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The Convex Hull Chapter 6 The convex hull The exp ected value function of an integer recourse program is in general non- convex. The lack of convexity precludes the use of many results that are in- disp ensable for eciently solving these mo dels. It is well-known that, given convexity, every lo cal 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 continu- ous recourse programs are based on the convexity of these problems. These considerations justify our interest in convex approximations of the exp ected value function of integer recourse mo dels. Obviously, if the exp ected value function is replaced by such an approximation, the resulting mo del is a convex minimization problem, since the rst-stage ob jective 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 exp ected value function Q, in case the recourse is simple and the technology matrix T is xed. For easy reference, we rep eat its de nition: n 1 ; Q(x) = E v (x; ! ); x 2 R ! + + + v (x; ! ) = inf fq y + q y : y p(! ) T x; y p(! ) T x; m m + 2 2 y 2 Z ; y 2 Z g; + + + 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 m 2 X n 1 Q(x) = Q (x); x 2 R ; i i=1 + + Q (x) = q E dp (! ) T xe + q E bp (! ) T xc ; i ! i i ! i i i i where p (! ) is the ith element of p(! ), and T is the ith row vector of T , i i resp ectively. Therefore, the function Q is completely determined by the one- ^ dimensional 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 = p (! ), z = T x, q = q and i i i ^ q = q , we have Q (x) = Q(z ). i i In Section 6.1 we show that every reasonable convex approximation of an integer simple recourse problem is equivalent to some continuous simple re- course problem. The remainder of this chapter is devoted to nding the convex hull of the exp ected value function. In Chapter 7 we use certain p erturbations 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 exp ected ~ value function Q, given by + + ~ Q(z ) = q E ( z ) + q E ( z ) ; z 2 R ; ^ provides a convex lower b ound as well as a convex upp er b ound for Q, since for all z 2 R + ~ ^ ~ Q(z ) Q(z ) Q(z ) + maxfq ; q g: (6.1) + ~ ^ Obviously, b oth b ounds 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 6.1 Relation to continuous simple recourse 113 + + 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 sharp er convex approximations. Nevertheless, the in- ~ equalities (6.1) and the prop erties of Q indicate that any reasonable convex ap- c c + ^ ^ ~ ^ ~ proximation of Q, say Q , should satisfy Q(z ) Q (z ) Q(z ) + maxfq ; q g c ^ for z 2 R . In particular, it follows that such a Q has asymptotes at 1 and c ~ ^ +1 with the same slop es as the asymptotes of Q. That is, Q has asymptotes + of the form q (d z ) at 1 and q (z d ) at +1, for some d 2 R and 1 2 1 c ^ d 2 R . Moreover, such a Q is non-linear, convex and Lipschitz continuous 2 + ~ with Lipschitz constant maxfq ; q g since Q has these prop erties. The following theorem is the fundamental to ol to establish a relation b e- tween convex approximations of the simple integer recourse exp ected value function and its continuous simple recourse analogue. Theorem 6.1.1 Let v b e a non-linear, convex, Lipschitz continuous function on R . De ne 0 0 a = lim v (z ); a = lim v (z ); 1 2 + + z !1 z !1 0 where v (z ) denotes the right derivative of v at z 2 R . Then the function + V : R 7! [0; 1] de ned by 0 v (s) + a 1 + V (s) = ; s 2 R ; a + a 1 2 is a cdf. If v has an asymptote at +1, say v (z ) a z + c as z ! 1, then 2 2 Z 1 v (z ) = a z + c + (a + a ) 1 V (s) ds; z 2 R : (6.2) 2 2 1 2 z If v has an asymptote at 1, say v (z ) a z + c as z ! 1, then 1 1 Z z V (s) ds; z 2 R : (6.3) v (z ) = a z + c + (a + a ) 1 1 1 2 1 If b oth asymptotes exist, then Z Z 1 z a c + a c 1 2 2 1 ; z 2 R : (6.4) v (z ) = a 1 V (s) ds + a V (s) ds + 1 2 a + a 1 2 z 1 Proof. First we note that b oth a and a exist since v is convex; they are nite 1 2 since v is Lipschitz continuous. By convexity of v we also have that a + a 0, 1 2 and since v is non-linear, even a + a > 0. Hence, V is well-de ned. 1 2 114 The convex hull The function V is non-decreasing since v is convex. Obviously, it is continu- 0 ous from the right since v is. Moreover, lim V (s) = 0 and lim V (s) = s!1 s!1 + 1, so that V is a cdf indeed. It is clear that if the function v has one or two asymptotes, they have to b e of the indicated form. To prove (6.2{6.4) we use that for all 1 < z z^ < 1 Z z^ 0 v (z ^) v (z ) = v (s) ds; + z so that Z z^ a + (a + a )V (s) ds v (z ^) v (z ) = 1 1 2 z Z Z z^ z^ = (a ) 1 V (s) ds + a V (s) ds: (6.5) 1 2 z z If a z + c is the asymptote of v at +1, then 2 2 v (z ^) (a z^ + c ) v (z ) (a z + c ) 2 2 2 2 = v (z ^) v (z ) a (z ^ z) 2 Z z^ = (a + a ) 1 V (s) ds; 1 2 z where the last equality follows from (6.5). Taking z^ ! 1 and replacing z by z , equation (6.2) follows. Similarly, if a z + c is an asymptote of v at 1, then 1 1 v (z ^) (a z^ + c ) v (z ) (a z + c ) 1 1 1 1 = v (z ^) v (z ) + a (z ^ z) 1 Z z^ V (s) ds; = (a + a ) 1 2 z where the last equality follows from (6.5). Takingz ! 1 and replacingz ^ by z , equation (6.3) follows. Finally, if b oth asymptotes exist, then b oth (6.2) and (6.3) hold. Hence, every ane combination of the right-hand sides of these equations is a formula for v (z ) to o. Weighing (6.2) with a =(a + a ) and (6.3) with a =(a + a ) 1 1 2 2 1 2 eliminates the linear term and gives (6.4). 2 Remark 6.1.1 The existence of the asymptotes do es not follow from the as- sumptions on the function v . For instance, the function p 2 v(z ) = 1 + jz j 1 ; z 2 R ; 6.1 Relation to continuous simple recourse 115 is convex and Lipschitz continuous, but is has no asymptotes. In general, the existence of asymptotes of the function v satisfying the conditions of Theo- rem 6.1.1 can b e expressed in terms of its conjugate function v (s) = sup fsz v (z )g ; s 2 R : z 2R If the asymptote at +1 exists then its slop e is a , and 2 lim (v (z ) a z ) = inf fv (z ) a z g 2 2 z !1 z 2R = sup fa z v (z )g 2 z 2R = v (a ): 2 0 The rst equality follows from v (z ) a for all z 2 R . Therefore, the asymp- 2 + tote at +1 exists (and is equal to a z v (a )) if and only if a 2 dom v = 2 2 2 fs 2 R : v (s) < 1g.
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