Feb 8, 1996 - (iv) Q has asymptotes q+(µ â z) as z â ââ and q. â. (z â µ) as z â â. It is easy to give convex lower and upper bounds on g, h and Q in terms ...
On the Convex Hull of the Simple Integer Recourse Objective Function Willem K. Klein Haneveld∗ Leen Stougie† Maarten H. van der Vlerk‡ February 8, 1996
Abstract We consider the objective function of a simple integer recourse problem with fixed technology matrix. Using properties of the expected value function, we prove a relation between the convex hull of this function and the expected value function of a continuous simple recourse program. We present an algorithm to compute the convex hull of the expected value function in case of discrete right-hand side random variables. Allowing for restrictions on the first stage decision variables, this result is then extended to the convex hull of the objective function. Keywords: Simple Integer Recourse, Convex Hull.
1
Introduction
The simple integer recourse model with fixed technology matrix is defined as
min cx + Q(x) : Ax = b, x ∈ IRn+ , x
∗
(1)
Department of Econometrics, University of Groningen, The Netherlands. Institute for Actuarial Sciences and Econometrics, University of Amsterdam, The Netherlands. ‡ Department of Econometrics, University of Groningen, The Netherlands. Supported by the Landelijk Netwerk Mathematische Besliskunde (LNMB). †
1
where the expected value function Q is Q(x) = Eξ v(ξ − T x), and v is the value function of the second stage problem v(s) = min{q + y + + q − y − : y + ≥ s, y y − ≥ −s, y + , y − ∈ ZZm + },
s ∈ IRm .
Here c, A, b, q + , q − and T are vectors/matrices of the appropriate size, q + ≥ 0, q − ≥ 0, and ξ is a random vector in IRm . In contrast to its continuous recourse counterpart, in which the second stage decision variables y = (y + , y − ) are nonnegative reals, the study of the integer problem is of a recent date. The expected value function of the continuous problem is Lipschitz continuous and convex, which makes the problem amenable to standard techniques from mathematical programming (disregarding possible difficulties in evaluating the integrals involved) [1] [6]. The expected value function of the integer problem lacks these properties in general. In [2] [3] and [5] (treating complete integer recourse) it is shown that this function is continuous if the distribution of ξ is absolutely continuous. In [2] [3] it is shown that convexity holds only under very special conditions on the distribution of ξ. Obviously, we could still solve the simple integer recourse problem by fairly standard means if we would be able to formulate the convex hull of the objective function cx+Q(x). By definition, the convex hull of a function f is the (pointwisely) greatest convex function majorized by f . The name indicates that it can be characterized via convex hulls of sets, in the following way (see e.g [4]). The epigraph of a function f : S → [−∞, ∞], S ⊂ IRd , is the set epif = {(x, t) ∈ S × IR|t ≥ f (x)}. The convex hull of a function f is the function f ∗∗ (x) = inf{t|(x, t) ∈ conv(epif )}. In the case of unconstrained infimization of extended real functions, it is not difficult to see that f and f ∗∗ have the same infimal value and that the infimizers of f form a subset of the infimizers of f ∗∗ . Therefore, it is 2
convenient to incorporate the restrictions in (1) in the objective function by defining Q(x) = ∞ if x is not feasible. However, it is clear that possible difficulties in formulating the convex hull of the objective function originate from the expected value function Q. Therefore, the main part of this paper concentrates on this function. Only in Section 4, when we apply the results from the other sections, we again consider the objective function itself and how to handle the restrictions. In Section 3 we prove that the convex hull of the expected value function of a simple integer recourse problem equals (up to a constant) the expected value function of a related continuous simple recourse problem. This is an extension of results in [2] [3]. In Section 4 an algorithm is presented for constructing the convex hull of the expected value function in case the random vector ξ is discretely distributed. Preliminary to these results we expose some relevant properties of the expected value function in Section 2. For proofs we refer to [2] [3]. Throughout the paper we will use the notation (ξ − x)+ = max{0, (ξ − x)}, (ξ − x)− = max{0, −(ξ − x)}, and dξ − xe is the integer roundup of ξ − x.
2
Properties of the expected value function
Since we consider simple recourse, the expected value function Q is separable. Moreover, we have a closed form expression for this function Q(x) =
m X
Qi (x),
i=1
where Qi (x) = qi+ Eξi dξi − Ti xe+ + qi− Eξi dTi x − ξi e+ . Here, ξi is the ith element of the random vector ξ, and Ti is the ith row of the matrix T . Properties of the expected value function follow from properties of its constituents, the functions Eξi dξi −Ti xe+ and Eξi dTi x−ξi e+ . It is convenient to introduce zi = Ti x as a new variable (in the literature zi is known as 3
tender variable). Dropping the index i we define as objects of our study the following nonnegative functions on IR: g(z) = Edξ − ze+ , h(z) = Edz − ξe+ , Q(z) = q + g(z) + q − h(z),
z ∈ IR,
where the expectation is taken over the random variable ξ. In the rest of this paper the term ‘expected value function’ refers to this function Q. Notice that ξ and Q are different from the definitions given above. There should not be any confusion, since in the sequel we only use the one-dimensional meaning. First we give an expression for g and h in terms of left- and right- continuous cumulative distribution functions (cdf) of the random variable ξ. Define Fξ (s) = Pr{ξ ≤ s} and Fˆξ (s) = Pr{ξ < s}, i.e. Fξ is the rightcontinuous cdf of ξ and Fˆξ is the left-continuous cdf of ξ. Then g(z) =
∞ X
(1 − Fξ (z + k)),
∀z ∈ IR,
(2)
k=0
and h(z) =
∞ X
Fˆξ (z − k),
∀z ∈ IR.
(3)
k=0
We have, for all z ∈ IR, g(z) < ∞ if and only if µ+ < ∞, h(z) < ∞ if and only if µ− < ∞, where µ+ = E(ξ)+ and µ− = E(ξ)− . It follows that, for all z ∈ IR, Q(z) = q + g(z) + q − h(z) < ∞ if and only if µ is finite, where µ = µ+ − µ− = Eξ. In the rest of this paper we assume that µ is finite indeed. Using (2) and (3) the following useful relation can be derived. For all z ∈ IR Q(z + 1) = Q(z) + q + (Fξ (z) − 1) + q − Fˆξ (z + 1). 4
(4)
In Section 4 we will use (4) in the following way. Assume that ξ has bounded support Ξ ⊂ [ξ l , ξ u ]. Then Q(z + 1) = Q(z) − q + ,
z ≤ ξ l − 1,
Q(z + 1) = Q(z) + q − ,
z ≥ ξu.
We will say that Q is semiperiodic with period 1 on the intervals (−∞, ξ l ] and [ξ u , ∞), meaning that Q can be decomposed in an affine function and a function which is periodic with period 1 on each of these intervals. The functions g and h are non-increasing and non-decreasing respectively. Since we also have that g is continuous from the right and h is continuous from the left, both functions are lower semicontinuous. This implies that the function Q is lower semicontinuous. The function g is continuous in a point z if and only if ∆g (z) = 0, where ∆g (z) = Pr{ξ ∈ z + ZZ+ }. Similarly, h is continuous in z if and only if ∆h (z) = 0, where ∆h (z) = Pr{ξ ∈ z − ZZ+ }. In case of a discontinuity at z, the jump equals −∆g (z) or ∆h (z) respectively. We see that Q is continuous at z if and only if q + ∆g (z) = q − ∆h (z).
(5)
We proceed with a necessary and sufficient condition for continuity of the function Q on IR. Corollary 2.1 Let ξ be a random variable. Then the function Q is continuous on IR if and only if ξ follows a continuous distribution. Proof. Sufficiency of the condition is trivial. To prove necessity we proceed as follows. We may assume that q + > 0 and q − > 0, since otherwise the result is implied by the results on the functions g and h mentioned above. Since Q is continuous by assumption, by (5) we have for all z ∈ IR q + ∆g (z) = q − ∆h (z), q + ∆g (z + 1) = q − ∆h (z + 1). 5
This implies
q + ∆g (z + 1) − ∆g (z) = q − ∆h (z + 1) − ∆h (z) , so that −q + Pr{ξ = z} = q − Pr{ξ = z + 1}. Since q + , q − > 0, it follows that Pr{ξ = z} = Pr{ξ = z + 1} = 0 for all z ∈ IR. 2 If ξ is a discrete random variable, then F and Fˆ are constant in between mass points. Using (2) and (3), it is easy to see that in this case Q is a discontinuous function which is constant in between discontinuity points. S By (5), the discontinuity points of Q are a subset of ξ i ∈Ξ {ξ i + ZZ}, where Ξ is the support of ξ.
3 3.1
The Convex Hull and Continuous Simple Recourse Continuous Simple Recourse
In the next section the convex hulls of the functions g, h and Q will be related to their analogues in continuous simple recourse models. To this purpose we define, for all z ∈ IR, g(z) = E(ξ − z)+ = h(z) = E(z − ξ)+ =
Z
∞
z
Z
(1 − Fξ (s)) ds,
z
−∞
Fξ (s) ds,
and Q(z) = q + g(z) + q − h(z). Under our assumption that µ is finite, we have that (i) (ii)
g, h and Q are finite, (Lipschitz) continuous and convex functions on IR, g has asymptotes µ − z as z → −∞ and 0 as z → ∞, 6
(iii) h has asymptotes 0 as z → −∞ and z − µ as z → ∞, (iv) Q has asymptotes q + (µ − z) as z → −∞ and q − (z − µ) as z → ∞. It is easy to give convex lower and upper bounds on g, h and Q in terms of g, h and Q, respectively. Indeed, since (ξ − z)+ ≤ dξ − ze+ ≤ (ξ − z)+ + 1 (z − ξ)+ ≤ dz − ξe+ ≤ (z − ξ)+ + 1 we get, for all z ∈ IR, f (z) ≤ f (z) ≤ f (z) + cf ,
f ∈ F,
(6)
where F = {g, h, Q}, cg = ch = 1 and cQ = q + + q − . In fact, the upper bound on Q can be sharpened by replacing q + + q − by max{q + , q − }. In order to see this, note that for any z ∈ IR g(z + 1) =
∞ X
(1 − Fξ (z + k))
k=1 Z ∞
= ≤
Zz ∞ z
(1 − Fξ (z + ds − ze)) ds (1 − Fξ (s)) ds = g(z),
so that g(z) = g(z + 1) + (1 − Fξ (z)) ≤ g(z) + (1 − Fξ (z)). Similarly, h(z − 1) = ≤
∞ X
Fˆξ (z − k) =
k=1 Z z −∞
Fˆξ (s) ds =
Z Z
z
−∞
Fˆξ (z + bs − zc) ds
z
−∞
Fξ (s) ds = h(z),
so that h(z) = h(z − 1) + Fˆξ (z) ≤ h(z) + Fˆξ (z). By combining terms we get Q(z) = q + g(z) + q − h(z) ≤ q + g(z) + q − h(z) + q + (1 − Fξ (z)) + q − Fˆξ (z) ≤ Q(z) + max{q + , q − }, where the last inequality follows from 0 ≤ Fˆξ (z) ≤ Fξ (z) ≤ 1. 7
3.2
The Convex Hull
In this section we consider the convex hull of the function Q. The convex hull of a function f is the largest (lower semicontinuous) convex function majorized by f . We denote the convex hull of a function f by f ∗∗ , since it is the biconjugate function of f (see e.g. [4]). We use f+0 to denote the right derivative of f . Note that if f is a finite convex function on IR the right derivative exists everywhere. First we prove a general result and then apply it to the function Q∗∗ . Similar, but less general results applicable to the functions g∗∗ and h∗∗ can be found in [2] [3]. Theorem 3.1 Let v be a finite, convex, Lipschitz continuous function on IR. Define 0 a1 = − lim v+ (z),
0 a2 = lim v+ (z). z→∞
z→−∞
Assume that a1 + a2 6= 0. Then V (s) =
0 (s) + a v+ 1 a1 + a2
is a cdf. If v has an asymptote for z → ∞, say v(z) ∼ a2 z + c2 as z → ∞, then Z
v(z) = a2 z + c2 + (a1 + a2 )
∞ z
(1 − V (s)) ds.
(7)
If v has an asymptote for z → −∞, say v(z) ∼ −a1 z + c1 as z → −∞, then v(z) = −a1 z + c1 + (a1 + a2 )
Z
z
−∞
V (s) ds.
(8)
If both asymptotes exist, then Z
v(z) = a1
z
∞
(1 − V (s)) ds + a2
Z
z
−∞
V (s) ds +
a1 c2 + a2 c1 . a1 + a2
(9)
Proof. First we note that both a1 and a2 are finite, since v is Lipschitz continuous. By convexity of v we also have that a1 + a2 ≥ 0. The function V is non-decreasing since v is convex. Trivially, it is continuous from the right. Moreover, lims→−∞ V (s) = 0 and lims→∞ V (s) = 1, so that V is a cdf. 8
It is clear that if the asymptotes of the function v exist, they have to be of the indicated form. To prove (7–9) we use that for all −∞ < z ≤ zˆ < ∞ v(ˆ z ) − v(z) = so that v(ˆ z ) − v(z) =
Z z
Z
zˆ
z
0 v+ (s) ds,
zˆ
(−a1 + (a1 + a2 )V (s)) ds Z
= (−a1 )
zˆ z
(1 − V (s)) ds + a2
Z
zˆ
V (s) ds.
(10)
z
If a2 z + c2 is an asymptote of v as z → ∞, then [v(ˆ z ) − (a2 zˆ + c2 )] − [v(z) − (a2 z + c2 )] = v(ˆ z ) − v(z) − a2 (ˆ z − z) = −(a1 + a2 )
Z
zˆ z
(1 − V (s)) ds,
where the last equality follows from (10). Taking zˆ → ∞ and replacing z by z, equation (7) follows. Similarly, if −a1 z + c1 is an asymptote of v as z → −∞, then [v(ˆ z ) − (−a1 zˆ + c1 )] − [v(z) − (−a1 z + c1 )] = v(ˆ z ) − v(z) + a1 (ˆ z − z) Z
= (a1 + a2 )
zˆ
V (s) ds, z
where the last equality follows from (10). Taking z → −∞ and replacing zˆ by z, equation (8) follows. Finally, if both asymptotes exist, then both (7) and (8) hold. Hence, 1 every affine combination of these equations holds. Weighing (7) by a1a+a 2 a2 and (8) by a1 +a2 eliminates the linear term and gives (9). 2 Remark. The existence of the asymptotes does not follow from the assumptions on the function v. Indeed, if the asymptote for z → ∞ exists then its slope is a2 , and lim (v(z) − a2 z) = inf (v(z) − a2 z)
z→∞
z
= − sup(a2 z − v(z)) z
= −v ∗ (a2 ). 9
(q − ) Q∗∗
(−q + )
c2 6 ?
6 c1 ?
µ
6 c ?
Figure 1: The function Q∗∗ , and asymptotes of Q∗∗ and Q. 0 (z) ≤ a , and v ∗ is the conjugate function The first equality follows from v+ 2 of v. Therefore, the asymptote for z → ∞ exists (and is equal to a2 z−v ∗ (a2 )) if and only if a2 ∈ dom v ∗ = {s ∈ IR|v ∗ (s) < ∞}. Similarly, the asymptote for z → −∞ exists (and equals −a1 z − v ∗ (a1 )) if and only if −a1 ∈ dom v ∗ . However, the assumptions on v only imply (−a1 , a2 ) ⊂ dom v ∗ ⊂ [−a1 , a2 ].
We will show that Q∗∗ satisfies the conditions of Theorem 3.1. It is convex by definition and finite on IR, since we assumed that µ is finite. To see that Q∗∗ is Lipschitz continuous, we use (6). Since Q is a convex minorant of Q and Q∗∗ is the largest convex minorant of Q, we have Q(z) ≤ Q∗∗ (z) ≤ Q(z) + max{q + , q − },
∀z ∈ IR.
From Section 3.1 we know that Q has asymptotes l1 = q + (µ−z) as z → −∞ and l2 = q − (z − µ) as z → ∞. This implies that Q∗∗ has asymptotes l1 + c1 as z → −∞ and l2 + c2 as z → ∞, where 0 ≤ ci ≤ max{q + , q − }, i ∈ {1, 2} (see Figure 1). Since the function Q∗∗ is convex, we have for all zˆ, z ∈ IR
|Q∗∗ (ˆ z ) − Q∗∗ (z)| ≤ max{−q + , q − }(ˆ z − z)
= max{−q + , q − } |ˆ z − z| . Hence we showed 10
Corollary 3.1 Let Q∗∗ be the convex hull of the function Q. Then W (s) =
(Q∗∗ )0+ (s) + q + q+ + q−
is a cdf, and Q∗∗ (z) = q +
Z z
∞
(1 − W (s)) ds + q −
Z
z
W (s) ds + c
−∞ +
= q + Eψ (ψ − z)+ + q − Eψ (z − ψ) + c. Here, ψ is any random variable with cdf W , and c=
q + c2 + q − c1 q+ + q−
where c1 = limz→−∞ (Q∗∗ (z) − Q(z)) and c2 = limz→∞ (Q∗∗ (z) − Q(z)).
2
Corollary 3.1 states that the convex hull of the function Q is equal to the function Q plus a constant c after a suitable transformation of the random variable ξ. Of course, taking q + = 1, q − = 0, or q + = 0, q − = 1, gives analogous results for the function g∗∗ and h∗∗ respectively. It is easy to see that in these special cases c = 0. Since continuous simple recourse problems are convex minimization problems, by Corollary 3.1 we can apply standard techniques of mathematical programming to solve the simple integer problem if we are able to find the convex hull of the objective function. This of course amounts to determining the convex hull of the function Q, since then we obtain the convex hull of the objective function by simply adding the linear function cx. However, in general, finding the convex hull of the function Q is not easy. Nevertheless, using some of the specific properties of the functions g and h, it was shown in [2] [3] how to find the convex hull of these functions if the cdf Fξ of ξ belongs to a special class. Of course, this does not give us the convex hull of Q, since in general the convex hull of the weighted sum of two or more non-convex functions is not equal to the weighted sum of their respective convex hulls. In the next section we present an algorithm that determines the convex hull of Q for discrete distributions. 11
4
Construction of the Convex Hull
We present an algorithm to find the convex hull of the expected value function Q in case ξ is a discrete random variable with bounded support and finitely many mass points. Notice that any distribution can be approximated by a distribution belonging to this class. See [5] for stability results with respect to such approximations. The algorithm is based on properties of the function Q, already presented in Section 2. Assume that ξ is a discrete random variable with support Ξ = {ξ 1 , . . . , ξ p } and ξ 1 < ξ 2 < . . . < ξ p . Then (i)
Q is a finite function, with discontinuity points in D∞ , where D∞ =
[
{ξ i + ZZ},
ξ i ∈Ξ
(ii) Q is constant in between discontinuity points, (iii) Q is lower semicontinuous, (iv) Q is semiperiodic with period 1 on the intervals (−∞, ξ 1 ] and [ξ p , ∞). Using (i)–(iii), it follows that the convex hull Q∗∗ is a continuous piecewise linear function with ‘knots’ in the set D∞ , where we define a knot as a point z ∈ IR such that Q∗∗ is not differentiable in z. In addition, (iv) and relation (4) imply that Q∗∗ is affine on the intervals (−∞, ψ 1 ] and [ψ ∗ , ∞), where ψ 1 ∈ [ξ 1 − 1, ξ 1 ] and ψ ∗ ∈ [ξ p , ξ p + 1]. We first give the algorithm and then present a theorem in which the correctness of the algorithm is proved and its complexity is discussed.
12
Algorithm 4.1 (Determine Convex Hull) Input: Ξ = {ξ 1 , . . . , ξ p }, Fξ . Output: Q∗∗ , Ψ, Fψ . [
Step 1. D =
{ξ i + ZZ} ∩ [ξ 1 − 1, ξ p + 1]
ξ i ∈Ξ
D1 = D ∩ [ξ 1 − 1, ξ 1 ] Dp = D ∩ [ξ p , ξ p + 1] Step 2. (Find first ‘knot’.) ψ 1 = max{argmax Q(di ) + q + di } di ∈D1
δ1 = −q
+
Step 3. (Find last ‘knot’.) ψ ∗ = min{argmin Q(di ) − q − di } di ∈Dp
δ∗ = q
−
Step 4. k = 1 D ← D ∩ (ψ 1 , ψ ∗ ] Step 5. (Find all other ‘knots’.) while D 6= ∅ do begin k =k+1 ψ k = max{argmin δk =
Q(di ) − Q(ψ k−1 ) } di − ψ k−1
di ∈D k Q(ψ ) − Q(ψ k−1 )
ψ k − ψ k−1 D ← D ∩ (ψ k , ψ ∗ ] end Step 6. m = k δm+1 = q − Step 7. (Determine cdf of ψ.) for i = 1 to m do Fψ (ψ i ) =
δi+1 + q + q+ + q−
13
Step 8. (Compute constant c.) c1 = Q(ψ 1 ) − q + (µ − ψ 1 ) c2 = Q(ψ m ) + q − (µ − ψ m ) q + c2 + q − c1 c= q+ + q− Step 9. For all z ∈ IR Q∗∗ (z) = q + Eψ (ψ − z)+ + q − Eψ (z − ψ)+ + c, where ψ is a random variable with cdf Fψ . Theorem 4.1 Let ξ be a discrete random variable with support Ξ = {ξ 1 , ξ 2 , . . . , ξ p }, where ξ 1 < ξ 2 < . . . < ξ p and p < ∞. Then Algorithm 4.1 determines (i) the convex hull Q∗∗ of the function Q, (ii) the constant c and the support and cdf of the random variable ψ, where c and ψ satisfy Q∗∗ (z) = q + Eψ (ψ − z)+ + q − Eψ (z − ψ)+ + c,
∀z ∈ IR.
Moreover, it does so in O(p2 r 2 ) time, where p = |Ξ| and r = dξ p − ξ 1 e. Proof. Since Q∗∗ is affine on (−∞, ψ 1 ], it has no knots to the left of ψ 1 . To fix ψ 1 we find di ∈ D1 = D∞ ∩ [ξ 1 − 1, ξ 1 ] such that (di , Q(di )) is closest to the line −q + z, which is parallel to the asymptote of Q∗∗ (see Section 3). In case of a tie, we set ψ1 equal to the largest minimizer. This is done in Step 2 of the algorithm. Note that we minimize over a finite set. In Step 7 of the algorithm it may turn out that ψ 1 is not a real knot, i.e. Q∗∗ is affine on (−∞, ψ k ] with ψ k > ψ 1 . In this case Fψ (ψ 1 ) = 0, meaning that ψ 1 is not in the support of the random variable ψ. By then ψ 1 will have served its purpose, i.e. to constitute a starting point for Step 5 of the algorithm. To find the largest knot ψ ∗ , we proceed in a similar way (Step 3). So far, we have determined the convex hull Q∗∗ on the half-open intervals (−∞, ψ 1 ] and [ψ ∗ , ∞). Assuming that ψ 1 6= ψ ∗ we still have to find the convex hull on the bounded interval (ψ 1 , ψ ∗ ). Since there is only a finite number of candidates left (the set D ← D ∩ (ψ 1 , ψ ∗ ] in Step 4), we can determine the knots (and implicitly Q∗∗ ) on this interval in a straightforward manner. Starting with ψ1 and working from left 14
to right, we iteratively determine the next knot ψ k by maximizing ψ k −ψ k−1 under the condition that no part of the function Q is ‘cut off’ by the line through the points (ψ k−1 , Q(ψ k−1 )) and (ψ k , Q(ψ k )), updating the set D as we go along. This part of the algorithm terminates when D = ∅ (Step 5). It is easy to see that the last knot found in Step 5 is ψ ∗ , so that we have defined Q∗∗ on the whole of IR indeed. Applying Corollary 3.1 we know that the set of knots constructed in the previous steps corresponds to the support of a random variable ψ with a cdf Fψ that is a simple transformation of the right derivative of the convex hull Q∗∗ (Step 7). Given Q∗∗ , computation of the constant c is straightforward (Step 8). To prove the complexity result we proceed as follows. It is clear that the complexity of the algorithm is determined by the computation of Q(di ), ∀di ∈ D, and/or Step 5. The former can be done in O(pr 2 ) time, since a single computation of Q consists of summation of O(r) terms (by (2), (3) and the boundedness of Ξ) and |D| ≤ pr. In each iteration in Step 5 we have to determine the maximum over a discrete set with cardinality at most pr, which can be done in O(pr) time. Since there are at most pr iterations, the result follows. 2 Before we illustrate Theorem 4.1 and Algorithm 4.1 with an example, we make two remarks. First of all, there may be numerical difficulties in computing Q(di ), di ∈ D. In these computations we need to evaluate Fξ and Fˆξ , which calls for comparing two reals which may be almost equal. If for a certain di it holds Q(di + ε) > Q(di ), for ε 6= 0 small, we may mis-specify the convex hull if we compute Q(di ) incorrectly. In the second place, it is not possible to generalize the algorithm so as to find the convex hull of a higher dimensional function, since it depends on the unique ordering of the points di on IR. Example 4.1 Consider the simple integer recourse program min { x + Q(x) : x ≥ α}, x∈IR
(11)
Q(x) = 3Eξ dξ − xe+ + 2Eξ dx − ξe+ , where the random variable ξ has support Ξ = {0.3, 3.4, 4.8} and takes on these values with probability 1/4, 1/3 and 5/12 respectively. The corre15
Figure 2: The function x + Q(x) and the convex hull. sponding continuous simple recourse program is min { x + Q∗∗ (x) : x ≥ α},
x∈IR ∗∗
Q (x) = 3Eψ (ψ − x)+ + 2Eψ (x − ψ)+ + 0.6, where ψ is a random variable with support Ψ = {−.2, .8, 2.8, 3.8, 4.8, 5.3} and respective probabilities .1, .15, .132, .198, .356, .064. Note that Ψ⊂
[
{ξ i + ZZ} ∩ [ξ 1 − 1, ξ 3 + 1].
ξ i ∈Ξ
As can be seen in Figure 2 these programs have the same optimal solution (x = 3.8) and optimal value if α ≤ 3.8. However, if α > 3.8 then the optimal solutions and/or optimal values do not agree (except for α ∈ {4.8} ∪ {5.3 + ZZ+ }). 2 From the example we see that if the optimal solution of the stochastic integer program is not an interior point of the feasible set, the optimal solution obtained by computation of the convex hull of Q may not agree. This shortcoming is clear once we realize that the computation of the convex hull of the objective function restricted to the feasible region is influenced 16
by points outside this region. To avoid this, we define the function (
Qf (x) =
Q(x) ∞
, if x is feasible; , otherwise.
Computation of the convex hull Q∗∗ f of this function requires a simple adaptation of Algorithm 4.1. The only adjustment is that in this case the points ψ 1 and/or ψ ∗ are not determined only by the semiperiodicity argument, but by (i) (ii)
ψ 1 = inf{x : x feasible} ψ ∗ = sup{x : x feasible} If ψ 1 = −∞ then it is set to a finite value by Step 2 of Algorithm 4.1. If ψ ∗ = ∞ then it is set to a finite value by Step 3 of Algorithm 4.1.
The optimal value and optimal solution of the unrestricted convex program min{cx + Q∗∗ f (x)}, x∈IR
are equal to the optimal value and optimal solution of the associated simple integer recourse program. However, in general the function Q∗∗ f does not correspond to the expected value function Q of a related continuous simple recourse program.
References [1] P. Kall. Stochastic Linear Programming. Springer-Verlag, 1976. [2] W.K. Klein Haneveld, L. Stougie, and M.H. van der Vlerk. Stochastic integer programming with simple recourse. Research Memorandum 455, Institute of Economic Research, University of Groningen, 1991. [3] F.V. Louveaux and M.H. van der Vlerk. Stochastic programming with simple integer recourse. Mathematical Programming, To appear. [4] R.T. Rockafellar. Convex Analysis. Princeton University Press, 1970. [5] R. Schultz. Continuity and stability in two-stage stochastic integer programming. In K. Marti, editor, Stochastic Optimization; Numerical Methods and Technical Applications, pages 81–92, Berlin, 1992. GAMM/IFIP Workshop, Springer-Verlag. [6] R. J-B. Wets. Solving stochastic programs with simple recourse. Stochastics, 10:219–242, 1984.
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