let W* be the polar matrixof the matrixW, then - W* is the polar
LEMMA4. K2t = {x I Tx =
PROPOSITION
PROGRAMMING UNDER UNCERTAINTY
1147
by 1,where1 = OitLL = W,I = 1
matrixof -W. W* is of dimensionmin
ifLt is a half-space,and so on.
{X I W*X < W*{&.
LEMMA 5. L=
Wy for
of the polar matrix,we have that {t t
Proof.By definition
some y > O} = {t I W*y < O}. Then, by translationof the cone so defined,we obtain the desiredresult.
infWi*t for
PROPOSITION 3. L =
{x I W*X < a, where a,i =
Proof.The resultis immediateif L = Jm, i.e., 1 = 0. Let us assume
that 1 ? 1. By definitionL = El.: Lt . For t in Z all the hyperplanes Wi*x = Wi, i = 1, ... , 1, are parallel. Thus x E L if
Wi*x < infE Wi* = ai*. If foreach i the linearformWi*t attains its
infimumon the convexset X, the set L is well-defined.If forsome i, no
infimumexists,the set of feasible solutionsis emptyand problem(1) is
not solvable.
We have thus provedthat L is a convexpolyhedron.L is a cone if and
only if there exists t in Wm such that W*t = a*. From Propositions2
and 3, we derivethe followingtheorem.
THEOREM 1. K2 = {x I W*Tx ? a*)} is a convexpolyhedron.
2.5. The solutionset. It now is easy to concludethat Proposition4 is
true.
PROPOSITION
4. K, the set of feasible solutions, is a convex polyhedron.
Proof.By Proposition1 and Theorem 1, K1 and K2 are convex polyhedrons;and since K = K1 n K2, the proofis immediate.
3. The equivalent convex program. In [8] it was shown that to each
problem,
problemof the form(1), thereexistsan equivalentdeterministic
in termsof the decisionvariables x, whichis a convex program,i.e., the
minimizationof a convex function(cx + Q(x)) on a convex set K. We
have shownhere that this set K is polyhedraland consequentlythat this
equivalentconvexprogramhas the generalform:
(3)
Minimizecx + Q(x) subject to
Ax = b,
W*Tx
< a*,
x > 0,
whereQ(x) is definedin [8] and W* and a* are as above.
4. Multi-stage linear program under uncertainty. In this last section,
we will show that the equivalentconvexprogram(in termsof the decision
variable x) of a multi-stagelinear programunder uncertaintyis also of
ROGER WETS
1148
the form(3), i.e., the minimizationof a convexfunctionsubject to linear
constraints.We firstconsiderthe followinggeneralizationof problem(1):
Minimizeex + Et{ minq(y)} subject to
Ax
(4)
=
b
on
Tx + Wy-i,
F)
y C D,
x > O,
whereq(y) is a conivexfunctionalin y on 9Jr,D is a convex polyhedron,
and A, T, W and t are as above. In [7],it was shown that problem(4)
possessesalso an equivalentconvexprogram,whose objectivereads:
Minimize cx + Q(x),
where
(5)
Q(x)
=
E{min q(y) I Wy
-
Tx, y > O}.
We now show that the solutionset of (4) is a convexpolyhedron.
Intuitively,we could argue as follows:Since D is a convexpolyhedron,
foreach t. Since
so is W(D) ={t I t = Wy,y C D}, and so is -W(D)
all polyhedronst - W(D) are the translateby t of a given polyhedron,
these polyhedronsare "parallel", and theirintersectionis a convex polyhedron of the same "form", with the possible exclusionof some faces.
Then by Proposition2, the set of induced constraints,K2, of problem(4)
is also a convexpolyhedron.Since K1 is the same as forproblem(1), we
have that K, the set of feasiblesolutions,is also polyhedral.
LEMMA 6. W(D) is a convexpolyhedron.
of W(D).
Proof.By definition
In orderto point out some of the propertiesof this set W(D), we give
more detailed constructionof the set W(D). By Lemma 3, everyconvex
polyhedronD can be expressedas the sum of a polytopeDp and a convex
polyhedralcone DC . Let
DC=
{r r = Vcz,z ? O}
Dp
{s I Vps > f},
anid
i.e., y C D if and only if y = p + q wherep C Dp and q C DC. Similarly
W(D) -W(D)p
+ W(D)c,
and it is easy to see that
W(D)p=
W(Dp)
and W(D)c = W(D,).
UNDER
PROGRAMMING
1149
UNCERTAINTY
Also
-W(D)
-
[W(Dv) + W(Dc)].
If we constructthe polar matrixof (WVc), we can find an explicit expressionfor the supportinghyperplanesof W(Dc) [9]. Similarly,the extremepoints of Dp can be foundby examiningthe basis of Vp, see [1].
The linear operatorW maps the extremepoints of Dp into the extreme
points of W(Dp). From these we can findthe boundinghyperplanesof
W(Dp). It is thus theoreticallypossible to findan expressionfor W(D)
of the form:
W(D) - {t I Wt < d}
(6)
forsome matrixW of dimensionm X 1 and some vectord. If 1 = 0, then
if l = 1, then W(D) is a half-space,and so on.
nm;
W(D) =
- W(D)} is a convexpolyhedron.
IX
PROPOSITION 5. L = nfEl {X
Proof.By (6),
Lt= {xIlWx _ a, where &j =i
+ ifi
If the linearformWN has no infimumon V, thenL -= .
COROLLARY
1. K2 = {x I Tx = x, x E L} is a convexpolyhedron.
COROLLARY 2. K, the set of feasiblesolutionsof (4), is a convexpolyhedron.
We now apply these resultsto the followiilgmulti-stageprogramminig
under uncertaintyproblem:
Minimize c1xl+ Et2 I minc2(x2)
(7)
+ E [nin
C3(X3)
+
(
+ Em-(min cm(xm)) ...
subjectto
= b,
Alx1
A21x1+ A22x=
A32x2+ A33x3
m=
Am,m_lxm+ Am,mx
X1 O0 X2 > 0x 3 > O
.
X
xm> O
i,m and the
wherethe t are independentrandomvectorsfori - 2,
This
in.
i
i,
problemis readily
,
ct(x') are convex functionsfor = *1
seen to be equivalentto the generalstructureof the multi-stageproblem
1150
ROGER
WETS
foundin [2, Chap. 25]. By Theorem 1, we obtain the equivalent mn- 1
underuncertaintyproblem:
stage programming
Minimizec'(x)' + Et{min c2(x2) + EJmin c3(x3)
+ E(min cm-(xml) + Qm,(xm')) .}
+
subject to
Allxl
b,
A2ix' + A22x2
A32x2+ A,8=JX
=
2
Am_l,m2Xm
x1 >
0, X2 > 0, X3 > 0
+
xm-2
Aml,milx
> 0
1
=
r-1
Xm-1 Dm,-1
whereQm-i(xm-l) is a convex functiondefinedon the convex polyhedral
Dm . Using repeatedly(5) and Corollary1, we can findthe equivalent
convex programto (7), which reads:
Minimizec(x') + Q(xl) subject to
Ax' = b,
X E
whereD' is a conivexpolyhedronand Q(xl) is convex.This completesthe
proofof the followingproposition.
PROPOSITION 6. The equivalent convex programof (7) is of the form:
Find an x whichminimizesa convexfunctionsubject to linearconstraints.
Acknowledgment.I am very gratefulto my colleague, Dr. David W.
Walkup of the Boeing ScientificResearch Laboratories,whose pertinent
remarksled me to investigatethe main resultof this paper.
REFERENCES
[1] M. L. BALINSKI, An algorithmfor finding all verticesof convex polyhedral sets,
[2] G. B.
[3] C.
thisJournal,9 (1961),pp. 72-88.
DANTZIG,
Linear Programming and Extensions, Princeton University
Press, Princeton,1963.
DAVIS,
Theory of positive linear dependence,Amer. J. Math., 76 (1954), pp.
733-746.
[4] D. GALE, Convex polyhedral cones and linear inequalities, Cowles Commission
MonographNo. 13,ActivityAnalysisof Productionand Allocation,T. C.
Koopmans,ed., JohnWiley,New York, 1951,pp. 287-297.
[5] M. GERSTENHABER, Theory of convex polyhedral cones, Ibid., pp. 298-316.
[6] G. L. THOMPSON, W. W. COOPER AND A. CHARNES, Characterization of chance-
PROGRAMMING
UNDER
UNCERTAINTY
1151
constrainedprogramming,
Recent Advances in Mathematical Programming,R. Graves and P. Wolfe,eds., McGraw-Hill,New York, 1963,pp.
113-120.
[71 R. VAN SLYKE
[8] R.
[9] R.
AND
R.
WETS,
Programmingunder uncertaintyand stochasticopti-
mal control,J. Soc. Indust. Appl. Math. Ser. A: Control,4 (1966), pp.
179-193.
WETS,
Programming under uncertainty: The equivalent convex program, this
Journal,14 (1966) pp. 89-105.
WETS
AND
C. WITZGALL, Towards an algebraic characterizationof convex poly-
hedral cones, Boeing documentD1-82-0525,Boeing ScientificResearch
Laboratories,Seattle, 1966.