On Some Generalizations of the Split Closure - IBM Research People ...
On Some Generalizations of the Split Closure Sanjeeb Dash1 , Oktay G¨ unl¨ uk1 , and Diego A. Mor´an R.2 1
2
IBM Research Georgia Institute of Technology
Abstract. Split cuts form a well-known class of valid inequalities for mixed-integer programming problems (MIP). Cook et al. (1990) showed that the split closure of a rational polyhedron P is again a polyhedron. In this paper, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We also show how this result can be used to prove that some generalizations of split cuts, namely cross cuts, also yield closures that are rational polyhedra.
Keywords: Cross cuts, closure, polyhedrality
1
Introduction
Cutting planes (or cuts, for short) are crucial for solving mixed-integer programs (MIPs), and currently the most effective cuts for general MIPs are the split cuts. In their seminal paper Cook, Kannan and Schrijver [8] studied split cuts which can also be seen as a class of disjunctive cuts that generalize GMI cuts. In [8], Cook et al. showed that the split closure of a rational polyhedron P – that is, the set of points in P satisfying all split cuts for P – is again a polyhedron. This is not a trivial result as one has to consider infinitely many split cuts associated with P . Recently there has been substantial work on generalizing split cuts in different ways to obtain new and more effective classes of cutting planes, and analogues of the polyhedrality of the split closure result have been obtained for some of these classes. Andersen et. al. [3] studied cuts obtained from two dimensional convex lattice-free sets, and Andersen, Louveaux and Weismantel [2] showed that the set of points in a rational polyhedron satisfying all cuts from lattice-free sets with bounded max-facet-width is a polyhedron. Averkov [4] recently gave a short proof of this latter result. Averkov, Wagner and Weismantel [5] showed that the closure with respect to integral lattice-free sets is a polyhedron. In another recent paper, Basu et. al. [7] show that the triangle closure (points satisfying cuts obtained from maximal lattice-free triangles) of a polyhedron in a specific family (the two-row continuous group relaxation) is a polyhedron. As a generalization of split cuts, recently Dash, Dey and G¨ unl¨ uk [10] studied cuts which are obtained by considering two split sets simultaneously. These cuts are called cross cuts and are equivalent to the 2-branch split cuts of Li and Richard [14]. In this paper, we generalize Cook et al’s result from a single rational polyhedron to the union of
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
a finite number of rational polyhedra and use this result to show that the cross cut closure of a rational polyhedron is a polyhedron. We next formally define split sets, split cuts for a given polyhedron (all polyhedra in this paper are assumed to be rational) and the split closure of a polyhedron. Let (π, π0 ) ∈ Zn × Z, then the split set associated with (π, π0 ) is defined to be S(π, π0 ) = {x ∈ Rn : π0 < π T x < π0 + 1}. Clearly, S(π, π0 ) ∩ Zn = ∅ and consequently the integer points contained in a polyhedron P ⊂ Rn are the same as the ones contained in conv(P \ S(π, π0 )) ⊂ Rn . Linear inequalities that are valid for conv(P \ S(π, π0 )) are called split cuts generated by the split set S(π, π0 ). Let S ∗ = {S(π, π0 ) : (π, π0 ) ∈ Zn × Z} denote the collection of all split sets and let S ⊆ S ∗ be given. The split closure of a set A ⊆ Rn , with respect to S is defined as \ SC(A, S) = conv (A \ S) , S∈S n
where for a given set X ⊆ R , we denote its convex hull by conv(X). We refer to SC(A, S ∗ ) as the split closure of A and denote it as SC(A). Cook, Kannan and Schrijver [8] showed that if P is a rational polyhedron, then SC(P ) is also a rational polyhedron. Several other proofs of this result can also be found in [1], [2], [11] and [17]. A crucial step in most of these proofs is b to show that there exists a finite set Sb ⊆ S ∗ such that SC(P, S) = SC(P, S). When such Sb exists, we say that the split closure is finitely generated. For a nonpolyhedral set the split closure is not necessarily polyhedral. However, in some cases it can be finitely generated (see for example [9]). We show the following generalization to a finite union of rational polyhedra: Theorem S 1. Let Pk be rational polyhedra for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any S ⊆ S ∗ . Note that Theorem 1 does not always implies that SC (P, S) is polyhedral as it is easy to see that for P1 = {(0, 0)} and P2 = {x ∈ R2 : x2 = 1} we have SC (P1 ∪ P2 , S ∗ ) = conv(P1 ∪ P2 ) which is not a polyhedron. As a generalization of split cuts, recently Dash, Dey and G¨ unl¨ uk [10] studied cross cuts. Let S1 , S2 ⊆ S ∗ be two collections of split sets. A cross disjunction is a pair (S1 , S2 ), where S1 ∈ S1 , S2 ∈ S2 . The cross closure of a set A ⊆ Rn , with respect to S1 , S2 , is defined as \ CC(A, S1 , S2 ) = conv (A \ (S1 ∪ S2 )) , S1 ∈S1 ,S2 ∈S2
and the cross closure of P is CC(A, S ∗ , S ∗ ), denoted simply by CC(A). In Section 5 we show another generalization of Cook, Kannan and Schrijver’s result, this time to cross cuts: Theorem 2. Let P be a rational polyhedron. Then CC(P ) is a polyhedron.
Generalizations of the Split Closure
2
3
Preliminaries
The two main ingredients that we use in this paper are the so-called GordanDickson Lemma, and, the analysis of intersection points of (closed) split sets and half-lines that have their end point contained in the split set. In [1], Anderson, Cornuejols and Li give an alternate proof of the polyhedrality of the split closure of polyhedra using a new proof technique. We next summarize the relevant results from [1], and state the Gordan-Dickson Lemma. We start with defining the point where a rational half-line H = {v + λr : λ ≥ 0}, where v, r ∈ Qn , intersects for the first time the complement of a split set S ∈ S ∗ that contains the end point v of H. Definition 1 (Intersection point step size). Let v, r ∈ Qn and S ∈ S ∗ such that v ∈ S, then λvr (S) = sup{λ : v + λr ∈ S}. Given a split set S = S(π, π0 ), the step size can be explicitly computed as follows: T π v−π0 πT r < 0 −πT r T v λvr (S) = π0 +1−π πT r > 0 πT r +∞ πT r = 0 Furthermore, notice that if π T r > 0, then the point p = v + λvr (S) r is the point where the half-line H = {v + λr : λ ≥ 0} intersects the hyperplane {x ∈ Rn : π T x = π0 }. If, on the other hand, π T r < 0 then p is the intersection point with the hyperplane {x ∈ Rn : π T x = π0 + 1}. We next review some properties of λvr (S) presented in [1] and [2]. Lemma 1 (Lemma 5 in [1]). Let S ∈ S ∗ and H = {v + λr : λ ≥ 0} where v, r ∈ Qn and v ∈ S. If λv,r (S) < +∞, then λvr (S) < min{z ∈ Z+ : zr ∈ Zn }. Lemma 2 (Lemma 6 in [1]). Let λ∗ > 0 and H = {v + λr : λ ≥ 0} where v, r ∈ Qn . Then there exists a finite set Λ ∈ R such that for all S ∈ S ∗ , λvr (S) ∈ Λ provided that ∞ > λvr (S) > λ∗ and v ∈ S. For a rational polyhedron P , we denote by V (P ) ⊆ Qn its set of vertices and by E(P ) ⊆ Qn its set of extreme rays. When V (P ) 6= ∅, we say that the polyhedron is pointed. Definition 2 (Relevant directions). For a vertex v ∈ V (P ), we define Dv (P ) = r ∈ E(P ) : {v + λr : λ ∈ R+ } is a 1-dimensional face of P ∪ v 0 − v : v 0 ∈ V (P ), and conv(v, v 0 ) is a 1-dimensional face of P to denote the set of relevant directions for the vertex v.
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
Observe that for v ∈ V (P ), the relevant directions are the extreme rays of the radial cone at the vertex v in the polyhedron P . The following result is originally presented in [1] for conic polyhedra and later generalized by Andersen, Louveaux, and Weismantel [2] to general polyhedra. Lemma 3 (Lemmas 2.3, 2.4, 4.2 in [1, 2]). Let P be a pointed rational polyhedron and let S ∈ S. If P \ S 6= ∅, then (1) conv (P \ S) is a rational polyhedron. (2) The extreme rays of conv (P \ S) are the same as the extreme rays of P . (3) If u is a vertex of of conv (P \ S), then either u ∈ V (P ) \ S, or, u = v + λvr (S) r, where v ∈ V (P ) ∩ S and r ∈ Dv (P ) satisfies one of the following: (i) r ∈ E(P ) and λvr (S) < +∞, or, (ii) r = v 0 − v for some v 0 ∈ V (P ) \ S such that conv(v, v 0 ) is an edge of P and λvr (S) < 1. Finally we state a very simple and useful lemma that shows that for any positive integer p, every set of p-tuples of natural numbers has finitely many minimal elements. Lemma 4 (Gordan-Dickson Lemma). Let X ⊆ Zp+ . Then there exists a finite set Y ⊆ X such that for every x ∈ X there exists y ∈ Y satisfying x ≥ y.
3
Split Closure of a Finite Collection of Polyhedral Sets
In this section, we show that given a finite collection of rational polyhedra, there exists a finite set of splits that define the split closure. We start with a simple observation based on Lemma 3. Corollary 1. Let P be a rational pointed polyhedron and S1 , S2 ∈ S ∗ such that V (P ) ∩ S1 = V (P ) ∩ S2 . If λvr (S1 ) ≥ λvr (S2 ), for all v ∈ V (P ) ∩ S1 and r ∈ Dv (P ),
(1)
then conv (P \ S1 ) ⊆ conv (P \ S2 ) . Proof. The claim clearly holds when conv (P \ S1 ) = ∅ and therefore we assume that conv (P \ S1 ) 6= ∅. Notice that by Lemma 3 conv(P \ S1 ) and conv(P \ S2 ) are polyhedral and have the same recession cone. Moreover, (1) implies that the vertices of conv(P \ S1 ) belong to conv(P \ S2 ). Therefore, conv (P \ S1 ) ⊆ conv (P \ S2 ) . Using Lemmas 1 and 2 we obtain the following result. Lemma 5. Let v, r ∈ Qn . There exists a function xvr : S ∗ → Z+ such that whenever v ∈ S1 , S2 ∈ S ∗ , we have, xvr (S1 ) ≤ xvr (S2 ) ⇔ λvr (S1 ) ≥ λvr (S2 ).
(2)
Generalizations of the Split Closure
5
Proof. Let Λ = {λvr (S) : v ∈ S and λvr (S) < +∞} and Mr = min{z ∈ Z+ : zr ∈ Zn }. Define ( 0, λvr (S) = +∞ xvr (S) = |Λ ∩ [λvr (S), Mr ]|, λvr (S) < +∞. Notice that xvr (S) is well defined for all S ∈ S ∗ as λvr (S) < Mr by Lemma 1 and |Λ ∩ [λvr (S), Mr ]| < +∞ by Lemma 2. When λvr (S1 ) = +∞ we have xvr (S1 ) = 0 and the equivalence (2) clearly holds. If, on the other hand, λvr (S1 ) < +∞, we obtain that λvr (S1 ) ≥ λvr (S2 ) is equivalent to xvr (S1 ) ≤ xvr (S2 ), since it is easy to see that the latter occurs if and only if |Λ ∩ [λvr (S1 ), Mr ]| ≤ |Λ ∩ [λvr (S2 ), Mr ]|. This observation together with the Gordan-Dickson Lemma (Lemma 4) can be used to show that the split closure of a polyhedron is again a polyhedron. In [4], Averkov uses a similar argument to show the polyhedrality of more general closures that include the split closure. We next use Lemma 5 for split closures of unions of polyhedra. Consider a finite collectionSof pointed rational polyhedra Pk , k ∈ K, where K is a finite set. For V 0 ⊆ k∈K V (Pk ) we denote S(V 0 ) = {S ∈ S : V 0 = S k∈K V (Pk ) ∩ S}. Proposition 1. Let S ⊆ S ∗ and {Pk }k∈K be a finite collection of pointed rational polyhedra. Then, there exists a finite set SY ⊆ S such that for all S1 ∈ S there exists S2 ∈ SY such that conv (Pk \ S2 ) ⊆ conv (Pk \ S1 ) for all k ∈ K. S Proof. Notice that sets S(V 0 ) for V 0 ⊆ k∈K V (Pk ) form a finite partition of S, that is, S(V 0 ) ∩ S(V 00 ) = ∅ if V 0 6= V 00 , and [ S(V 0 ). S= S V 0 ⊆ k∈K V (Pk ) 0 0 Consequently, S it suffices to show the existence of finite sets SY0 (V ) ⊆ S(V ) for 0 each V ⊆ k∈K V (Pk ) that satisfy the claimSwhen S1 ∈ S(V ). We now consider an arbitrary set V 0 ⊆ k∈K V (Pk ). If V 0 = ∅, then it is easy to see that for all S ∈ S(V 0 ) we have conv(P \ S) = P . Thus, it is sufficient to take SY = {S}, where S ∈ S(V 0 ). If V 0 6= ∅, let X X p= |Dv (Pk )|. k∈K v∈V 0 ∩V (Pk )
For each S ∈ S(V 0 ) we now define a p-tuple t(S), where for each k ∈ K, v ∈ V 0 ∩ V (Pk ), and r ∈ Dv (Pk ), the tuple has a unique entry that equals xvr (S) (Lemma 5). Collection of these p-tuples gives the following set contained in Zp+ : n o X = t(S) : S ∈ S(V 0 ) .
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
By Lemma 4, there exists a finite set Y ⊆ X such that for every x ∈ X there exists y ∈ Y satisfying x ≥ y. In particular, there exists a finite set SY (V 0 ) ⊆ S(V 0 ) such that for any S1 ∈ S(V 0 ) there exists S2 ∈ SY (V 0 ) satisfying xvr (S2 ) ≤ xvr (S1 ) for all k ∈ K, v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ).
(3)
By Lemma 5, the above inequality implies that λvr (S2 ) ≥ λvr (S1 ) for all k ∈ K, v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ).
(4)
As both S1 , S2 ∈ S(V 0 ), we have V (Pk ) ∩ S2 = V (Pk ) ∩ S1 for all k ∈ K and applying Corollary 1 we conclude that conv (Pk \ S2 ) ⊆ conv (Pk \ S1 ) for all k ∈ K. To conclude the proof it suffices to let [ SY = SY (V 0 ) S V 0 ⊆ k∈K V (Pk )
Lemma 6. Let P Sk be a rational pointed polyhedron for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any S ⊆ S ∗ . More precisely, \ SC (P, S) = conv (P \ S) S∈Sˆ
where Sˆ ⊂ S is a finite set. S S Proof. First note that for any S ∈ S we have: conv ( P )\S = conv k∈K (Pk \ 2 k 2 k∈K S S2 ) = conv k∈K conv(Pk \ S2 ) . Furthermore, by Proposition 1, there is a finite set SY ⊂ S such that for each S1 ∈ S there exists S2 ∈ SY that satisfies [ [ [ conv conv(Pk \ S2 ) ⊆ conv conv(Pk \ S1 ) = conv Pk \ S 1 . k∈K
k∈K
k∈K
As SY is finite, to complete the proof, it suffices to observe that [ [ \ \ SC(P, S) = conv Pk \ S = conv Pk \ S . S∈S
k∈K
S∈SY
k∈K
We next relax the assumption of pointedness in the previous result; due to space restrictions we only give a sketch of the proof below. Theorem 1.SLet Pk be a rational polyhedron for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any S ⊆ S ∗ . Proof. (sketch) We first observe that a general rational polyhedron P can be written as P = Q + L, where L is a rational lineal subspace and Q ⊆ L⊥ is a rational pointed polyhedron. It can be shown that the split closure of P only depends on splits sets S such that L is contained in the lineality space of S. The projection of such splits onto L⊥ can be seen as splits sets in L⊥ . Therefore, the split closure of P only depends on the split closure of the rational pointed polyhedron Q in the subspace L⊥ , and, as a consequence, it suffices to prove the result for the rational pointed polyhedral case.
Generalizations of the Split Closure
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7
Split Closure of a Union of Mixed-integer Sets
Consider a mixed-integer set defined by a polyhedron P LP ∈ Rn+l and the mixed-integer lattice Zn × Rl where n and l are positive integers: P I = P LP ∩ (Zn × Rl )
(5)
An inequality is called a split cut for P LP with respect to the lattice Zn × Rl if ∗ where it is valid for conv(P LP \ S) for some S ∈ Sn,l ∗ Sn,l = {S(π, π0 ) ∈ S ∗ : π ∈ Zn × {0}l }.
The split closure is then defined in the usual way as the intersection of all such split cuts. A straightforward extension of Theorem 1 is the following: Corollary 2. Let Pk ∈ S Rn+l be a rational polyhedron for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any ∗ S ⊆ Sn,l .
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Cross Closure of a Polyhedral Set
In this section, we show that the cross closure of a rational polyhedron is again a polyhedron. We give the complete proof for the pointed case (see Lemma 14 below). The extension to the general case can be done using Lemma 14 in the same fashion as explained at the beginning of Section 3 for the split closure. We will present the proof of this case in the journal version of the paper. We combine the proof technique of Cook, Kannan and Schrijver [8] for showing that the split closure of a polyhedron is polyhedral along with the results we derived in earlier sections based on proof techniques of Anderson, Cornu´ejols, Li [1], and Averkov [4]. We need some definitions to discuss the overall techniques used. Lets denote by k · k the usual euclidean norm. Define the width of a split set S(π, π0 ) as w(S(π, π0 )) = 1/kπk (this is the geometric distance between the parallel hyperplanes bounding the split set). Then w(S(π, π0 )) > η for some η > 0 implies that kπk < 1/η. Therefore, for any fixed η > 0 and π0 ∈ Zn , there are only a finite number of π ∈ Zn such that w(S(π, π0 )) > η. Cook, Kannan, Schrijver (roughly) prove their polyhedrality result using the following idea. Assume P is a polyhedron, L is a finite list of split sets and let SC(P, L) = ∩S∈L conv(P \S). Suppose that for every face F of P , SC(P, L)∩F ⊆ SC(F ). Then (i) there are only finitely many split sets beyond the ones contained in L which yield split cuts cutting off points of SC(P, L) (they show that if S(π, π0 ) is such a split set, then π must have bounded norm). Therefore, (ii) if one assumes (by induction on dimension) that the number of split sets needed to define the split closure of each face of a polyhedron is finite, then so is the number of split sets needed to define the split closure of the polyhedron. Santanu Dey [12] observed that idea (i) in the Cook, Kannan, Schrijver proof technique can also be used in the case of some disjunctive cuts which generalize
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
split cuts. We apply a modification of idea (i) to cross cuts; namely we show in Lemma 13 that if L is a finite list of cross disjunctions (represented as a pair of split sets) such that CC(P, L) = ∩(S1 ,S2 )∈L conv(P \ (S1 ∪ S2 ))
(6)
intersected with each face of P is equal to the cross closure of each face, then cross disjunctions (S1 , S2 ) where both w(S1 ) and w(S2 ) are at most some η > 0 can only yield cross cuts valid for CC(P, L), and are therefore not needed to define the cross closure of P . We then only need to consider cross disjunctions (S1 , S2 ) where one of w(S1 ), w(S2 ) is greater than η (such cross disjunctions are still infinitely many in number). We first need a generalization of Lemma 3, property (2.). Let rec(P ) denote the recession cone of P , aff(P ) denote the affine hull of P , and P I denote the integer hull of P . Lemma 7. Let P be a polyhedron in Rn , and let S1 , S2 ∈ S ∗ be any two split sets. If conv(P \ (S1 ∪ S2 )) is nonempty, then its recession cone equals rec(P ). Proof. Let P 0 = conv(P \ (S1 ∪ S2 )). As P 0 ⊆ P and P is closed, we obtain rec(P 0 ) ⊆ rec(P ). To prove the reverse inclusion, let v be a point in P 0 , and let r ∈ rec(P ) 6= ∅. Let v1 , v2 be two points in P \ (S1 ∪ S2 ) such that v is a convex combination of v1 , v2 (we will choose v1 = v2 = v if v 6∈ S1 ∪ S2 ). Consider the half lines H1 = {v1 + λr : λ ≥ 0} and H2 = {v2 + λr : λ ≥ 0}. As v1 6∈ S1 ∪ S2 , either H1 does not intersect S1 ∪ S2 , or else sup{λ ≥ 0 : v1 + λr ∈ S1 ∪ S2 } is finite. In either case, the half line H1 ⊆ P 0 , and similarly H2 ⊆ P 0 and therefore conv(H1 ∪ H2 ) ⊆ P 0 . But then the half line {v + λr : λ ≥ 0} ⊆ P 0 and therefore r ∈ rec(P 0 ) =⇒ rec(P ) ⊆ rec(P 0 ). Now we generalize Lemma 3, property (1.). For a set A ⊆ Rn , let conv(A) denote the topological closure of conv(A). The following result is a direct consequence of Theorem 3.5 in [13]. Lemma 8 ([13]). Let A ⊆ Rn be a nonempty closed set. Then every vertex of conv(A) belongs to A. Lemma 9. Let P be a pointed polyhedron, and let S1 , S2 ∈ S ∗ . Then conv (P \ (S1 ∪ S2 )) is a polyhedron. Proof. The claim trivially holds if conv(P \ (S1 ∪ S2 )) is empty. Therefore we assume that conv(P \ (S1 ∪ S2 )) is nonempty. Since conv (P \ (S1 ∪ S2 )) is a closed convex set not containing a line, by a standard result in convex analysis (see for example Theorem 18.5 in [15]) we have that conv (P \ (S1 ∪ S2 )) can be written as the Minkowski sum of its set of vertices and its recession cone. By Lemma 8 we have that the vertices of conv (P \ (S1 ∪ S2 )) belong to P \ (S1 ∪ S2 ). Thus, since P \ (S1 ∪ S2 ) is a finite union of polyhedra, we obtain that conv (P \ (S1 ∪ S2 )) has a finite number of vertices.
Generalizations of the Split Closure
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On the other hand, by Lemma 7 we obtain that rec(conv (P \ (S1 ∪ S2 ))) = rec(conv (P \ (S1 ∪ S2 ))) = rec(P ). This implies that conv (P \ (S1 ∪ S2 )) is a polyhedron. Moreover, since all its vertices belong to P \(S1 ∪S2 ) ⊆ conv (P \ (S1 ∪ S2 )), we conclude that conv (P \ (S1 ∪ S2 )) = conv (P \ (S1 ∪ S2 )). Therefore, conv (P \ (S1 ∪ S2 )) is a polyhedron. Observe that Lemma 7 and Lemma 9 implies that CC(P, L) is a polyhedron with rec(CC(P, L)) = rec(P ) if CC(P, L) is defined as in (6). The proof of the following lemma is ommited. Lemma 10. Let P be a polyhedron and let F be a face of P . For any set B, conv(P \ B) ∩ F = conv(F \ B). The next result is essentially contained in Cook, Kannan and Schrijver [8], though our statement and proof are slightly different. Lemma 11. Let P and P 0 be pointed, full-dimensional polyhedra in Rn with P ⊂ P 0 . Then there exists a number r > 0 such that for any c ∈ Rn satisfying (i) max{cT x : x ∈ P } = d < max{cT x : x ∈ P 0 } < ∞ and (ii) the first maximum is attained at a vertex of P contained in the interior of P 0 , there exists a ball of radius r in P 0 with each point x in the ball satisfies cT x > d. Proof. Let P 0 = {x : aTi x ≤ bi for i = 1, . . . , m} where ai ∈ Rn and let η = maxi {||ai ||}. Let V = {v1 , . . . , vk } be the set of vertices of P contained in the interior of P 0 , and let = mini,j {bi − aTi vj } > 0. Let c satisfy the conditions of the lemma and let cT vj = d for some vertex vj ∈ V . Further, let max{cT x : 0 x ∈ P 0 } = d0 < ∞ be attained at a vertex v 0 of PP . By LP duality, there Pm exists m multipliersP0 ≤ λ = (λ1 , . . . , λm ) such that c = i=1 λai andP d0 = i=1 λi bi m m ¯ ¯ λ) ¯ = (c, d0 , λ)/τ . Then c¯ = and c, d, i=1 λi > 0. Let (¯ i=1 λai where Pm τ ¯= λ = 1 and therefore ||¯ c || ≤ η. Further i i=1 d¯ − c¯T vj =
m X
¯ i (bi − aT vj ) ≥ . λ i
i=1
By definition, max{¯ cT x : x ∈ P } is attained at vj , max{¯ cT x : x ∈ P 0 } is attained at v 0 , and the distance between the hyperplanes c¯T x = c¯T vj and c¯T x = d¯ is at least /||¯ c|| > /(η +1). Therefore, any point z in the ball B(v 0 , /(η +1)) satisfies c¯T z > c¯T vj (and also cT z > cT vj ). We can find an r > 0 such that B(v, /(η+1)) contains a ball of radius r for each vertex v of P 0 . In the proof above, we can assume that we construct a fixed set B of balls, one per each vertex of P 0 , such that one of these balls satisfies the desired property in Lemma 11. A strip in Rn is the set of points between a pair of parallel hyperplanes (and including the hyperplanes) and the width of a strip is the distance between its
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
bounding hyperplanes. Thus the topological closure of a split set is a strip. If the minimum width of a closed, compact, convex set A is defined as the minimum width of a strip containing A, then it is known (Bang [6]) that the sum of widths of a collection of strips containing A must exceed its minimum width. The following statement is a trivial consequence of Bang’s result. Lemma 12. Let B be a ball of radius r > 0, and let S1 , S2 be split sets such that B ⊆ (S1 ∪ S2 ). Then, w(S1 ) + w(S2 ) ≥ 2r. Lemma 13. Let P be a pointed, full-dimensional polyhedron. Let L ⊆ S ∗ ×S ∗ be a finite list of cross disjunctions. Let CC(P, L) be defined as in (6). If CC(P, L)∩ F ⊆ CC(F ) for all faces F of P , then there exists η > 0 such that all cross cuts obtained from cross disjunctions (S1 , S2 ) ∈ S ∗ × S ∗ with w(S1 ), w(S2 ) ≤ η are valid for CC(P, L). Proof. As P is pointed, CC(P, L) is also pointed as it is contained in P . Let r > 0 be the number given by Lemma 11 when applied to P and CC(P, L). Let 0 < η < r. Let cT x ≤ γ be a cross cut obtained from a cross disjunction (S1 , S2 ) with w(S1 ), w(S2 ) ≤ η. We will prove that cT x ≤ γ is valid for CC(P, L). As γ ≥ max{cT x : x ∈ conv(P \ (S1 ∪ S2 ))} < ∞ and since by Lemma 7 we have rec(CC(P, L)) = rec(P ) = rec(conv(P \ (S1 ∪ S2 ))), we obtain that d := max{cT x : x ∈ CC(P, L)} < ∞. We have two cases. Case 1: The maximum is attained in a face F of P . Then, since CC(P, L) ∩ F ⊆ CC(F ) ⊆ {x ∈ Rn : cT x ≤ γ}, we infer that d ≤ γ. Therefore cT x ≤ γ is valid for CC(P, L). Case 2: The maximum is attained in a vertex of CC(P, L) in the interior of P . This implies that d < max{cT x : x ∈ P }. By Lemma 11, there exists a ball B of radius r in P with all points in the ball satisying cT x > d. Since w(S1 )+w(S2 ) < 2r, Lemma 12 implies that B \ (S1 ∪ S2 ) 6= ∅. Let x ¯ ∈ B \ (S1 ∪ S2 ). Then x ¯ ∈ conv(P \(S1 ∪S2 )) with cT x ¯ > d. Since cT x ≤ γ is valid for conv(P \(S1 ∪S2 )), it follows that d < γ. Therefore cT x ≤ γ is valid for CC(P, L). The discussion after Lemma 11 implies that the ball B in the proof above can be assumed to be a member of B. Further, the proof implies that even if w(S1 ), w(S2 ) ≥ r, if B \ (S1 ∪ S2 ) 6= ∅, then (assuming P, L satisfy the conditions of the Lemma) cross cuts from the disjunction (S1 , S2 ) are valid for CC(P, L). We will need the following basic properties of unimodular matrices (matrices with determinant ±1). If V is a rational affine subspace of Rn with dimension k < n such that V ∩ Zn 6= ∅, then there is a n × n integral unimodular matrix U and vector v ∈ Zn such that the one-to-one mapping σ(x) = U x + v maps V to Rk × {0}n−k ; also split sets are mapped to split sets. Further, the intersection of any split set S in Rn with Rk × {0}n−k is either empty, Rk × {0}n−k or equals S 0 × {0}n−k where S 0 is a split set in Rk .
Generalizations of the Split Closure
11
Lemma 14. Let P be a rational pointed polyhedron. Then CC(P ) is a polyhedron. More precisely, CC(P ) =
\
conv (P \ (S1 ∪ S2 ))
(S1 ,S2 )∈L∗
where L∗ ⊂ S ∗ × S ∗ is a finite set. Proof. We use standard techniques to show that P can be assumed to be fulldimensional. Assume P has dimension k < n. As aff(P ) is a rational, affine subspace of Rn , if aff(P ) ∩ Zn = ∅, then it is well-known (see [16, Corollary 4.1a]) that P ⊆ aff(P ) ⊆ S for some split set S. Then CC(P ) ⊆ SC(P ) ⊆ conv(P \ S) = ∅ = P I . Therefore we assume aff(P ) ∩ Zn 6= ∅. There is a function σ(x) (as discussed before the theorem) that maps aff(P ) to a polyhedron in Rk × {0}n−k , i.e., to P 0 × {0}n−k , where P 0 is a full-dimensional polyhedron. Let L0 be a subset (not necessarily finite) of all cross disjunctions in Rk which yield the cross closure of P 0 ; Then for each cross disjunction (or split set pair) (S10 , S20 ) in L0 , if we define a cross set (S1 , S2 ) in Rn as (S10 × {0}n−k , S20 × {0}n−k ), and then apply the inverse function σ −1 (x) (of σ(x)) to (S1 , S2 ), we get a list of cross disjunctions in Rn which yield the cross closure of P . Therefore we can work on P 0 . The proof is by induction on dim(P ). The case dim(P ) = 0 is straightforward. Now, lets assume that for all polyhedra Q of dimension strictly less than dim(P ), CC(Q) is defined by a finite number of cross disjunctions. Let F be a face of P . Since dim(F ) < dim(P ), by the induction hypothesis (and the argument in the previous paragraph) we infer that there exists a finite set of cross disjunctions L(F ) in Rn such that CC(F ) = CC(F, L(F )). Define L = ∪F is a face of P L(F ). By the induction hypothesis, L is a finite list of cross disjunctions. Also, for any face F of P , CC(P, L) ⊆ CC(P, L(F )) and therefore CC(P, L) ∩ F = CC(F, L) ⊆ CC(F, L(F )) = CC(F ), where the first equality follows from Lemma 10. Therefore, Lemma 13 implies the existence of a number η > 0 such that all cross cuts obtained from cross disjunctions (S1 , S2 ) with w(S1 ), w(S2 ) ≤ η are valid for PL . This implies that to define the cross closure of P , it suffices to consider cross disjunctions with (S1 , S2 ) with either w(S1 ) > η or w(S2 ) > η. Further, the discussion after Lemma 13 implies that one of S1 , S2 satisfies the property: w(S) > η and S ∩ B 6= ∅ for a ball B ∈ B defined after Lemma 11. Define Sη = {S ∈ S ∗ : w(S) > η and S ∩ (∪B B) 6= ∅} and observe that it is a finite set. For S ∈ Sη the set P \ S is a union of two pointed rational polyhedra (possibly empty). Therefore, Theorem 1 implies that SC(P \ S) = SC(P \ S, S ∗ ) is finitely
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
generated. On the other hand, we have \ CC(P ) = conv (P \ (S1 ∪ S2 )) S1 ∈S ∗ ,S2 ∈S ∗
\
=
conv (P \ (S1 ∪ S2 )) ∩
conv (P \ (S1 ∪ S2 ))
S1 ∈Sη ,S2 ∈S ∗
(S1 ,S2 )∈L
= CC(P, L) ∩
\
\
SC(P \ S, S ∗ ).
S∈Sη
Therefore, we conclude that CC(P ) is finitely generated and, by Lemma 9, is a polyhedron.
References 1. Kent Andersen, G´erard Cornu´ejols, and Yanjun Li, Split closure and intersection cuts, Mathematical Programming 102 (2005), no. 3, 457–493. 2. Kent Andersen, Quentin Louveaux, and Robert Weismantel, An analysis of mixed integer linear sets based on lattice point free convex sets, Math. Oper. Res. 35 (2010), no. 1, 233–256. 3. Kent Andersen, Quentin Louveaux, Robert Weismantel, and Lawrence Wolsey, Inequalities from two rows of a simplex tableau, IPCO proceedings, Lecture Notes in Computer Science, vol. 4513, 2007, pp. 1–15. 4. Gennadiy Averkov, On finitely generated closures in the theory of cutting planes, Discrete Optimization 9 (2012), no. 1, 209–215. 5. Gennadiy Averkov, Christian Wagner, and Robert Weismantel, Maximal latticefree polyhedra: finiteness and an explicit description in dimension three, Mathematics of Operations Research 36 (2011), no. 4, 721–742. 6. Thoger Bang, A solution of the plank problem, Proc. American Mathematical Society 2 (1951), no. 6, 990–993. 7. Amitabh Basu, Robert Hildebrand, and Matthias K¨ oppe, The triangle closure is a polyhedron, manuscript, 2011. 8. William J. Cook, Ravi Kannan, and Alexander Schrijver, Chv´ atal closures for mixed integer programming problems, Math. Program. 47 (1990), 155–174. 9. Daniel Dadush, Santanu S. Dey, and Juan Pablo Vielma, The split closure of a strictly convex body, Oper. Res. Lett. 39 (2011), no. 2, 121–126. 10. Sanjeeb Dash, Santanu Dey, and Oktay G¨ unl¨ uk, Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra, Math. Program. 135 (2012), 221–254. 11. Sanjeeb Dash, Oktay G¨ unl¨ uk, and Andrea Lodi, MIR closures of polyhedral sets, Math. Program. 121 (2010), no. 1, 33–60. 12. Santanu S. Dey, Personal communication, (2010). 13. V. L. Klee, Extremal structure of convex sets, Archiv der Mathematik 8 (1957), 234– 240. 14. Yanyun Li and J. P. P. Richard, Cook, kannan and schrijver’s example revisited, Discrete Optimization 5 (2008), 724–734. 15. G. T. Rockafeller, Convex analysis, Princeton University Press, New Jersey, NJ, 1970.
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16. Alexander Schrijver, Theory of linear and integer programming, John Wiley and Sons, New York, 1986. 17. Juan Pablo Vielma, A constructive characterization of the split closure of a mixed integer linear program, Oper. Res. Lett. 35 (2007), no. 1, 29–35.
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A
Appendix: Proof of Theorem 1 in the general case
A.1
On unimodular matrices and split sets
We next review some basic properties of unimodular matrices, i.e., square matrices with determinant ±1. If U is an n × n unimodular matrix, and v ∈ Rn , the affine transformation σ(x) = U x + v is a one-to-one, invertible, mapping of Rn to Rn with σ −1 (x) = U −1 (x − v) and this transformation preserves volumes. If U is an integral unimodular matrix, then so is U −1 ; if in addition v ∈ Zn , then the function σ(x) is a one-to-one, invertible, mapping of Zn to Zn . Further, if a ∈ Zn , b ∈ Z, the set {x ∈ Rn : aT x = b} is mapped to the set {x0 ∈ Rn : aT U −1 (x0 − v) = b} ≡ {x0 ∈ Rn : aT U −1 x0 = b + aT U −1 v} and aT U −1 ∈ Zn . Therefore, given a split set S(a, b), σ(S(a, b)) and σ −1 (S(a, b)) are both split sets. If a ∈ Zn and the g.c.d. of the coefficients of a is one, then there is a unimodular matrix U such that aT U = (0, . . . , 0, 1) and aT x = b has an integral solution for any integer b, say v b (see [16, Corollary 4.1c]). Note that the previous statement implies that aT is the last row of U −1 . Then, under the linear transformation x → U x (with inverse transformation x → U −1 x + v b ), there is a one-to-one mapping of {x ∈ Rn : aT x = b} to the set {x ∈ Rn : xn = b}, and of the integer points in the respective sets. Further, any linear subspace L of {x : aT x = 0} is mapped to a subspace of {x ∈ Rn : xn = 0}, and therefore the intersection of a split set and L⊥ is a split set. We summarize this result in the next lemma. The lineality space of a split set S(π, π0 ), where (π, π0 ) ∈ Zn , is denoted ls(S) := {x ∈ Rn : π T x = 0}. We also denote ProjL⊥ (S) := S ∩ L⊥ . Lemma 15. Let L be a rational subspace such that L ⊆ ls(S). Then ProjL⊥ (S) is a split set of L⊥ . A.2
Proof of Theorem 1
Let L ⊆ Rn be a rational linear subspace and let P be a pointed polyhedron such that P ⊆ L⊥ . Then a general polyhedron is a polyhedron of the form Q = P +L. Lemma 16. Let L be a rational linear subspace, P ⊆ L⊥ be a pointed polyhedra, and let Q = P + L. Let S be a split set. Then 1. If L * ls(S), then conv(Q \ S) = Q. 2. If L ⊆ ls(S), then conv(Q \ S) = conv(P \ ProjL⊥ (S)) + L. Proof. 1. Let x ∈ Q and l ∈ L such that π T l 6= 0. Since π T l 6= 0, we obtain that the line x + {λl : λ ∈ R} is contained in conv(Q \ S). Therefore, x ∈ conv(Q \ S). 2. Since L ⊆ ls(S), we can write S = ProjL⊥ (S) + L. Therefore, we can write conv(Q \ S) = conv((P + L) \ (ProjL⊥ (S) + L)) = conv(P \ ProjL⊥ (S)) + L, where the last equality is given by the fact that P, ProjL⊥ (S) ⊆ L⊥ .
Generalizations of the Split Closure
15
By Lemma 16 applied to Q and by Lemma 3 applied to P , we conclude that conv(Q \ S) is characterized in terms of intersections points defined by the split set ProjL⊥ (S) ⊆ L⊥ and the pointed polyhedron P ⊆ L⊥ . These facts allow us to applied in the general case the same proof techniques as before. The crucial result in Section 3 is Proposition 1. We extend this proposition to the general case next. We begin with some notation. Consider a finite collection of pointed rational polyhedra for k ∈ K Qk = Pk + Lk ,
(7)
where Lk are rational linear subspaces, Pk = conv (V (Pk )) + cone (E(Pk )) are pointed polyhedra with V (Pk ), E(Pk ) ⊆ Qn denoting the corresponding set of S vertices and extreme rays of polyhedron Pk . For K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) we denote S(K0 , V 0 ) = {S ∈ S : V 0 =
[ k∈K0
[V (Pk )∩ProjL⊥ (S)], Lk ⊆ ls(S) ∀ k ∈ K0 , Lk * ls(S) ∀ k ∈ / K0 }. k
We need the following corollary of Lemma 5 and Lemma 15. Corollary 3. Let v, r ∈ Qn . Let L be a rational linear subspace. Let SL∗ be the set of all splits in S ∗ such that L ⊆ ls(S). Then there exists a function ∗ ∗ xL vr : SL → Z+ such that whenever v ∈ S1 , S2 ∈ SL , we have, L xL vr (ProjL⊥ (S1 )) ≤ xvr (ProjL⊥ (S2 )) ⇔ λvr (ProjL⊥ (S1 )) ≥ λvr (ProjL⊥ (S2 )). (8)
Proposition 2. Let S ⊆ S ∗ and {Qk }k∈K be a finite collection of rational polyhedra of the form (7). Then, there exists a finite set SY ⊆ S such that for all S1 ∈ S there exists S2 ∈ SY such that conv (Qk \ S2 ) ⊆ conv (Qk \ S1 )
for all k ∈ K. S Proof. Notice that sets S(K0 , V 0 ) for K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) form a finite partition of S. Consequently, it suffices to show S the existence of finite sets SY (K0 , V 0 ) ⊆ S(K0 , V 0 ) for each K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) that satisfy the claim when S1 ∈ S(K0 , V 0 ). S We now consider a arbitrary sets K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) and let X X p= |Dv (Pk )|. k∈K0 v∈V 0 ∩V (Pk )
For each S ∈ S(K0 , V 0 ) we now define a p-tuple t(S), where for each k ∈ K0 , v ∈ V 0 ∩ V (Pk ), and r ∈ Dv (Pk ), the tuple has a unique entry that equals k xL (S)) (Corollary 3). Collection of these p-tuples gives the following vr (ProjL⊥ k set contained in Zp+ : n o X = t(S) : S ∈ S(K0 , V 0 ) .
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
By Lemma 4, there exists a finite set Y ⊆ X such that for every x ∈ X there exists y ∈ Y satisfying x ≥ y. In particular, there exists a finite set SY (K0 , V 0 ) ⊆ S(K0 , V 0 ) such that for any S1 ∈ S(K0 , V 0 ) there exists S2 ∈ SY (K0 , V 0 ) satisfying k k xL (S2 )) ≤ xL (S1 )) for all k ∈ K0 , v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ). vr (ProjL⊥ vr (ProjL⊥ k k (9) By Corollary 3, the above inequality implies that
λvr (ProjL⊥ (S2 )) ≥ λvr (ProjL⊥ (S1 )) for all k ∈ K0 , v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ). k k (10) As both S1 , S2 ∈ S(K0 , V 0 ), we have V (Pk ) ∩ ProjLk (S2 ) = V (Pk ) ∩ ProjLk (S1 ) for all k ∈ K0 and applying Corollary 1 we conclude that conv Pk \ ProjLk (S2 ) ⊆ conv Pk \ ProjLk (S1 ) for all k ∈ K0 . By Lemma 16 this implies that conv(Qk \ S2 ) ⊆ conv(Qk \ S1 ) for all k ∈ K0 . Now, for k ∈ / K0 , by Lemma 16 and since S1 , S2 ∈ S(K0 , V 0 ) we have that conv(Qk \ S2 ) = conv(Qk \ S1 ) = Qk . Therefore, we conclude that conv (Qk \ S2 ) ⊆ conv (Qk \ S1 )
for all k ∈ K.
Letting [
SY =
K0 ⊆K,V 0 ⊆
S
SY (K0 , V 0 )
k∈K0
V (Pk )
concludes the proof. We now present the proof of Theorem 1 for the general case. Theorem 1. Let Qk beSa general rational polyhedron for k ∈ K where K is a finite set and let Q = k∈K Qk . Then SC (Q, S) is finitely generated for any S ⊆ S ∗ . More precisely, \ SC (Q, S) = conv (Q \ S) S∈Sˆ
where Sˆ ⊂ S is a finite set. Proof. By Proposition 2, there is a finite set SY ⊂ S such that for each S1 ∈ S there exists S2 ∈ SY with the property that [ [ [ conv Qk \ S2 = conv (Qk \ S2 ) = conv conv(Qk \ S2 ) k∈K
k∈K
⊆ conv
k∈K
[
[ conv(Qk \ S1 ) = conv Qk \ S1 .
k∈K
k∈K
Therefore, SC(Q, S) =
\ S∈S
conv
[
[ \ Qk \ S = conv Qk \ S .
k∈K
As SY is finite, the proof is complete.
S∈SY
k∈K
2
IBM Research Georgia Institute of Technology
Abstract. Split cuts form a well-known class of valid inequalities for mixed-integer programming problems (MIP). Cook et al. (1990) showed that the split closure of a rational polyhedron P is again a polyhedron. In this paper, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We also show how this result can be used to prove that some generalizations of split cuts, namely cross cuts, also yield closures that are rational polyhedra.
Keywords: Cross cuts, closure, polyhedrality
1
Introduction
Cutting planes (or cuts, for short) are crucial for solving mixed-integer programs (MIPs), and currently the most effective cuts for general MIPs are the split cuts. In their seminal paper Cook, Kannan and Schrijver [8] studied split cuts which can also be seen as a class of disjunctive cuts that generalize GMI cuts. In [8], Cook et al. showed that the split closure of a rational polyhedron P – that is, the set of points in P satisfying all split cuts for P – is again a polyhedron. This is not a trivial result as one has to consider infinitely many split cuts associated with P . Recently there has been substantial work on generalizing split cuts in different ways to obtain new and more effective classes of cutting planes, and analogues of the polyhedrality of the split closure result have been obtained for some of these classes. Andersen et. al. [3] studied cuts obtained from two dimensional convex lattice-free sets, and Andersen, Louveaux and Weismantel [2] showed that the set of points in a rational polyhedron satisfying all cuts from lattice-free sets with bounded max-facet-width is a polyhedron. Averkov [4] recently gave a short proof of this latter result. Averkov, Wagner and Weismantel [5] showed that the closure with respect to integral lattice-free sets is a polyhedron. In another recent paper, Basu et. al. [7] show that the triangle closure (points satisfying cuts obtained from maximal lattice-free triangles) of a polyhedron in a specific family (the two-row continuous group relaxation) is a polyhedron. As a generalization of split cuts, recently Dash, Dey and G¨ unl¨ uk [10] studied cuts which are obtained by considering two split sets simultaneously. These cuts are called cross cuts and are equivalent to the 2-branch split cuts of Li and Richard [14]. In this paper, we generalize Cook et al’s result from a single rational polyhedron to the union of
2
Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
a finite number of rational polyhedra and use this result to show that the cross cut closure of a rational polyhedron is a polyhedron. We next formally define split sets, split cuts for a given polyhedron (all polyhedra in this paper are assumed to be rational) and the split closure of a polyhedron. Let (π, π0 ) ∈ Zn × Z, then the split set associated with (π, π0 ) is defined to be S(π, π0 ) = {x ∈ Rn : π0 < π T x < π0 + 1}. Clearly, S(π, π0 ) ∩ Zn = ∅ and consequently the integer points contained in a polyhedron P ⊂ Rn are the same as the ones contained in conv(P \ S(π, π0 )) ⊂ Rn . Linear inequalities that are valid for conv(P \ S(π, π0 )) are called split cuts generated by the split set S(π, π0 ). Let S ∗ = {S(π, π0 ) : (π, π0 ) ∈ Zn × Z} denote the collection of all split sets and let S ⊆ S ∗ be given. The split closure of a set A ⊆ Rn , with respect to S is defined as \ SC(A, S) = conv (A \ S) , S∈S n
where for a given set X ⊆ R , we denote its convex hull by conv(X). We refer to SC(A, S ∗ ) as the split closure of A and denote it as SC(A). Cook, Kannan and Schrijver [8] showed that if P is a rational polyhedron, then SC(P ) is also a rational polyhedron. Several other proofs of this result can also be found in [1], [2], [11] and [17]. A crucial step in most of these proofs is b to show that there exists a finite set Sb ⊆ S ∗ such that SC(P, S) = SC(P, S). When such Sb exists, we say that the split closure is finitely generated. For a nonpolyhedral set the split closure is not necessarily polyhedral. However, in some cases it can be finitely generated (see for example [9]). We show the following generalization to a finite union of rational polyhedra: Theorem S 1. Let Pk be rational polyhedra for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any S ⊆ S ∗ . Note that Theorem 1 does not always implies that SC (P, S) is polyhedral as it is easy to see that for P1 = {(0, 0)} and P2 = {x ∈ R2 : x2 = 1} we have SC (P1 ∪ P2 , S ∗ ) = conv(P1 ∪ P2 ) which is not a polyhedron. As a generalization of split cuts, recently Dash, Dey and G¨ unl¨ uk [10] studied cross cuts. Let S1 , S2 ⊆ S ∗ be two collections of split sets. A cross disjunction is a pair (S1 , S2 ), where S1 ∈ S1 , S2 ∈ S2 . The cross closure of a set A ⊆ Rn , with respect to S1 , S2 , is defined as \ CC(A, S1 , S2 ) = conv (A \ (S1 ∪ S2 )) , S1 ∈S1 ,S2 ∈S2
and the cross closure of P is CC(A, S ∗ , S ∗ ), denoted simply by CC(A). In Section 5 we show another generalization of Cook, Kannan and Schrijver’s result, this time to cross cuts: Theorem 2. Let P be a rational polyhedron. Then CC(P ) is a polyhedron.
Generalizations of the Split Closure
2
3
Preliminaries
The two main ingredients that we use in this paper are the so-called GordanDickson Lemma, and, the analysis of intersection points of (closed) split sets and half-lines that have their end point contained in the split set. In [1], Anderson, Cornuejols and Li give an alternate proof of the polyhedrality of the split closure of polyhedra using a new proof technique. We next summarize the relevant results from [1], and state the Gordan-Dickson Lemma. We start with defining the point where a rational half-line H = {v + λr : λ ≥ 0}, where v, r ∈ Qn , intersects for the first time the complement of a split set S ∈ S ∗ that contains the end point v of H. Definition 1 (Intersection point step size). Let v, r ∈ Qn and S ∈ S ∗ such that v ∈ S, then λvr (S) = sup{λ : v + λr ∈ S}. Given a split set S = S(π, π0 ), the step size can be explicitly computed as follows: T π v−π0 πT r < 0 −πT r T v λvr (S) = π0 +1−π πT r > 0 πT r +∞ πT r = 0 Furthermore, notice that if π T r > 0, then the point p = v + λvr (S) r is the point where the half-line H = {v + λr : λ ≥ 0} intersects the hyperplane {x ∈ Rn : π T x = π0 }. If, on the other hand, π T r < 0 then p is the intersection point with the hyperplane {x ∈ Rn : π T x = π0 + 1}. We next review some properties of λvr (S) presented in [1] and [2]. Lemma 1 (Lemma 5 in [1]). Let S ∈ S ∗ and H = {v + λr : λ ≥ 0} where v, r ∈ Qn and v ∈ S. If λv,r (S) < +∞, then λvr (S) < min{z ∈ Z+ : zr ∈ Zn }. Lemma 2 (Lemma 6 in [1]). Let λ∗ > 0 and H = {v + λr : λ ≥ 0} where v, r ∈ Qn . Then there exists a finite set Λ ∈ R such that for all S ∈ S ∗ , λvr (S) ∈ Λ provided that ∞ > λvr (S) > λ∗ and v ∈ S. For a rational polyhedron P , we denote by V (P ) ⊆ Qn its set of vertices and by E(P ) ⊆ Qn its set of extreme rays. When V (P ) 6= ∅, we say that the polyhedron is pointed. Definition 2 (Relevant directions). For a vertex v ∈ V (P ), we define Dv (P ) = r ∈ E(P ) : {v + λr : λ ∈ R+ } is a 1-dimensional face of P ∪ v 0 − v : v 0 ∈ V (P ), and conv(v, v 0 ) is a 1-dimensional face of P to denote the set of relevant directions for the vertex v.
4
Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
Observe that for v ∈ V (P ), the relevant directions are the extreme rays of the radial cone at the vertex v in the polyhedron P . The following result is originally presented in [1] for conic polyhedra and later generalized by Andersen, Louveaux, and Weismantel [2] to general polyhedra. Lemma 3 (Lemmas 2.3, 2.4, 4.2 in [1, 2]). Let P be a pointed rational polyhedron and let S ∈ S. If P \ S 6= ∅, then (1) conv (P \ S) is a rational polyhedron. (2) The extreme rays of conv (P \ S) are the same as the extreme rays of P . (3) If u is a vertex of of conv (P \ S), then either u ∈ V (P ) \ S, or, u = v + λvr (S) r, where v ∈ V (P ) ∩ S and r ∈ Dv (P ) satisfies one of the following: (i) r ∈ E(P ) and λvr (S) < +∞, or, (ii) r = v 0 − v for some v 0 ∈ V (P ) \ S such that conv(v, v 0 ) is an edge of P and λvr (S) < 1. Finally we state a very simple and useful lemma that shows that for any positive integer p, every set of p-tuples of natural numbers has finitely many minimal elements. Lemma 4 (Gordan-Dickson Lemma). Let X ⊆ Zp+ . Then there exists a finite set Y ⊆ X such that for every x ∈ X there exists y ∈ Y satisfying x ≥ y.
3
Split Closure of a Finite Collection of Polyhedral Sets
In this section, we show that given a finite collection of rational polyhedra, there exists a finite set of splits that define the split closure. We start with a simple observation based on Lemma 3. Corollary 1. Let P be a rational pointed polyhedron and S1 , S2 ∈ S ∗ such that V (P ) ∩ S1 = V (P ) ∩ S2 . If λvr (S1 ) ≥ λvr (S2 ), for all v ∈ V (P ) ∩ S1 and r ∈ Dv (P ),
(1)
then conv (P \ S1 ) ⊆ conv (P \ S2 ) . Proof. The claim clearly holds when conv (P \ S1 ) = ∅ and therefore we assume that conv (P \ S1 ) 6= ∅. Notice that by Lemma 3 conv(P \ S1 ) and conv(P \ S2 ) are polyhedral and have the same recession cone. Moreover, (1) implies that the vertices of conv(P \ S1 ) belong to conv(P \ S2 ). Therefore, conv (P \ S1 ) ⊆ conv (P \ S2 ) . Using Lemmas 1 and 2 we obtain the following result. Lemma 5. Let v, r ∈ Qn . There exists a function xvr : S ∗ → Z+ such that whenever v ∈ S1 , S2 ∈ S ∗ , we have, xvr (S1 ) ≤ xvr (S2 ) ⇔ λvr (S1 ) ≥ λvr (S2 ).
(2)
Generalizations of the Split Closure
5
Proof. Let Λ = {λvr (S) : v ∈ S and λvr (S) < +∞} and Mr = min{z ∈ Z+ : zr ∈ Zn }. Define ( 0, λvr (S) = +∞ xvr (S) = |Λ ∩ [λvr (S), Mr ]|, λvr (S) < +∞. Notice that xvr (S) is well defined for all S ∈ S ∗ as λvr (S) < Mr by Lemma 1 and |Λ ∩ [λvr (S), Mr ]| < +∞ by Lemma 2. When λvr (S1 ) = +∞ we have xvr (S1 ) = 0 and the equivalence (2) clearly holds. If, on the other hand, λvr (S1 ) < +∞, we obtain that λvr (S1 ) ≥ λvr (S2 ) is equivalent to xvr (S1 ) ≤ xvr (S2 ), since it is easy to see that the latter occurs if and only if |Λ ∩ [λvr (S1 ), Mr ]| ≤ |Λ ∩ [λvr (S2 ), Mr ]|. This observation together with the Gordan-Dickson Lemma (Lemma 4) can be used to show that the split closure of a polyhedron is again a polyhedron. In [4], Averkov uses a similar argument to show the polyhedrality of more general closures that include the split closure. We next use Lemma 5 for split closures of unions of polyhedra. Consider a finite collectionSof pointed rational polyhedra Pk , k ∈ K, where K is a finite set. For V 0 ⊆ k∈K V (Pk ) we denote S(V 0 ) = {S ∈ S : V 0 = S k∈K V (Pk ) ∩ S}. Proposition 1. Let S ⊆ S ∗ and {Pk }k∈K be a finite collection of pointed rational polyhedra. Then, there exists a finite set SY ⊆ S such that for all S1 ∈ S there exists S2 ∈ SY such that conv (Pk \ S2 ) ⊆ conv (Pk \ S1 ) for all k ∈ K. S Proof. Notice that sets S(V 0 ) for V 0 ⊆ k∈K V (Pk ) form a finite partition of S, that is, S(V 0 ) ∩ S(V 00 ) = ∅ if V 0 6= V 00 , and [ S(V 0 ). S= S V 0 ⊆ k∈K V (Pk ) 0 0 Consequently, S it suffices to show the existence of finite sets SY0 (V ) ⊆ S(V ) for 0 each V ⊆ k∈K V (Pk ) that satisfy the claimSwhen S1 ∈ S(V ). We now consider an arbitrary set V 0 ⊆ k∈K V (Pk ). If V 0 = ∅, then it is easy to see that for all S ∈ S(V 0 ) we have conv(P \ S) = P . Thus, it is sufficient to take SY = {S}, where S ∈ S(V 0 ). If V 0 6= ∅, let X X p= |Dv (Pk )|. k∈K v∈V 0 ∩V (Pk )
For each S ∈ S(V 0 ) we now define a p-tuple t(S), where for each k ∈ K, v ∈ V 0 ∩ V (Pk ), and r ∈ Dv (Pk ), the tuple has a unique entry that equals xvr (S) (Lemma 5). Collection of these p-tuples gives the following set contained in Zp+ : n o X = t(S) : S ∈ S(V 0 ) .
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
By Lemma 4, there exists a finite set Y ⊆ X such that for every x ∈ X there exists y ∈ Y satisfying x ≥ y. In particular, there exists a finite set SY (V 0 ) ⊆ S(V 0 ) such that for any S1 ∈ S(V 0 ) there exists S2 ∈ SY (V 0 ) satisfying xvr (S2 ) ≤ xvr (S1 ) for all k ∈ K, v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ).
(3)
By Lemma 5, the above inequality implies that λvr (S2 ) ≥ λvr (S1 ) for all k ∈ K, v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ).
(4)
As both S1 , S2 ∈ S(V 0 ), we have V (Pk ) ∩ S2 = V (Pk ) ∩ S1 for all k ∈ K and applying Corollary 1 we conclude that conv (Pk \ S2 ) ⊆ conv (Pk \ S1 ) for all k ∈ K. To conclude the proof it suffices to let [ SY = SY (V 0 ) S V 0 ⊆ k∈K V (Pk )
Lemma 6. Let P Sk be a rational pointed polyhedron for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any S ⊆ S ∗ . More precisely, \ SC (P, S) = conv (P \ S) S∈Sˆ
where Sˆ ⊂ S is a finite set. S S Proof. First note that for any S ∈ S we have: conv ( P )\S = conv k∈K (Pk \ 2 k 2 k∈K S S2 ) = conv k∈K conv(Pk \ S2 ) . Furthermore, by Proposition 1, there is a finite set SY ⊂ S such that for each S1 ∈ S there exists S2 ∈ SY that satisfies [ [ [ conv conv(Pk \ S2 ) ⊆ conv conv(Pk \ S1 ) = conv Pk \ S 1 . k∈K
k∈K
k∈K
As SY is finite, to complete the proof, it suffices to observe that [ [ \ \ SC(P, S) = conv Pk \ S = conv Pk \ S . S∈S
k∈K
S∈SY
k∈K
We next relax the assumption of pointedness in the previous result; due to space restrictions we only give a sketch of the proof below. Theorem 1.SLet Pk be a rational polyhedron for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any S ⊆ S ∗ . Proof. (sketch) We first observe that a general rational polyhedron P can be written as P = Q + L, where L is a rational lineal subspace and Q ⊆ L⊥ is a rational pointed polyhedron. It can be shown that the split closure of P only depends on splits sets S such that L is contained in the lineality space of S. The projection of such splits onto L⊥ can be seen as splits sets in L⊥ . Therefore, the split closure of P only depends on the split closure of the rational pointed polyhedron Q in the subspace L⊥ , and, as a consequence, it suffices to prove the result for the rational pointed polyhedral case.
Generalizations of the Split Closure
4
7
Split Closure of a Union of Mixed-integer Sets
Consider a mixed-integer set defined by a polyhedron P LP ∈ Rn+l and the mixed-integer lattice Zn × Rl where n and l are positive integers: P I = P LP ∩ (Zn × Rl )
(5)
An inequality is called a split cut for P LP with respect to the lattice Zn × Rl if ∗ where it is valid for conv(P LP \ S) for some S ∈ Sn,l ∗ Sn,l = {S(π, π0 ) ∈ S ∗ : π ∈ Zn × {0}l }.
The split closure is then defined in the usual way as the intersection of all such split cuts. A straightforward extension of Theorem 1 is the following: Corollary 2. Let Pk ∈ S Rn+l be a rational polyhedron for k ∈ K where K is a finite set and let P = k∈K Pk . Then SC (P, S) is finitely generated for any ∗ S ⊆ Sn,l .
5
Cross Closure of a Polyhedral Set
In this section, we show that the cross closure of a rational polyhedron is again a polyhedron. We give the complete proof for the pointed case (see Lemma 14 below). The extension to the general case can be done using Lemma 14 in the same fashion as explained at the beginning of Section 3 for the split closure. We will present the proof of this case in the journal version of the paper. We combine the proof technique of Cook, Kannan and Schrijver [8] for showing that the split closure of a polyhedron is polyhedral along with the results we derived in earlier sections based on proof techniques of Anderson, Cornu´ejols, Li [1], and Averkov [4]. We need some definitions to discuss the overall techniques used. Lets denote by k · k the usual euclidean norm. Define the width of a split set S(π, π0 ) as w(S(π, π0 )) = 1/kπk (this is the geometric distance between the parallel hyperplanes bounding the split set). Then w(S(π, π0 )) > η for some η > 0 implies that kπk < 1/η. Therefore, for any fixed η > 0 and π0 ∈ Zn , there are only a finite number of π ∈ Zn such that w(S(π, π0 )) > η. Cook, Kannan, Schrijver (roughly) prove their polyhedrality result using the following idea. Assume P is a polyhedron, L is a finite list of split sets and let SC(P, L) = ∩S∈L conv(P \S). Suppose that for every face F of P , SC(P, L)∩F ⊆ SC(F ). Then (i) there are only finitely many split sets beyond the ones contained in L which yield split cuts cutting off points of SC(P, L) (they show that if S(π, π0 ) is such a split set, then π must have bounded norm). Therefore, (ii) if one assumes (by induction on dimension) that the number of split sets needed to define the split closure of each face of a polyhedron is finite, then so is the number of split sets needed to define the split closure of the polyhedron. Santanu Dey [12] observed that idea (i) in the Cook, Kannan, Schrijver proof technique can also be used in the case of some disjunctive cuts which generalize
8
Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
split cuts. We apply a modification of idea (i) to cross cuts; namely we show in Lemma 13 that if L is a finite list of cross disjunctions (represented as a pair of split sets) such that CC(P, L) = ∩(S1 ,S2 )∈L conv(P \ (S1 ∪ S2 ))
(6)
intersected with each face of P is equal to the cross closure of each face, then cross disjunctions (S1 , S2 ) where both w(S1 ) and w(S2 ) are at most some η > 0 can only yield cross cuts valid for CC(P, L), and are therefore not needed to define the cross closure of P . We then only need to consider cross disjunctions (S1 , S2 ) where one of w(S1 ), w(S2 ) is greater than η (such cross disjunctions are still infinitely many in number). We first need a generalization of Lemma 3, property (2.). Let rec(P ) denote the recession cone of P , aff(P ) denote the affine hull of P , and P I denote the integer hull of P . Lemma 7. Let P be a polyhedron in Rn , and let S1 , S2 ∈ S ∗ be any two split sets. If conv(P \ (S1 ∪ S2 )) is nonempty, then its recession cone equals rec(P ). Proof. Let P 0 = conv(P \ (S1 ∪ S2 )). As P 0 ⊆ P and P is closed, we obtain rec(P 0 ) ⊆ rec(P ). To prove the reverse inclusion, let v be a point in P 0 , and let r ∈ rec(P ) 6= ∅. Let v1 , v2 be two points in P \ (S1 ∪ S2 ) such that v is a convex combination of v1 , v2 (we will choose v1 = v2 = v if v 6∈ S1 ∪ S2 ). Consider the half lines H1 = {v1 + λr : λ ≥ 0} and H2 = {v2 + λr : λ ≥ 0}. As v1 6∈ S1 ∪ S2 , either H1 does not intersect S1 ∪ S2 , or else sup{λ ≥ 0 : v1 + λr ∈ S1 ∪ S2 } is finite. In either case, the half line H1 ⊆ P 0 , and similarly H2 ⊆ P 0 and therefore conv(H1 ∪ H2 ) ⊆ P 0 . But then the half line {v + λr : λ ≥ 0} ⊆ P 0 and therefore r ∈ rec(P 0 ) =⇒ rec(P ) ⊆ rec(P 0 ). Now we generalize Lemma 3, property (1.). For a set A ⊆ Rn , let conv(A) denote the topological closure of conv(A). The following result is a direct consequence of Theorem 3.5 in [13]. Lemma 8 ([13]). Let A ⊆ Rn be a nonempty closed set. Then every vertex of conv(A) belongs to A. Lemma 9. Let P be a pointed polyhedron, and let S1 , S2 ∈ S ∗ . Then conv (P \ (S1 ∪ S2 )) is a polyhedron. Proof. The claim trivially holds if conv(P \ (S1 ∪ S2 )) is empty. Therefore we assume that conv(P \ (S1 ∪ S2 )) is nonempty. Since conv (P \ (S1 ∪ S2 )) is a closed convex set not containing a line, by a standard result in convex analysis (see for example Theorem 18.5 in [15]) we have that conv (P \ (S1 ∪ S2 )) can be written as the Minkowski sum of its set of vertices and its recession cone. By Lemma 8 we have that the vertices of conv (P \ (S1 ∪ S2 )) belong to P \ (S1 ∪ S2 ). Thus, since P \ (S1 ∪ S2 ) is a finite union of polyhedra, we obtain that conv (P \ (S1 ∪ S2 )) has a finite number of vertices.
Generalizations of the Split Closure
9
On the other hand, by Lemma 7 we obtain that rec(conv (P \ (S1 ∪ S2 ))) = rec(conv (P \ (S1 ∪ S2 ))) = rec(P ). This implies that conv (P \ (S1 ∪ S2 )) is a polyhedron. Moreover, since all its vertices belong to P \(S1 ∪S2 ) ⊆ conv (P \ (S1 ∪ S2 )), we conclude that conv (P \ (S1 ∪ S2 )) = conv (P \ (S1 ∪ S2 )). Therefore, conv (P \ (S1 ∪ S2 )) is a polyhedron. Observe that Lemma 7 and Lemma 9 implies that CC(P, L) is a polyhedron with rec(CC(P, L)) = rec(P ) if CC(P, L) is defined as in (6). The proof of the following lemma is ommited. Lemma 10. Let P be a polyhedron and let F be a face of P . For any set B, conv(P \ B) ∩ F = conv(F \ B). The next result is essentially contained in Cook, Kannan and Schrijver [8], though our statement and proof are slightly different. Lemma 11. Let P and P 0 be pointed, full-dimensional polyhedra in Rn with P ⊂ P 0 . Then there exists a number r > 0 such that for any c ∈ Rn satisfying (i) max{cT x : x ∈ P } = d < max{cT x : x ∈ P 0 } < ∞ and (ii) the first maximum is attained at a vertex of P contained in the interior of P 0 , there exists a ball of radius r in P 0 with each point x in the ball satisfies cT x > d. Proof. Let P 0 = {x : aTi x ≤ bi for i = 1, . . . , m} where ai ∈ Rn and let η = maxi {||ai ||}. Let V = {v1 , . . . , vk } be the set of vertices of P contained in the interior of P 0 , and let = mini,j {bi − aTi vj } > 0. Let c satisfy the conditions of the lemma and let cT vj = d for some vertex vj ∈ V . Further, let max{cT x : 0 x ∈ P 0 } = d0 < ∞ be attained at a vertex v 0 of PP . By LP duality, there Pm exists m multipliersP0 ≤ λ = (λ1 , . . . , λm ) such that c = i=1 λai andP d0 = i=1 λi bi m m ¯ ¯ λ) ¯ = (c, d0 , λ)/τ . Then c¯ = and c, d, i=1 λi > 0. Let (¯ i=1 λai where Pm τ ¯= λ = 1 and therefore ||¯ c || ≤ η. Further i i=1 d¯ − c¯T vj =
m X
¯ i (bi − aT vj ) ≥ . λ i
i=1
By definition, max{¯ cT x : x ∈ P } is attained at vj , max{¯ cT x : x ∈ P 0 } is attained at v 0 , and the distance between the hyperplanes c¯T x = c¯T vj and c¯T x = d¯ is at least /||¯ c|| > /(η +1). Therefore, any point z in the ball B(v 0 , /(η +1)) satisfies c¯T z > c¯T vj (and also cT z > cT vj ). We can find an r > 0 such that B(v, /(η+1)) contains a ball of radius r for each vertex v of P 0 . In the proof above, we can assume that we construct a fixed set B of balls, one per each vertex of P 0 , such that one of these balls satisfies the desired property in Lemma 11. A strip in Rn is the set of points between a pair of parallel hyperplanes (and including the hyperplanes) and the width of a strip is the distance between its
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
bounding hyperplanes. Thus the topological closure of a split set is a strip. If the minimum width of a closed, compact, convex set A is defined as the minimum width of a strip containing A, then it is known (Bang [6]) that the sum of widths of a collection of strips containing A must exceed its minimum width. The following statement is a trivial consequence of Bang’s result. Lemma 12. Let B be a ball of radius r > 0, and let S1 , S2 be split sets such that B ⊆ (S1 ∪ S2 ). Then, w(S1 ) + w(S2 ) ≥ 2r. Lemma 13. Let P be a pointed, full-dimensional polyhedron. Let L ⊆ S ∗ ×S ∗ be a finite list of cross disjunctions. Let CC(P, L) be defined as in (6). If CC(P, L)∩ F ⊆ CC(F ) for all faces F of P , then there exists η > 0 such that all cross cuts obtained from cross disjunctions (S1 , S2 ) ∈ S ∗ × S ∗ with w(S1 ), w(S2 ) ≤ η are valid for CC(P, L). Proof. As P is pointed, CC(P, L) is also pointed as it is contained in P . Let r > 0 be the number given by Lemma 11 when applied to P and CC(P, L). Let 0 < η < r. Let cT x ≤ γ be a cross cut obtained from a cross disjunction (S1 , S2 ) with w(S1 ), w(S2 ) ≤ η. We will prove that cT x ≤ γ is valid for CC(P, L). As γ ≥ max{cT x : x ∈ conv(P \ (S1 ∪ S2 ))} < ∞ and since by Lemma 7 we have rec(CC(P, L)) = rec(P ) = rec(conv(P \ (S1 ∪ S2 ))), we obtain that d := max{cT x : x ∈ CC(P, L)} < ∞. We have two cases. Case 1: The maximum is attained in a face F of P . Then, since CC(P, L) ∩ F ⊆ CC(F ) ⊆ {x ∈ Rn : cT x ≤ γ}, we infer that d ≤ γ. Therefore cT x ≤ γ is valid for CC(P, L). Case 2: The maximum is attained in a vertex of CC(P, L) in the interior of P . This implies that d < max{cT x : x ∈ P }. By Lemma 11, there exists a ball B of radius r in P with all points in the ball satisying cT x > d. Since w(S1 )+w(S2 ) < 2r, Lemma 12 implies that B \ (S1 ∪ S2 ) 6= ∅. Let x ¯ ∈ B \ (S1 ∪ S2 ). Then x ¯ ∈ conv(P \(S1 ∪S2 )) with cT x ¯ > d. Since cT x ≤ γ is valid for conv(P \(S1 ∪S2 )), it follows that d < γ. Therefore cT x ≤ γ is valid for CC(P, L). The discussion after Lemma 11 implies that the ball B in the proof above can be assumed to be a member of B. Further, the proof implies that even if w(S1 ), w(S2 ) ≥ r, if B \ (S1 ∪ S2 ) 6= ∅, then (assuming P, L satisfy the conditions of the Lemma) cross cuts from the disjunction (S1 , S2 ) are valid for CC(P, L). We will need the following basic properties of unimodular matrices (matrices with determinant ±1). If V is a rational affine subspace of Rn with dimension k < n such that V ∩ Zn 6= ∅, then there is a n × n integral unimodular matrix U and vector v ∈ Zn such that the one-to-one mapping σ(x) = U x + v maps V to Rk × {0}n−k ; also split sets are mapped to split sets. Further, the intersection of any split set S in Rn with Rk × {0}n−k is either empty, Rk × {0}n−k or equals S 0 × {0}n−k where S 0 is a split set in Rk .
Generalizations of the Split Closure
11
Lemma 14. Let P be a rational pointed polyhedron. Then CC(P ) is a polyhedron. More precisely, CC(P ) =
\
conv (P \ (S1 ∪ S2 ))
(S1 ,S2 )∈L∗
where L∗ ⊂ S ∗ × S ∗ is a finite set. Proof. We use standard techniques to show that P can be assumed to be fulldimensional. Assume P has dimension k < n. As aff(P ) is a rational, affine subspace of Rn , if aff(P ) ∩ Zn = ∅, then it is well-known (see [16, Corollary 4.1a]) that P ⊆ aff(P ) ⊆ S for some split set S. Then CC(P ) ⊆ SC(P ) ⊆ conv(P \ S) = ∅ = P I . Therefore we assume aff(P ) ∩ Zn 6= ∅. There is a function σ(x) (as discussed before the theorem) that maps aff(P ) to a polyhedron in Rk × {0}n−k , i.e., to P 0 × {0}n−k , where P 0 is a full-dimensional polyhedron. Let L0 be a subset (not necessarily finite) of all cross disjunctions in Rk which yield the cross closure of P 0 ; Then for each cross disjunction (or split set pair) (S10 , S20 ) in L0 , if we define a cross set (S1 , S2 ) in Rn as (S10 × {0}n−k , S20 × {0}n−k ), and then apply the inverse function σ −1 (x) (of σ(x)) to (S1 , S2 ), we get a list of cross disjunctions in Rn which yield the cross closure of P . Therefore we can work on P 0 . The proof is by induction on dim(P ). The case dim(P ) = 0 is straightforward. Now, lets assume that for all polyhedra Q of dimension strictly less than dim(P ), CC(Q) is defined by a finite number of cross disjunctions. Let F be a face of P . Since dim(F ) < dim(P ), by the induction hypothesis (and the argument in the previous paragraph) we infer that there exists a finite set of cross disjunctions L(F ) in Rn such that CC(F ) = CC(F, L(F )). Define L = ∪F is a face of P L(F ). By the induction hypothesis, L is a finite list of cross disjunctions. Also, for any face F of P , CC(P, L) ⊆ CC(P, L(F )) and therefore CC(P, L) ∩ F = CC(F, L) ⊆ CC(F, L(F )) = CC(F ), where the first equality follows from Lemma 10. Therefore, Lemma 13 implies the existence of a number η > 0 such that all cross cuts obtained from cross disjunctions (S1 , S2 ) with w(S1 ), w(S2 ) ≤ η are valid for PL . This implies that to define the cross closure of P , it suffices to consider cross disjunctions with (S1 , S2 ) with either w(S1 ) > η or w(S2 ) > η. Further, the discussion after Lemma 13 implies that one of S1 , S2 satisfies the property: w(S) > η and S ∩ B 6= ∅ for a ball B ∈ B defined after Lemma 11. Define Sη = {S ∈ S ∗ : w(S) > η and S ∩ (∪B B) 6= ∅} and observe that it is a finite set. For S ∈ Sη the set P \ S is a union of two pointed rational polyhedra (possibly empty). Therefore, Theorem 1 implies that SC(P \ S) = SC(P \ S, S ∗ ) is finitely
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Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
generated. On the other hand, we have \ CC(P ) = conv (P \ (S1 ∪ S2 )) S1 ∈S ∗ ,S2 ∈S ∗
\
=
conv (P \ (S1 ∪ S2 )) ∩
conv (P \ (S1 ∪ S2 ))
S1 ∈Sη ,S2 ∈S ∗
(S1 ,S2 )∈L
= CC(P, L) ∩
\
\
SC(P \ S, S ∗ ).
S∈Sη
Therefore, we conclude that CC(P ) is finitely generated and, by Lemma 9, is a polyhedron.
References 1. Kent Andersen, G´erard Cornu´ejols, and Yanjun Li, Split closure and intersection cuts, Mathematical Programming 102 (2005), no. 3, 457–493. 2. Kent Andersen, Quentin Louveaux, and Robert Weismantel, An analysis of mixed integer linear sets based on lattice point free convex sets, Math. Oper. Res. 35 (2010), no. 1, 233–256. 3. Kent Andersen, Quentin Louveaux, Robert Weismantel, and Lawrence Wolsey, Inequalities from two rows of a simplex tableau, IPCO proceedings, Lecture Notes in Computer Science, vol. 4513, 2007, pp. 1–15. 4. Gennadiy Averkov, On finitely generated closures in the theory of cutting planes, Discrete Optimization 9 (2012), no. 1, 209–215. 5. Gennadiy Averkov, Christian Wagner, and Robert Weismantel, Maximal latticefree polyhedra: finiteness and an explicit description in dimension three, Mathematics of Operations Research 36 (2011), no. 4, 721–742. 6. Thoger Bang, A solution of the plank problem, Proc. American Mathematical Society 2 (1951), no. 6, 990–993. 7. Amitabh Basu, Robert Hildebrand, and Matthias K¨ oppe, The triangle closure is a polyhedron, manuscript, 2011. 8. William J. Cook, Ravi Kannan, and Alexander Schrijver, Chv´ atal closures for mixed integer programming problems, Math. Program. 47 (1990), 155–174. 9. Daniel Dadush, Santanu S. Dey, and Juan Pablo Vielma, The split closure of a strictly convex body, Oper. Res. Lett. 39 (2011), no. 2, 121–126. 10. Sanjeeb Dash, Santanu Dey, and Oktay G¨ unl¨ uk, Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra, Math. Program. 135 (2012), 221–254. 11. Sanjeeb Dash, Oktay G¨ unl¨ uk, and Andrea Lodi, MIR closures of polyhedral sets, Math. Program. 121 (2010), no. 1, 33–60. 12. Santanu S. Dey, Personal communication, (2010). 13. V. L. Klee, Extremal structure of convex sets, Archiv der Mathematik 8 (1957), 234– 240. 14. Yanyun Li and J. P. P. Richard, Cook, kannan and schrijver’s example revisited, Discrete Optimization 5 (2008), 724–734. 15. G. T. Rockafeller, Convex analysis, Princeton University Press, New Jersey, NJ, 1970.
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16. Alexander Schrijver, Theory of linear and integer programming, John Wiley and Sons, New York, 1986. 17. Juan Pablo Vielma, A constructive characterization of the split closure of a mixed integer linear program, Oper. Res. Lett. 35 (2007), no. 1, 29–35.
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A
Appendix: Proof of Theorem 1 in the general case
A.1
On unimodular matrices and split sets
We next review some basic properties of unimodular matrices, i.e., square matrices with determinant ±1. If U is an n × n unimodular matrix, and v ∈ Rn , the affine transformation σ(x) = U x + v is a one-to-one, invertible, mapping of Rn to Rn with σ −1 (x) = U −1 (x − v) and this transformation preserves volumes. If U is an integral unimodular matrix, then so is U −1 ; if in addition v ∈ Zn , then the function σ(x) is a one-to-one, invertible, mapping of Zn to Zn . Further, if a ∈ Zn , b ∈ Z, the set {x ∈ Rn : aT x = b} is mapped to the set {x0 ∈ Rn : aT U −1 (x0 − v) = b} ≡ {x0 ∈ Rn : aT U −1 x0 = b + aT U −1 v} and aT U −1 ∈ Zn . Therefore, given a split set S(a, b), σ(S(a, b)) and σ −1 (S(a, b)) are both split sets. If a ∈ Zn and the g.c.d. of the coefficients of a is one, then there is a unimodular matrix U such that aT U = (0, . . . , 0, 1) and aT x = b has an integral solution for any integer b, say v b (see [16, Corollary 4.1c]). Note that the previous statement implies that aT is the last row of U −1 . Then, under the linear transformation x → U x (with inverse transformation x → U −1 x + v b ), there is a one-to-one mapping of {x ∈ Rn : aT x = b} to the set {x ∈ Rn : xn = b}, and of the integer points in the respective sets. Further, any linear subspace L of {x : aT x = 0} is mapped to a subspace of {x ∈ Rn : xn = 0}, and therefore the intersection of a split set and L⊥ is a split set. We summarize this result in the next lemma. The lineality space of a split set S(π, π0 ), where (π, π0 ) ∈ Zn , is denoted ls(S) := {x ∈ Rn : π T x = 0}. We also denote ProjL⊥ (S) := S ∩ L⊥ . Lemma 15. Let L be a rational subspace such that L ⊆ ls(S). Then ProjL⊥ (S) is a split set of L⊥ . A.2
Proof of Theorem 1
Let L ⊆ Rn be a rational linear subspace and let P be a pointed polyhedron such that P ⊆ L⊥ . Then a general polyhedron is a polyhedron of the form Q = P +L. Lemma 16. Let L be a rational linear subspace, P ⊆ L⊥ be a pointed polyhedra, and let Q = P + L. Let S be a split set. Then 1. If L * ls(S), then conv(Q \ S) = Q. 2. If L ⊆ ls(S), then conv(Q \ S) = conv(P \ ProjL⊥ (S)) + L. Proof. 1. Let x ∈ Q and l ∈ L such that π T l 6= 0. Since π T l 6= 0, we obtain that the line x + {λl : λ ∈ R} is contained in conv(Q \ S). Therefore, x ∈ conv(Q \ S). 2. Since L ⊆ ls(S), we can write S = ProjL⊥ (S) + L. Therefore, we can write conv(Q \ S) = conv((P + L) \ (ProjL⊥ (S) + L)) = conv(P \ ProjL⊥ (S)) + L, where the last equality is given by the fact that P, ProjL⊥ (S) ⊆ L⊥ .
Generalizations of the Split Closure
15
By Lemma 16 applied to Q and by Lemma 3 applied to P , we conclude that conv(Q \ S) is characterized in terms of intersections points defined by the split set ProjL⊥ (S) ⊆ L⊥ and the pointed polyhedron P ⊆ L⊥ . These facts allow us to applied in the general case the same proof techniques as before. The crucial result in Section 3 is Proposition 1. We extend this proposition to the general case next. We begin with some notation. Consider a finite collection of pointed rational polyhedra for k ∈ K Qk = Pk + Lk ,
(7)
where Lk are rational linear subspaces, Pk = conv (V (Pk )) + cone (E(Pk )) are pointed polyhedra with V (Pk ), E(Pk ) ⊆ Qn denoting the corresponding set of S vertices and extreme rays of polyhedron Pk . For K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) we denote S(K0 , V 0 ) = {S ∈ S : V 0 =
[ k∈K0
[V (Pk )∩ProjL⊥ (S)], Lk ⊆ ls(S) ∀ k ∈ K0 , Lk * ls(S) ∀ k ∈ / K0 }. k
We need the following corollary of Lemma 5 and Lemma 15. Corollary 3. Let v, r ∈ Qn . Let L be a rational linear subspace. Let SL∗ be the set of all splits in S ∗ such that L ⊆ ls(S). Then there exists a function ∗ ∗ xL vr : SL → Z+ such that whenever v ∈ S1 , S2 ∈ SL , we have, L xL vr (ProjL⊥ (S1 )) ≤ xvr (ProjL⊥ (S2 )) ⇔ λvr (ProjL⊥ (S1 )) ≥ λvr (ProjL⊥ (S2 )). (8)
Proposition 2. Let S ⊆ S ∗ and {Qk }k∈K be a finite collection of rational polyhedra of the form (7). Then, there exists a finite set SY ⊆ S such that for all S1 ∈ S there exists S2 ∈ SY such that conv (Qk \ S2 ) ⊆ conv (Qk \ S1 )
for all k ∈ K. S Proof. Notice that sets S(K0 , V 0 ) for K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) form a finite partition of S. Consequently, it suffices to show S the existence of finite sets SY (K0 , V 0 ) ⊆ S(K0 , V 0 ) for each K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) that satisfy the claim when S1 ∈ S(K0 , V 0 ). S We now consider a arbitrary sets K0 ⊆ K, and V 0 ⊆ k∈K0 V (Pk ) and let X X p= |Dv (Pk )|. k∈K0 v∈V 0 ∩V (Pk )
For each S ∈ S(K0 , V 0 ) we now define a p-tuple t(S), where for each k ∈ K0 , v ∈ V 0 ∩ V (Pk ), and r ∈ Dv (Pk ), the tuple has a unique entry that equals k xL (S)) (Corollary 3). Collection of these p-tuples gives the following vr (ProjL⊥ k set contained in Zp+ : n o X = t(S) : S ∈ S(K0 , V 0 ) .
16
Sanjeeb Dash, Oktay G¨ unl¨ uk, and Diego A. Mor´ an R.
By Lemma 4, there exists a finite set Y ⊆ X such that for every x ∈ X there exists y ∈ Y satisfying x ≥ y. In particular, there exists a finite set SY (K0 , V 0 ) ⊆ S(K0 , V 0 ) such that for any S1 ∈ S(K0 , V 0 ) there exists S2 ∈ SY (K0 , V 0 ) satisfying k k xL (S2 )) ≤ xL (S1 )) for all k ∈ K0 , v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ). vr (ProjL⊥ vr (ProjL⊥ k k (9) By Corollary 3, the above inequality implies that
λvr (ProjL⊥ (S2 )) ≥ λvr (ProjL⊥ (S1 )) for all k ∈ K0 , v ∈ V 0 ∩ V (Pk ), r ∈ Dv (Pk ). k k (10) As both S1 , S2 ∈ S(K0 , V 0 ), we have V (Pk ) ∩ ProjLk (S2 ) = V (Pk ) ∩ ProjLk (S1 ) for all k ∈ K0 and applying Corollary 1 we conclude that conv Pk \ ProjLk (S2 ) ⊆ conv Pk \ ProjLk (S1 ) for all k ∈ K0 . By Lemma 16 this implies that conv(Qk \ S2 ) ⊆ conv(Qk \ S1 ) for all k ∈ K0 . Now, for k ∈ / K0 , by Lemma 16 and since S1 , S2 ∈ S(K0 , V 0 ) we have that conv(Qk \ S2 ) = conv(Qk \ S1 ) = Qk . Therefore, we conclude that conv (Qk \ S2 ) ⊆ conv (Qk \ S1 )
for all k ∈ K.
Letting [
SY =
K0 ⊆K,V 0 ⊆
S
SY (K0 , V 0 )
k∈K0
V (Pk )
concludes the proof. We now present the proof of Theorem 1 for the general case. Theorem 1. Let Qk beSa general rational polyhedron for k ∈ K where K is a finite set and let Q = k∈K Qk . Then SC (Q, S) is finitely generated for any S ⊆ S ∗ . More precisely, \ SC (Q, S) = conv (Q \ S) S∈Sˆ
where Sˆ ⊂ S is a finite set. Proof. By Proposition 2, there is a finite set SY ⊂ S such that for each S1 ∈ S there exists S2 ∈ SY with the property that [ [ [ conv Qk \ S2 = conv (Qk \ S2 ) = conv conv(Qk \ S2 ) k∈K
k∈K
⊆ conv
k∈K
[
[ conv(Qk \ S1 ) = conv Qk \ S1 .
k∈K
k∈K
Therefore, SC(Q, S) =
\ S∈S
conv
[
[ \ Qk \ S = conv Qk \ S .
k∈K
As SY is finite, the proof is complete.
S∈SY
k∈K
