Chapter 6
The convex
hull
The expected value function of an integer recourse program is in general nonconvex. The lack of convexity precludes the use of many results that are indispensable for eciently solving these models. It is well-known that, given
convexity, every local minimum is a global minimum. Moreover, the duality
theory for convex optimization problems provides a wealth of results, resulting
in the fact that such problems are well-solved, at least in theory (see Grotschel
et al. [14]). For example, the entire theory and all algorithms to solve continuous recourse programs are based on the convexity of these problems.
These considerations justify our interest in convex approximations of the
expected value function of integer recourse models. Obviously, if the expected
value function is replaced by such an approximation, the resulting model is a
convex minimization problem, since the rst-stage objective function is linear
and the constraints are linear equalities and non-negativities.
In this and the following chapter we deal with convex approximations of
the integer expected value function Q, in case the recourse is simple and the
technology matrix T is xed. For easy reference, we repeat its denition:
Q(x) = E! v(x; !); x 2 R n ;
v(x; !) = inf fq+y+ + q y : y+ p(!) Tx;
y p(!) Tx;
y+ 2 Zm+ ; y 2 Zm+ g;
1
2
2
where (q+ ; q ) and T are xed vectors/matrices of compatible dimensions,
q+ 0, q 0, and p(!) is a random vector.
111
112
The convex hull
Due to the inherent separability, as explained in Section 4.1, we may write
Q(x) =
m
X
2
i=1
Qi (x); x 2 R n ;
1
Qi (x) = qi+ E! dpi (!) Ti xe+ + qi E! bpi (!) Ti xc ;
where pi (!) is the ith element of p(!), and Ti is the ith row vector of T ,
respectively. Therefore, the function Q is completely determined by the onedimensional generic function Q^ , given by
Q^ (z ) = q+ g(z ) + q h(z ); z 2 R ;
where q+ ; q 2 R , with q+ 0, q 0,
g(z ) = E d z e+; z 2 R ;
h(z ) = E b z c ; z 2 R ;
with a random variable. Obviously, for = pi (!), z = Ti x, q+ = qi+ and
q = qi , we have Qi (x) = Q^ (z ).
In Section 6.1 we show that every reasonable convex approximation of an
integer simple recourse problem is equivalent to some continuous simple recourse problem. The remainder of this chapter is devoted to nding the convex
hull of the expected value function. In Chapter 7 we use certain perturbations
of the distribution of to obtain convex approximations.
6.1 Relation to continuous simple recourse
By Theorem 4.4.1, the one-dimensional continuous simple recourse expected
value function Q~ , given by
Q~ (z ) = q+ E ( z )+ + q E ( z ) ; z 2 R ;
provides a convex lower bound as well as a convex upper bound for Q^ , since
for all z 2 R
Q~ (z ) Q^ (z ) Q~ (z ) + maxfq+; q g:
(6.1)
Obviously, both bounds are sharp if q+ = q = 0: then Q~ = Q^ = 0. They
are also sharp if q+ > 0 and E + = 1 or if q > 0 and E = 1: then
Q~ = Q^ = 1. Leaving out these uninteresting cases, we assume that
= E 2 R ; q+ + q > 0:
This is no loss, since already in Section 4.1 it is assumed that q+ 0 and q 0. Under these assumptions, Q~ is a non-linear, convex, Lipschitz continuous
113
6.1 Relation to continuous simple recourse
function with Lipschitz constant maxfq+; q g having q+ ( z ) as asymptote
at 1 and q (z ) as asymptote at +1 (see Lemma 2.4.3).
The inequalities (6.1) show, that Q~ is a convex approximation of Q^ , with
error of at most maxfq+; q g. This error is not necessarily small. Therefore it
is worthwhile to consider sharper convex approximations. Nevertheless, the inequalities (6.1) and the properties of Q~ indicate that any reasonable convex approximation of Q^ , say Q^ c , should satisfy Q~ (z ) Q^ c(z ) Q~ (z ) + maxfq+ ; q g
for z 2 R . In particular, it follows that such a Q^ c has asymptotes at 1 and
+1 with the same slopes as the asymptotes of Q~ . That is, Q^ c has asymptotes
of the form q+ (d1 z ) at 1 and q (z d2 ) at +1, for some d1 2 R and
d2 2 R . Moreover, such a Q^ c is non-linear, convex and Lipschitz continuous
with Lipschitz constant maxfq+; q g since Q~ has these properties.
The following theorem is the fundamental tool to establish a relation between convex approximations of the simple integer recourse expected value
function and its continuous simple recourse analogue.
Theorem 6.1.1 Let v be a non-linear, convex, Lipschitz continuous function
on R . Dene
a1 = z!lim1 v+0 (z );
0
a2 = zlim
!1 v+ (z );
where v+0 (z ) denotes the right derivative of v at z 2 R . Then the function
V : R 7! [0; 1] dened by
0
V (s) = v+a(s+) +a a1 ; s 2 R ;
1
2
is a cdf.
If v has an asymptote at +1, say v(z ) a2 z + c2 as z ! 1, then
v(z ) = a2 z + c2 + (a1 + a2 )
Z1
z
1 V (s) ds; z 2 R :
(6.2)
If v has an asymptote at 1, say v(z ) a1 z + c1 as z ! 1, then
v(z ) = a1 z + c1 + (a1 + a2 )
Zz
1
V (s) ds; z 2 R :
(6.3)
If both asymptotes exist, then
v(z ) = a1
Z1
z
1
Zz
a2 c 1 ; z 2 R :
V (s) ds + a2
V (s) ds + a1ac2 +
1 + a2
1
(6.4)
Proof. First we note that both a1 and a2 exist since v is convex; they are nite
since v is Lipschitz continuous. By convexity of v we also have that a1 + a2 0,
and since v is non-linear, even a1 +a2 > 0. Hence, V is well-dened.
114
The convex hull
The function V is non-decreasing since v is convex. Obviously, it is continuous from the right since v+0 is. Moreover, lims! 1 V (s) = 0 and lims!1 V (s) =
1, so that V is a cdf indeed.
It is clear that if the function v has one or two asymptotes, they have to be
of the indicated form. To prove (6.2{6.4) we use that for all 1 < z z^ < 1
v(^z) v(z ) =
so that
v(^z) v(z ) =
Z z^
Z z^
z
v+0 (s) ds;
a1 + (a1 + a2 )V (s) ds
z
= ( a1 )
Z z^
z
1 V (s) ds + a2
If a2 z + c2 is the asymptote of v at +1, then
v(^z ) (a2 z^ + c2 ) v(z) (a2 z + c2 )
= v(^z) v(z ) a2 (^z z)
=
(a1 + a2 )
Z z^
z
Z z^
z
V (s) ds:
(6.5)
1 V (s) ds;
where the last equality follows from (6.5). Taking z^ ! 1 and replacing z by
z , equation (6.2) follows.
Similarly, if a1z + c1 is an asymptote of v at 1, then
v(^z ) ( a1 z^ + c1 ) v(z ) ( a1 z + c1 )
= v(^z ) v(z ) + a1 (^z z)
= (a1 + a2 )
Z z^
z
V (s) ds;
where the last equality follows from (6.5). Taking z ! 1 and replacing z^ by
z , equation (6.3) follows.
Finally, if both asymptotes exist, then both (6.2) and (6.3) hold. Hence,
every ane combination of the right-hand sides of these equations is a formula
for v(z ) too. Weighing (6.2) with a1 =(a1 + a2 ) and (6.3) with a2 =(a1 + a2 )
eliminates the linear term and gives (6.4).
2
Remark 6.1.1 The existence of the asymptotes does not follow from the assumptions on the function v. For instance, the function
v(z ) =
p
2
1 + jz j 1 ; z 2 R ;
115
6.1 Relation to continuous simple recourse
is convex and Lipschitz continuous, but is has no asymptotes. In general, the
existence of asymptotes of the function v satisfying the conditions of Theorem 6.1.1 can be expressed in terms of its conjugate function
v (s) = sup fsz v(z )g ; s 2 R :
z2R
If the asymptote at +1 exists then its slope is a2 , and
lim (v(z ) a2 z ) = inf fv(z ) a2 z g
z!1
=
z2R
sup fa2 z v(z )g
z2R
v (a2 ):
The rst equality follows from v+0 (z ) a2 for all z 2 R . Therefore, the asymptote at +1 exists (and is equal to a2 z v (a2 )) if and only if a2 2 dom v =
fs 2 R : v (s) < 1g. Similarly, the asymptote at 1 exists (and equals
a1 z v ( a1 )) if and only if a1 2 dom v . However, the assumptions on v
only imply ( a1 ; a2 ) dom v [ a1 ; a2 ].
The conjugate function of the example function v is
s2=(1 jsj); if jsj < 1;
v (s) = +
1;
if jsj 1;
so that v ( a1 ) = v (a2 ) = +1, since a1 = a2 = 1 in this case.
=
At the outset of this section we showed that any reasonable convex approximation of Q^ c satises the conditions of Theorem 6.1.1. Hence we proved the
following corollary.
Corollary 6.1.1 Let Q^ c be a convex function on R bounded from below by Q~
and bounded from above by Q~ + maxfq+ ; q g. Then
+
^c 0
W (s) = (Q q)++ (+s)q+ q ; s 2 R ;
is a cdf, and for all z 2 R
Q^ c(z ) = q+
Z1
z
(1 W (s)) ds + q
Zz
1
W (s) ds + c
= q+ E ( z )+ + q E (z )+ + c:
Here, is any random variable with cdf W , and
+ + q c1
c = q qc+2 +
q ;
where c1 = limz! 1 (Q^ c (z ) Q~ (z )) and c2 = limz!1 (Q^ c (z ) Q~ (z )).
2
116
The convex hull
(q )
( q+ )
Q^ c
6c2
?
6?c1
6?c
Figure 6.1: The function Q^ c , and asymptotes of Q^ c and Q~ .
Corollary 6.1.1 states that every convex function Q^ c between Q~ and Q~ +
maxfq+; q g is equal to the one-dimensional expected value function of some
continuous simple recourse program (plus a constant). In other words, every reasonable convex approximation of an integer simple recourse problem is
equivalent to some continuous simple recourse problem. Consequently, we can
solve simple integer recourse problems (at least approximately) by algorithms
developed for continuous simple recourse problems.
However, Corollary 6.1.1 is non-constructive in the sense that the distribution of the random variable is a transformation of the right derivative of
Q^ c, which is not known. In the remainder of this chapter and in the following
chapter we consider techniques for constructing particular convex approximations Q^ c or, equivalently, for nding the distribution of that replaces in the
equivalent continuous simple recourse model.
6.2 Denition and properties of the convex hull
For reasons that we shall reveal shortly, the convex hull of a function f is the
most interesting candidate among all possible convex approximations of f .
Denition 6.2.1 The convex hull of a function f , denoted by conv f , is the
pointwise greatest convex function majorized by f .
As suggested by its name, the convex hull of f can be characterized via the
convex hull of sets. Let epi f denote the epigraph of the function f : R n 7!
6.2 Definition and properties of the convex hull
117
[ 1; 1],
epi f = f(x; s) : x 2 R n ; s 2 R ; s f (x)g:
Then (see e.g. [30] page 36)
conv f (x) = inf fs : (x; s) 2 conv(epi f )g;
where conv S denotes the convex hull of a set S .
The convex hull of f is intimately related to the biconjugate function f .
Since the conjugate function f of f is dened by
(n
)
X
f () = supn
i xi f (x) ; 2 R n ;
x2R i=1
the biconjugate f = (f ) is given by
(n
X
f (x) = supn
i xi
2R i=1
)
f ( ) ; x 2 R n :
It can be shown (see [30] Theorem 12.2) that under a condition to be discussed
f is the lower semicontinuous hull of conv f , that is
epi f = cl(epi(conv f )) = cl(conv(epi f ))
where cl S denotes the topological closure of a set S . The condition is, that
(conv f )(x) > 1 for all x 2 R n . Since we will only consider non-negative
functions f this condition is satised automatically. Moreover, since we only
consider non-negative functions f that are nite everywhere, also conv f is
nite everywhere. Since nite convex functions on R n are continuous, taking
the lower semicontinuous hull is superuous. Hence, we will always have
f = conv f:
For that reason we will denote the convex hulls of functions as biconjugates.
The following properties of the convex hull make it very attractive in the context
of minimization:
inf f (x) = inf n f (x);
x2R n
x2R
and
argmin
f (x) argmin
f (x):
n
n
x2R
x2R
118
The convex hull
Figure 6.2: The function jx2 4j (dashed) and its convex hull (x2 4)+ (solid).
See Figure 6.2 for an illustration of these facts. The rst statement is equivalent
to f (0) = (f ) (0), and this is true since f is lower semicontinuous and
convex (see [30] Theorem 12.2). The second one is a direct consequence of the
fact that f f . Notice that these properties have to do with unconstrained
minimization of f over R n . If one is interested in inmization of f over a nonempty closed convex subset C of R n , one should replace f by f, given by
f(x) =
f (x);
if x 2 C ;
+1; if x 26 C:
Obviously, unconstrained minimization of f, or rather of f , is equivalent
to the original constrained minimization. We will return to such constrained
optimization problems at the end of Section 6.4.
In the following sections we concentrate on nding the convex hull of the
expected value function, instead of the convex hull of the objective function
cx + Q(x). This approach is justied by the following lemma, which implies
that (cx + Q(x)) = cx + Q (x), x 2 R n .
1
Lemma 6.2.1 Let f1 be a function on R n, and dene f (x) = ax + f1(x) for
some xed (row) vector a. Then
f (x) = ax + f1 (x);
x 2 R n:
6.3 The convex hull of the one-dimensional expected value function
119
Proof. First we compute the conjugate of f . For 2 R n ,
f () = supn x ax + f1 (x)
x2R
= supn f( a)x f1 (x)g
x2R
= f1 ( a):
Conjugation of this function given the biconjugate of f .
f (x) = supnfx f1 ( a)g
2R
= supnf( + a)x f1()g
2R
= ax + supnfx f1()g
2R
= ax + f1 (x):
2
6.3 The convex hull of the one-dimensional expected value function
Finding the convex hull Q^ of the one-dimensional expected value function Q^
is in general a dicult problem. The characterization via the epigraph of Q^
gives a clear idea of the concept, but is of little use for practical computations.
On the other hand, to determine Q^ as the biconjugate of Q^ presupposes the
computation of the conjugate Q^ () for all 2 R . Using the denition of the
conjugate function we have
Q^ () = sup fz Q^ (z )g
=
z2R
inf f z + Q^ (z )g; 2 R ;
z2R
which we recognize as a one-dimensional (unrestricted) simple integer recourse
problem. In other words, now we are faced with a parameterized version of the
original problem, which we have to solve for all 2 R . Clearly this is not a
practical approach.
Consequently, we need to rely on the specic properties of the function
Q^ in order to nd its convex hull. Even then we are not able to solve the
problem in all cases. Indeed, for the case that follows a continuous distribution
no general technique is known: so far we can only handle the exponential
120
The convex hull
distribution. As an illustration of the diculties involved, this instance is
treated in Section 6.3.1.
Below we restrict ourselves to the case that follows a discrete distribution
with nitely many mass points. This is a signicant loss of generality, but on the
other hand we note that any distribution can be approximated arbitrarily close
by a distribution belonging to this class. We refer to Schultz [35] for stability
results with respect to such approximations.
In the rest of this section we assume that is a discrete random variable
with support = f 1 ; : : : ; p g, and also that the mass points are ordered, that
is, 1 < 2 < : : : < p . Then, by Theorem 4.4.1, the function Q^ has the
following properties.
(a) Q^ is a nite function, that is discontinuous at all points of D1 , where
D1 =
[p
i=1
f i + Zg:
Q^ is constant in between discontinuity points;
Q^ is lower semicontinuous;
Q^ is semi-periodic with period 1 on the intervals ( 1; 1 ] and [ p ; 1),
with slope q+ and q , respectively.
(e) On every interval [ i ; i+1 ], i = 1; : : : ; p 1, the function Q^ is semi-periodic
with period 1 and slope q+ Prf > i g + q Prf < i+1 g.
(b)
(c)
(d)
See Example 4.4.2 and Figure 4.13 for an illustration of these properties.
The crucial observation to make is that, due to properties (a){(c) above, the
convex hull of the function Q^ depends simply and solely on the points (d; Q^ (d)),
d 2 D1 . This can be seen as follows. Since Q^ is lower semicontinuous on R
and constant in between any two neighboring discontinuity points di < di+1 ,
it follows that
^ i+1 ^ i
Q^ (z ) Q^ (di ) + (z di ) Q(d ) Q(d ) 8z 2 [di ; di+1 ]:
(6.6)
di+1 di
That is to say, on [di ; di+1 ] the function Q^ is bounded from below by the linear
function through (di ; Q^ (di )) and (di+1 ; Q^ (di+1 )), which is obviously independent of the function values of Q^ on the interior of this interval. At the same
time, this linear function is an upper bound for all convex lower bounds of Q^
restricted to the interval [di ; di+1 ].
It follows that the convex hull of Q^ is equal to the convex hull of the piecewise
linear function Q^ pl that is implicitly dened by the right-hand side of (6.6).
Explicitly this function is given by
Q^ pl (z ) = Q^ (z ); z 2 D1 ;
6.3 The convex hull of the one-dimensional expected value function
121
and if di < di+1 are neighboring points in D1 then
i+1
i
Q^ pl (z ) = ddi+1 dzi Q^ (di ) + diz+1 d di Q^ (di+1 ); z 2 [di ; di+1 ]:
The points where the slope of Q^ pl changes will be called the knots of Q^ pl .
For convenience, we will use the term `knot' not only to refer to the point
d 2 D1 but also to the corresponding point (d; Q^ pl (d)) of the graph of Q^ pl .
The meaning will always be clear from the context.
Since the set of knots of Q^ pl is D1 , there is only a nite number of knots in
each bounded interval. However, usually Q^ pl has innitely many knots on the
whole of R . Intuitively, it is clear that the convex hull of Q^ pl is again piecewise
linear, such that its set of knots is a subset of the set D1 which contains the
knots of Q^ pl itself. This is the idea behind the following algorithm.
This algorithm (known as the Graham scan, Graham [13]) determines the
convex hull of a piecewise linear function f . Let f 0 and f+0 denote the left
and right derivative of f . Obviously, a necessary condition for a knot di to be
on the convex hull is that f 0 (di ) f+0 (di ). Consequently, if this condition is
not satised then we may eliminate the knot di , and redene the function f
on [di 1 ; di+1 ] as the linear function connecting the points (di 1 ; f (di 1 )) and
(di+1 ; f (di+1 )). It is easy to see that by repeating this procedure until all knots
satisfy the condition mentioned above, eventually we end up with the convex
hull of the function f (see Figure 6.3).
Of course, this algorithm only terminates in nite time if only a nite number of knots has to be to considered. For the function Q^ pl niteness of the
initial set of knots is due to its semi-periodicity on the intervals ( 1; 1 ] and
[ p ; 1) (property (d) above), as we will show now.
It follows immediately from property (d) that the convex hull Q^ is ane on
( 1; 1 1] and [ p +1; 1) with slopes q+ and q , respectively. Consequently,
we only need to determine the convex hull of Q^ pl restricted to [ 1 1; p + 1]
which depends only on knots in the set D = D1 \ [ 1 1; p +1], which is nite
indeed. Next, we extend this function dened on [ 1 1; p + 1] to the convex
hull of Q^ pl on R as follows. Among the remaining knots d 2 D we select the
largest one such that (Q^ pl )0 (d) q+ (Q^ pl )0+ (d), and the smallest one such
that (Q^ pl )0 (d) q (Q^ pl )0+ (d), say d and d, respectively. Then
(Q^ pl ) (z ) = Q^ pl (d) q+ (z d);
z 2 ( 1; d];
and
(Q^ pl ) (z ) = Q^ pl (d) + q (z d);
z 2 [d; 1);
see Figure 6.4. Consequently, all knots outside [d; d] can be eliminated at once.
122
The convex hull
r
r
r
]1
r
r
]2
]3
Figure 6.3: Iterative deletion of knots not on the convex hull. Knots to be
deleted at iteration ]n are indicated with a .
Thus, we have established a basic algorithm to construct the convex hull
of Q^ in case follows a nite discrete distribution. Next we will show how to
improve the algorithm by using property (e) above and the formula that we
have to compute function dierences.
In the rst place, note that the procedure to eliminate knots of the piecewise linear function Q^ pl uses only one-sided derivatives. To compute these
derivatives, we only need to know the dierence between function values in
neighboring discontinuity points of Q^ . Let d1 < d2 be any two such points.
Then by Theorem 4.4.1 we have
Q^ (d2 ) Q^ (d1 ) =
q+
1
X
k=0
+q
Prfd1 + k < d2 + kg
1
X
k=0
Prfd1 k < d2 kg:
Now suppose that ^ 2 and k 2 Z exist such that d1 + k < ^ < d2 + k. Then
^ k 2 D1 \ (d1 ; d2 ). This can not be true since d1 and d2 are neighboring
6.3 The convex hull of the one-dimensional expected value function
123
Figure 6.4: Here it is assumed that Prf = 1g = 0:7, Prf = 5:5g = 0:3, and
q+ = 1, q = 1:2. Knots of Q^ pl are indicated with , and the solid line denotes
the convex hull of Q^ pl (c.q. Q^ ) restricted to [ 1 1; 2 + 1] = [0; 6:5]. Its
extension to R is denoted by the dashed line. In this case d = 1 and d = 6.
Hence, the convex hull has only knots in d = 1, 5 and 6.
points in D1 . Hence,
Q^ (d2 ) Q^ (d1 ) =
=
=
=
q+
1
X
Prf = d2 + kg + q
1
X
Prf = d1 kg
k=0
k=0
+
2
q Prf 2 d + Z+ g + q Prf 2 d1 Z+ g
q+ Prfd2 2 Z+ g + q Prfd1 2 + Z+ g
p X
+
q
Prf = i g : d2 2 i Z+
i=1
p X
+q
Prf = i g : d1 2 i + Z+ :
i=1
It follows that computing the required function dierences is merely a matter of
book-keeping: no function values have to be calculated. For each discontinuity
point we simply have to record by which mass point(s) it is generated. This
comes at virtually no extra eort, since in practice the set D is constructed by
taking each mass point i 2 , and adding to the set points i + k and i k,
124
The convex hull
k 2 Z, if they are in [ 1 1; p +1].
The second improvement of the algorithm is based on the fact that Q^ is
semi-periodic on every interval [ i ; i+1 ], i = 1; : : : ; p 1.
Instead of constructing the convex hull of Q^ on the entire interval [ 1
p
1; +1] as a whole, we rst consider each subinterval [ i ; i+1 ], i = 1; : : : ; p 1,
separately. By Theorem 4.4.1 the function Q^ is semi-periodic on each of these
intervals, which means that if both z and z +n, n 2 Z+ , are in [ i ; i+1 ], then
Q^ (z + n) Q^ (z ) = ni ;
where i = q+ Prf > i g + q Prf < i+1 g. In particular, this holds true
for every pair of knots d 2 [ i ; i + 1] and d + ni 2 [ i+1 1; i+1 ], where
ni = b i+1 i 1c+ .
Now consider the convex hull of Q^ restricted to the interval [ i ; i + 1],
^ i
^ i
which we will denote by Q^ [i] . Since Q( + 1) Q( ) = i , it follows from
Lebourg's Mean Value Theorem (see e.g. Clarke [6]) that Q^ [i] has a knot
i
i
^
^
di 2 [ ; + 1] such that i 2 @ Q[i] (di ). Next, using that Q is semi-periodic
on [ i ; i+1 ], it follows that the line segment with endpoints (di ; Q^ (di )) and
(di + ni ; Q^ (di + ni )) and slope i is a part of the convex hull of Q^ restricted
to [ i ; i+1 ]. Consequently, all knots in ( i ; i+1 ) can be discarded without
inuencing the computation of the convex hull of Q^ on R (see Figure 6.5).
Obviously, this local procedure is only eective if ni 1, that is, if i+1
i
2. In any case, we conclude that after applying this procedure, there
remain at most 2p + 1 knots between any two neighboring mass points that
need to be considered: at most p + 1 knots in [ i ; i + 1] and at most p knots in
(di + ni ; i+1 ] since i+1 (di + ni ) < 1. As we will show below, this implies
that our algorithm to construct the convex hull of Q^ is strongly ecient. For
the explanation of this term see e.g. Garey and Johnson [12].
Summarizing, the improved algorithm consists of the following steps.
(i) Local construction of convex hulls on intervals
^
For each i = 1; : : : ; p 1, construct the convex hull Q^ [i] of Q restricted
i
i
^
to the interval [ ; + 1]. Determine di such that i 2 @ Q[i] (di ), where
i = q+ Prf > i g + q Prf < i+1 g is the slope of the semi-periodic
function Q^ .
(ii) Elimination of knots in intervals
For each i = 1; : : : ; p 1, eliminate all knots in (di ; di + ni ), where
ni = b i+1 i 1c+ . Obviously, Q^ (di + ni ) Q^ (di ) = ni i .
(iii) `Global' construction of the convex hull
Let D be the set containing all remaining knots in [ 1 1; p + 1]. Dene
6.3 The convex hull of the one-dimensional expected value function
125
Figure 6.5: Here is discretely distributed with support = f 1 ; 2 ; 3 g =
f0:3; 4:8; 6g with Prf = 0:3g = 0:1, Prf = 4:8g = 0:3, Prf = 6g = 0:6, and
q+ = 1:1, q = 1. Knots of Q^ are indicated with , and the solid line denotes
Q^ [1] . The dashed line segment with slope 1 = 0:89 indicates the lower bound
for Q^ on [d1 ; d1 + n1 ] = [1; 4].
Q^ pl on this interval as the piecewise linear function with knots in D, and
determine its convex hull using one-sided derivatives.
(iv) Extension of the convex hull to R
Extend the convex hull of Q^ pl on [ 1 1; p + 1] to the convex hull of Q^
on R .
Finally, we notice that every step of the improved algorithm is executed
without calculating any function values of the piecewise linear functions involved. Only knots and sloped are used. As a result, after completing step (iv)
we know all knots of Q^ and its slope in between two successive knots. That
is, Q^ is determined up to a constant. By Corollary 6.1.1 this information is
sucient to determine the distribution of the random variable that replaces
in the equivalent continuous simple recourse formulation. Indeed, it follows
that has a cdf that is a simple transformation of the known right derivative of Q^ . Since Q^ is piecewise linear, we see that is a discrete random
variable with mass points i that correspond to the knots of Q^ . Moreover,
Prf = i g = i =(q+ + q ), where i is the increase of the slope of Q^ at i .
126
The convex hull
It remains to compute the constant c as dened in Corollary 6.1.1. We use that
Q^ equals Q^ at each of its knots. Hence,
i )+ + q E (
i) ;
c^ = Q^ ( i ) q+ E (
where i is an arbitrary point in the support of . This is the only time that
we actually calculate a function value of Q^ .
Computational complexity of the algorithm
Here we discuss the computational complexity of the algorithm. We will treat
the complexity of the basic parts of the algorithm separately.
(1) Elimination of points not on the convex hull (both locally and globally).
Given a piecewise linear function f with ordered knots di , i = 1; 2; : : : ; n,
its convex hull can be determined in O(n) operations by the procedure
based on one-sided derivatives (see [13]).
To eliminate points in between i and i+1 based on the semi-periodicity
argument, we need to consider at most p +1 points in the interval [ i ; i +
1]. Therefore, this step takes O(p) operations.
We have argued above that there are at most 2p + 1 knots left in any
interval [ i ; i+1 ], i = 1; 2; : : : ; p 1, after the rst elimination. In addition, the knots in [ 1 1; 1 ] and [ p ; p + 1] need to be considered, so
that there are all together O(p2 ) knots that enter the global elimination
procedure. Therefore, this part of the algorithm takes O(p2 ) operations.
(2) Generation of the set D.
As we have seen, it is sucient to consider only points in [ i 1; i + 1],
i = 1; 2; : : :; p. If i+1 i < 2 then all discontinuity points in between
these two mass points are covered. On the other hand, if i+1 i 2, we
know from the discussion above that we only need knots in the indicated
intervals to determine di and di + ni , such that all knots in the interval
(di ; di + ni ) can be discarded right away.
It is easy to see that there are at most 2p + 1 discontinuity points in
any interval of length two. The order of the points in every interval
[ i 1; i + 1], i = 1; 2; : : :; p, is dictated by the order of the fractions
of the mass points 1 mod 1; 2 mod 1; : : : ; p mod 1. Thus, we need to
order the fractions once, which can be done in O(p log p) time, and then
the points in each interval [ i 1; i + 1] can be enumerated in O(p)
operations. It follows that it takes O(p2 ) operations to form the set D.
(3) Computation of the constant c.
i )+ + q E (
i) .
This is done by evaluating Q^ ( i ) and q+ E (
Both functions are the sum of n terms, not depending on p, where n is
O(p) and O(p2 ), respectively. Hence, this step takes O(p2 ) operations.
6.3 The convex hull of the one-dimensional expected value function
127
We conclude that the overall computational complexity of the algorithm is
O(p2 ), the maximum over the separate steps.
6.3.1 Example: the exponential distribution
In this section we consider the convex hull of the function Q^ for the case that
is exponentially distributed with parameter > 0. As indicated in the
previous section, this is the only continuous distribution for which we are able
to construct the convex hull. The techniques used below heavily depend on
properties of Q^ that are specic for the case with the exponential distribution,
at least in the sense that we have not been able to prove similar properties for
other distributions.
First we repeat some results already presented in Example 4.4.1.
(z+d ze+ )
Q^ (z ) = q+ d z e+ + e 1 e e
!
(z bzc)
e (z+1) I
[0;1) (z ); z 2 R :
1 e This function is continuous, and it is dierentiable on all open intervals (k; k +
1), k 2 Z. In fact, Q^ is innitely dierentiable on all such intervals. For
z 2 (k; k +1), the nth derivative, n = 1; 2; : : :, is given by
+q
bz c + 1
8
k
>
n q + e z e
(
)
>
;
<
1
e
(
n
)
Q^ (z ) = >
> ( )n e z q+ q (ek e ) ;
:
1 e if k 1;
if k 0:
(6.7)
This formula reveals that on each interval (k; k + 1) each derivative is equal to
e z up to a multiplicative constant. The following results are direct conse-
quences of this property. Dene
1 + q
k = ln e + q
;
(6.8)
that is, k is the smallest integer such that q+ q (ek e ) 0. Since we
assume that q+ =q > 0 it is easy to see that k 0.
Lemma 6.3.1 The function Q^ is monotone on each interval [k; k + 1]; k 2 Z.
In particular
(i) Q^ is strictly decreasing on [k; k + 1] for k < k ;
(ii) Q^ is strictly increasing on [k; k + 1] for k > k ;
128
The convex hull
(iii) Q^ is strictly increasing on [k ; k +1] except if k = (1=) ln(q+ =q +e ),
that is, if the round up in (6.8) has no eect. In that case Q^ is constant
on [k ; k + 1].
Consequently, Q^ attains its minimum value at k .
Proof. Immediate from (6.7) with n = 1. The exception in the third statement follows from the fact that the condition k = (1=) ln(q+ =q + e ) is
equivalent to q+ q (ek e ) = 0.
2
Lemma 6.3.2 The function Q^ is convex or concave on each interval [k; k +
1]; k 2 Z. In particular
(i) Q^ is strictly convex on [k; k + 1] for k < k ;
(ii) Q^ is strictly concave on [k; k + 1] for k > k ;
(iii) Q^ is strictly concave on [k ; k + 1] except if k = (1=) ln(q+ =q + e ),
that is, if the round up in (6.8) has no eect. In that case Q^ is ane on
[k ; k + 1].
Proof. Immediate from (6.7) with n = 2.
2
Corollary 6.3.1 The function Q^ is strictly convex on [k; k + 1], k 2 Z, if and
only if it is decreasing on this interval. The function Q^ is strictly concave on
[k; k + 1] if and only if it is increasing on this interval. Moreover, Q^ is ane on
[k; k + 1] if and only if it is constant on this interval.
Proof. Immediate from Lemmas 6.3.1 and 6.3.2.
2
^
Thus, we have completely characterized the behavior of Q on all intervals
[k; k + 1], k 2 Z. Since we are interested in the convex hull of Q^ on R , we also
need information on the behavior of Q^ 0(z ) as z passes from one such interval to
the next. From (6.7), or again using results derived in Example 4.4.1 we have,
for k 2 Z,
8 q+;
if k 1;
<
Q^ 0+ (k) Q^ 0 (k) = : ( q+ + q ); if k = 0;
q ;
if k 1:
We conclude that Q^ is not dierentiable at k 6= 0, since it is assumed that
q+ and q are strictly positive. If k < 0 then Q^ 0 (k) > Q^ 0+ (k); if k > 0
then Q^ 0 (k) < Q^ 0+ (k). The function Q^ is dierentiable at 0 if and only if
q+ = q .
Now we are ready to calculate the convex hull Q^ of Q^ . In fact, we will
compute the convex hulls of Q^ restricted to the intervals ( 1; a], [a; b] and
[b; 1), where a and b depend on q+ , q and . This approach is justied by
the following corollary.
6.3 The convex hull of the one-dimensional expected value function
129
Corollary 6.3.2 Let 1+ = a1 < a2 < : : : < an < an+1 = +1 and I =
f1; : : : ; ng. Let f : R 7! R and dene for i 2 I
f (s); if s 2 [a ; a ] \ R ;
fi (s) = +1; otherwisei ; i+1
so that f (s) = inf i2I fi (s) for all s 2 R .
If, for i = 2; : : : ; n, fi1 (ai ) = fi(ai ) and (fi1 )0 (ai ) (fi )0+ (ai ) then
f (s) = inf
f (s) 8s 2 R :
i2I i
Proof. Immediate from Theorem 5.6 in [30].
2
Thus, if we choose a and b such that the conditions mentioned in Corollary 6.3.2
are satised, then the convex hull on R follows trivially from the convex hulls
of the restrictions of Q^ to the respective intervals.
First, we derive a formula for Q^ (z ) for small values of z . Recall that
^
Q = q+ g + q h, where g and h denote the integer expected surplus and shortage
function, respectively . Since h(z ) = 0 for z 2 ( 1; 0] and h(z ) > 0 for
z 2 (0; 1), it holds
Q^ (z ) = q+ g(z ); z 2 ( 1; 0];
and
Q^ (z ) > q+ g(z ); z 2 (0; 1):
It follows that for small values of z the convex hull Q^ is simply q+ g. Here
g denotes the convex hull of g, which we will compute now. In Example 4.2.4
we have seen that the function g is semi-periodic with slope 1 on the interval
( 1; 1). Moreover, g is convex on [0; 1) since the pdf of is non-increasing on
this interval. It follows that on ( 1; 1 +1] the convex hull of g is given by the
ane function l(z ) = g(1 ) + 1 z , where 1 2 ( 1; 0) is the unique number
such that g0(1 ) = 1. Trivially, g = g on [1 +1; 1).
Straightforward calculation gives
1 = 1 ln 1 e 1:
We conclude that on ( 1; 1 ] the convex hull of Q^ is given by q+ l(z ) = Q^ (1 )+
q+ (1 z ).
Next, consider Q^ on [k ; 1). By Lemma 6.3.1 the function Q^ is decreasing
on ( 1; k ] and non-decreasing on [k ; 1). It follows that on [k ; 1) the
convex hull Q^ is completely determined by function values Q^ (z ), z 2 [k ; 1).
Moreover, by Lemma 6.3.2 Q^ is concave on every interval [k; k +1], k k , and
130
The convex hull
by Theorem 4.4.1 the piecewise linear function with knots (k; Q^ (k)), k 2 Z, is
convex. We conclude that on [k ; 1) the convex hull Q^ is piecewise linear
with knots (k; Q^ (k)), k k .
In summary, we have derived the following result on the tails of Q^ .
Lemma 6.3.3 Let follow an exponential distribution with parameter > 0.
Then
Q^ (z ) = Q^ (1 ) + q+ (1 z ); z 2 ( 1; 1 ];
and
Q^ (z ) = Q^ (k) + (z k) Q^ (k + 1) Q^ (k) ; k z k + 1; k k ;
where
and
1 = 1 ln 1 e 1
+
:
k = 1 ln e + qq
In particular, the convex hull Q^ coincides with Q^ at the points z = 1 and
z = k .
2
It remains to calculate Q^ for all z 2 (1 ; k ). It appears that two dierent
cases have to be considered, depending on the value of q+ =q . If this value
is at most 1, it appears that Q^ coincides with Q^ on [1 ; k ]; if this value is
larger than 1, Q^ diers from Q^ in the neighborhood of 0, due to the fact
that Q^ is non-convex in that neighborhood then. In the following we work
out both cases, giving a complete description of Q^ and illustrating it with a
numerical example.
Case 1 0 < q+ q . As shown in Lemma 6.3.2, the function Q^ is convex on
each interval [1 ; 0], [0; 1], : : : , [k 1; k]. Moreover, Q^ 0+ (0) Q^ 0 (0) = ( q+ +
q ) 0 in this case, and also Q^ 0+ (k) Q^ 0 (k) = q > 0 for k 2 f1; : : : ; k 1g.
Therefore, Q^ is convex on [1 ; k ]. Hence, using Lemma 6.3.3, it follows that
8 Q^ ( ) + q+( z);
z 1 ;
>
1
< ^ 1
^
1 z k ;
Q (z ) = > Q(z );
: Q^ (k) + (z k) Q^ (k + 1) Q^ (k) ; k z k + 1; k k:
See Figure 6.6. Note that if q+ =q = 1 e then (1=) ln(e +(q+ =q )) = 0,
so that Q^ is constant on [0; 1] by Lemma 6.3.1.
6.3 The convex hull of the one-dimensional expected value function
131
Figure 6.6: Illustration of Case 1: q+ = 0:9, q = 1, = 0:4, hence 1 =
0:5166 and k = 2. The dashed line indicates the function Q^ whereas the
solid line denotes its convex hull Q^ .
Case 2 0 < q < q+. In this case we have k 1. Again, Lemma 6.3.1
shows that Q^ is convex on each interval [1 ; 0], [0; 1], : : : , [k 1; k ]. Also we
have Q^ 0+ (k) Q^ 0 (k) = q > 0 for k = 1; : : : ; k 1, so that Q^ is convex on
[1 ; 0] and on [0; k]. But Q^ is non-convex on [1 ; k] since Q^ 0+ (0) Q^ 0 (0) =
( q+ + q ) < 0. It follows that the convex hull is ane on the interval [2 ; 3 ],
where 2 2 ( 1; 0) and 3 2 (0; 1) satisfy
Q^ 0 (2 ) = Q^ 0 (3 );
Q^ (3 ) = Q^ (2 ) + Q^ 0 (2 )(3 2 ):
Straightforward calculation gives
2 = 1 ln q+ ln q+ q (1 e ) ln q+ + 1 ln(q+ q ) + 1 e 1
and
3 = 2 + 1 ln q+ q (1 e ) ln q+ + 1:
132
The convex hull
Figure 6.7: Illustration of Case 2: q+ = 9, q = 1, = 2, hence k = 2,
1 = 0:5807, 2 = 0:5477, 3 = 0:4018. The dashed line indicates the
function Q^ whereas the solid line denotes its convex hull Q^ .
It now follows from Lemma 6.3.3 that
8 Q^( ) + q+( z);
1
1
>
>
^
Q
(
z
)
;
>
<
Q^ (z ) = > Q^ (2 ) + Q^ 0(2 )(z 2 );
Q^ (z );
>
>
: Q^(k) + (z k) Q^(k + 1) Q^ (k) ;
z 1 ;
1 z 2 ;
2 z 3 ;
3 z k ;
k z k + 1; k k :
See Figure 6.7.
Finally, we give one example of the cdf F of the random variable that
replaces in the equivalent continuous simple recourse problem. By Corollary 6.1.1 this cdf is given by
+
^ 0
F (s) = q +q+(Q+ q)+ (s) ;
s 2 R:
To compute the right derivative (Q^ )0+ we use that it is either locally constant
or equal to the right derivative of Q^ itself, which according to Example 4.4.1
133
6.4 The convex hull of the expected value function
is given by
8 e (z bzc) >
q+
;
>
<
1 e Q^ 0+(z ) = > z >
: q+ 1 ee + q e
if z < 0;
e (z+1) ; if z 0:
1 e (z bzc)
If q+ and q are as considered in Case 2, then
8 0;
s 1 ;
>
>
>
>
v(s);
1 s < 2 ;
>
>
>
>
v(2 );
2 s < 3 ;
>
>
>
<
s
3 s < 1;
F (s) = > v(s 1) + qq+e
+q ;
>
>
>
q e s e e >
v(s 1) + (q+ + q )(1 e ) ; 1 s < 2;
>
>
>
>
+ >
>
: 1 e k q q+++e q q ;
k s < k + 1; k 2;
where
v(s) =
q+ 1 e e (s+1)
(q+ + q )(1 e ) ; s 2 R :
Figure 6.8 shows this cdf F in case q+ = 9, q = 1 and = 2.
The approach above is obviously tailored to the case with exponentially
distributed. Most likely, it is not possible to generalize this method, so that
the problem of nding the convex hull of the function Q^ for general continuous
distributions remains open.
6.4 The convex hull of the expected value function
In the previous sections we have discussed the convex hull of the one-dimensional
function Q^ . However, recall that the actual expected value function Q :
R n 7! R of a simple integer recourse problem with xed technology matrix
1
134
The convex hull
Figure 6.8: The cdf F in case q+ = 9, q = 1, and = 2.
is given by
Q(x) =
m
X
2
i=1
Qi (x) =
m
X
^
2
i=1
Qi (Ti x); x 2 R n ;
1
with
Q^ i (Ti x) = qi+ Ei di Ti xe+ + qi Ei bi Ti xc ;
where the scalars qi+ and qi denote the ith components of vectors q+ and q , respectively, i denotes the ith component of the random vector = (1 ; : : : ; m ),
and Ti is the ith row of the matrix T . In this section we will show that, at least
if the matrix T has linear independent rows,
2
Q (x) =
m
X
^ 2
i=1
Qi (Ti x)
8x 2 R n :
1
That is, the convex hull of Q is determined by the convex hulls of the functions
Q^ i in precisely the same way as Q itself is determined by the functions Q^ i .
In general it is not true that the convex hull of a sum of functions is equal to
the sum of the convex hulls of the individual functions. Consider for example
x2 ; if x 2 ( 1; 0) [ (0; 1);
f1 (x) =
1; if x = 0;
6.4 The convex hull of the expected value function
f2 (x) =
0;
135
if x 2 ( 1; 0) [ (0; 1);
1; if x = 0;
and f = f1 + f2. It is easy to see that f (x) = f (x) = x2 , whereas f1 (x) = x2
and f2 (x) = 1, so that f1 (x)+ f2 (x) = x2 1.
Also, it is not true in general, that the convex hull of a composition of
a function f with a linear transformation is equal to the composition of the
convex hull of f with that linear transformation. Consider
f (x1 ; x2 ) = ex x ; (x1 ; x2 ) 2 R2 ; fS (y) = f (Sy); y 2 R ; S = 11 :
2
1
2
2
Then f (x1 ; x2 ) = 0 for all (x1 ; x2 ) 2 R 2 , so that f (Sy) = 0 for all y 2 R ,
whereas fS (y) = fS (y) = 1 for all y 2 R .
We now will show, that under certain conditions on the real functions fi ,
i = 1; : : : ; m, and the matrix S with rows Si , i = 1; : : : ; m, it holds that for
all x 2 R n
m
X
i=1
fi (Si )
!
(x) =
m
X
i=1
fi(Si x):
Remark 6.4.1 Until now there was no need to make explicit whether a vector
is a row or a column vector. For example, it is understood that cx = c1 x1 +
: : : + cn xn so that evidently c is an n-dimensional row vector whereas x is an
n-dimensional column vector. However, the presentation of the general results
below does not allow such ambiguity. Therefore, for the time being we agree
that all vectors are column vectors. Also, we will use the notation hx; yi to
denote the inner product of vectors x and y.
To begin with, we assume that the functions are proper.
Denition 6.4.1 A function f : R n 7! ( 1; 1] is proper if
(i) 9 2 R n 9 2 R : f (x) h; xi 8x 2 R n ;
(ii) 9 2 R n 9 2 R : f ( ) .
Thus a function f is proper if there exists a linear minorant and if f is nite
somewhere. This rather weak assumption on f is justied by the fact that it
guarantees that its conjugate function f is proper, which in turn implies that
f is proper, too. Moreover, summation of proper functions is well-dened,
since the combination 1 1 is excluded.
First we consider the (closed) convex hull of separable functions.
136
The convex hull
Theorem 6.4.1 Let fi,mi = 1; : : : ; m be proper functions on R . Dene the
separable function f : R 7! ( 1; 1] by
f (x) =
m
X
i=1
fi (xi );
Then
f (x) =
m
X
i=1
x = (x1 ; : : : ; xm ):
fi (xi )
8x 2 R m :
Proof. First we determine the conjugate function f . For all = (1 ; : : : ; m )0
2 R m we have
n
o
f () = supm h; xi f (x)
x2R
m
X
= sup
(i xi fi (xi )):
x1 ;:::;xm i=1
Since each fi is proper, the supremum can be calculated term by term, so that
f () =
=
m
X
n
sup i xi fi (xi )
o
i=1 xi 2R
m
X
i=1
fi (i ):
We see that the conjugate function f is again separable. Since each fi is
proper, this result can be applied to f as well. This proves the claim of the
theorem, since (f ) = f and (fi ) = fi , i = 1; : : : ; m.
2
The next result deals with functions that are dened in terms of another
function and a linear transformation of the argument. First we introduce some
notation. Let S : R n 7! R m be a linear transformation. Denote S : R m 7! R n
the adjoint linear transformation, characterized by
hy; Sxi = hS y; xi 8x 2 R n and 8y 2 R m ;
where on the left we have the inner product in R m and on the right the inner
product in R n . With these linear transformations we associate matrix representations S and S (the transpose of S ), respectively.
The null space of S is denoted by N (S ) = fx 2 R n : Sx = 0g, and the range
of S by R(S ) = fy 2 R m : 9x 2 R n such that Sx = yg. Similar denitions
hold for the null space and range of S : N (S ) = fy 2 R m : S y = 0g and
R(S ) = fx 2 R n : 9y 2 R m such that S y = xg.
137
6.4 The convex hull of the expected value function
Let X be a subspace in R m or R n . Then by X ? we denote its orthogonal
complement. In particular, we have N (S )? = R(S ) and N (S )? = R(S ).
The restriction of the linear transformation S to N (S )? is invertible. The
inverse map is denoted by S + : R(S ) 7! N (S )? . Similarly, the linear transformation S restricted to N (S )? is invertible, and the inverse map is denoted
by (S )+ : R(S ) 7! N (S )? .
Below C denotes the indicator function of the set C , that is, C (x) = 0 if
x 2 C and C (x) = +1 otherwise.
Theorem
6.4.2 Let f : R m 7! ( 1; 1] be an proper
function. Dene fS :
n
R 7! ( 1; 1] by fS (x) = f (Sx), where S : R 7! R m is a linear map. Then
fS(x) = f + R(S) (Sx); x 2 R n :
Proof. If R(S ) \ dom f = ; the result is trivial, since fS +1 and f +
R(S) +1 in that case. Without loss of generality, therefore, we assume that
R(S ) \ dom f 6= ; so that fS is proper.
Now we derive a formula for the conjugate function fS .
n
fS () = supn h; xi f (Sx)
=
=
x2R
n
sup
x2N (S )
x^2R(S )
o
h; (x + x^)i f (S (x + x^))
n
o
o
sup h; xi + sup h; x^i f (S x^) ;
x2N (S )
x^2R(S )
(6.9)
where the second equality follows from the fact that every x 2 R n can be
represented uniquely as x = x + x^, x 2 N (S ), x^ 2 R(S ), since R(S ) =
N (S )? , and where the last equality is true since fS is proper.
Consider the rst term of (6.9). It is easy to see that it is equal to +1 if 62
N (S )? and 0 otherwise. It follows that fS () = 1 if 62 N (S )? = R(S ).
If 2 R(S ) then
fS () =
=
=
n
o
n
o
sup h; x^i f (S x^)
x^2R(S )
sup
y2R(S )
sup
y2R(S )
n
h; S + yi f (y)
h(S + ) ; yi f (y)
o
where we used that Sx = y is equivalent to x = S + y for all x 2 R(S ). We
now have
+ if 2 R(S );
fS () = 1f ;+ R(S) ((S ) ); otherwise
:
138
The convex hull
Conjugation of this function gives
n
fS(x) = supn h; xi fS ()
=
=
2R
n
o
o
sup h; xi fS ()
2R(S )
n
o
f + R(S) (S + ) S y :
supm hS y; xi
y2R
Since (S + ) = (S )+ we get
n
fS(x) = supm hy; Sxi
=
y2R
o
f + R(S) (y)
f + R(S) (Sx)
8x 2 R n :
This completes the proof.
2
The following theorem combines the previous results, and covers the expected value function in particular.
Theorem 6.4.3 Let fi,mi = 1; : : : ; m, be proper functions on R . Dene the
separable function f : R 7! ( 1; 1] by
f (y) =
m
X
i=1
fi (yi );
y = (y1 ; : : : ; ym):
Dene fS : R n 7! ( 1; 1] by
fS (x) = f (Sx) =
m
X
i=1
fi (Si x);
where S : R n 7! R m is a linear transformation such that rank S = m, and Si
denotes the ith row of the matrix S . Then
fS (x) =
m
X
i=1
fi(Si x)
8x 2 R n :
2
Proof. Immediate from Theorems 6.4.1 and 6.4.2.
In terms of the expected value function Q the result reads as follows.
Corollary 6.4.1 Let Q(x) = Pmi=1 Q^ i(Tix), x 2 R n , be the expected value
function of a simple integer recourse program with xed technology matrix T .
2
1
139
6.4 The convex hull of the expected value function
If rank T = m2 , i.e., if the rows Ti , i = 1; : : : ; m2 , of T are linearly independent,
then
Q (x) =
m
X
^ 2
i=1
Qi (Ti x)
8x 2 R n :
1
P Q^ (T x) for all x 2 R n .
Independent of rank T it holds Q (x) m
i=1 i i
Proof. The case with rank T = m2 is covered by Theorem 6.4.3. The second
claim holds trivially, since byPdenition of Q it is the greatest convex lower
bound of Q, and obviously m
i=1 Q^ i (Ti x) is also a convex lower bound of
Q.
2
Consequently, we can actually compute the convex hull of the expected
value function Q, at least if T has full row rank and if the components of
the random right-hand side vector = (1 ; : : : ; m ) follow a nite discrete
distribution. First we compute the convex hull of the one-dimensional functions
Q^ i , i = 1; : : : ; m2 , using the algorithm described in Section 6.3. For each i, the
convex hull Q
i of the n1 -dimensional function Qi given by Qi (x) = Q^ i (Ti x),
n
x 2 R , is then dened by
n
Q
i (x) = Q^ i (Ti x); x 2 R :
2
1
2
2
1
1
We see that Q
function that is non-dierentiable at a point
i is a polyhedral
j
x if and only if Ti x = i for some j , where ij is a knot of (Q^ i ) . The
convex hull Q is simply the sum of the m2 polyhedral functions Q
i , and
hence polyhedral itself. It follows that Q is non-dierentiable at x if and
only if Ti x = ij for some
j . Finally, we observe that computing the
P i and
2
convex hull Q takes O( m
i=1 pi ) operations, where pi , i = 1; : : : ; m2 , denotes
the number of mass points of the random variable i . This complexity result
2
follows trivially from the fact that computing Q^ i takes O(pi ) operations (see
Section 6.3).
In other words, under the above-mentioned conditions on T and and
for the moment disregarding possible complications due to the rst-stage constraints, Corollary 6.4.1 implies that solving a simple integer recourse problem
amounts to solving a convex polyhedral problem. Moreover, it follows immediatelyPfrom Corollary 6.1.1 that the latter problem is equivalent (up to a constant
c = mi=1 ci in the objective function) to the continuous simple recourse problem that is obtained if the random right-hand side vector is replaced by .
Consequently, once we have computed the distribution of and the constant
c, we can use any algorithm for continuous simple recourse programs to solve
our integer recourse problem, disregarding possible diculties related to the
rst-stage constraints. Because there exist several special purpose computer
codes that exploit the specic structure of continuous simple recourse models,
2
2
140
The convex hull
Figure 6.9: Constrained minimization of f (dashed) and f (solid) over the
set K = f0 x bg. Then inf x2K f (x) inf x2K f (x), where equality holds
only if b 1 or b 3.
this is an ecient approach. A preliminary version of the computer code providing the distribution of and the constant c is implemented in the computer
package SLP-IOR, see Kall and Mayer [16].
Finally, it remains to be discussed how to handle the constraint set K =
fx 2 R n+ : Ax = bg, which is completely determined by the rst-stage constraints because there are no induced constraints. The point is that, while we
are interested in the minimum of cx + Q(x) over the set K , the approximation cx + Q (x) depends also on function values of Q outside K . It is not
dicult to see that this provides no hardship if the constrained optimum coincides with the free optimum. However, if this is not the case then it may
happen that inf x2K fcx + Q(x)g > inf x2K fcx + Q (x)g, so that we only obtain a lower bound on the optimal value. See Figure 6.9, that also illustrates
that even in this case we may still have equality of the two optimal values.
Indeed, if there exists a x 2 argminx2K fcx + Q(x)g such that Q(x) = Q (x)
then we have
cx + Q(x) = cx + Q(x) cx + Q (x) cx + Q(x) 8x 2 K;
so that x 2 argminx2K fcx + Q(x)g.
As a matter of fact, the natural approach to solve the constrained optimization problem via convex hulls would be to incorporate the constraints
1
6.4 The convex hull of the expected value function
141
in the objective function, as was already suggested in Section 6.2. That is,
to dene
+ Q(x); if x 2 K ;
f (x) = cx
+1;
otherwise,
and solve the equivalent problem of unconstrained minimization of the convex
hull f instead. However, it is not dicult to see that f is separable only if T
is a multiple of the identity matrix I and K = \m
i=1 fai xi bi g. Therefore,
we can not apply the technique presented above to compute the convex hull of
f in general, since it is based on the very fact that f is separable.
The conclusion is that at the moment we are not able to handle binding
rst-stage constraints.
2