A probabilistic comparison of split and type 1 triangle cuts for two row mixed-integer programs

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1 A probabilistic comparison of split and type 1 triangle cuts for two row mixed-integer programs Qie He, Shabbir Ahmed, George L. Nemhauser H. Milton Stewart School of Industrial & Systems Engineering Georgia Institute of Technology, Atlanta, GA333 December 3, 1 Abstract We provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables in terms of cut coefficients and volume cut off. nder a specific probabilistic model of the problem parameters, we show that for the above measure, the probability that a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut is better than a split cut. 1 Introduction This paper is concerned with valid inequalities for a two-row mixed-integer program (MIP) with two integer variables of the form x f + r j y j x Z, y j, (1) where f Q \ Z and r j Q \ {} for all j. Let X denote the set of solutions to (1). It has been shown (e.g Andersen et al. [1]) that any valid inequality for conv(x) that cuts off the infeasible point (x, y) (f, ) is an intersection cut (Balas []), corresponding to a convex set L R with int(l) Z (i.e. integer-free) and f int(l). Such a cut is of the form where ψ L : Q R is given by ψ L (r j )y j 1, () { r rec.cone(l) ψ L (r) λ >, f + λr boundary(l). 1 λ (3) Furthermore, minimal inequalities of the form () can be derived from maximal integer-free sets in R with non-empty interior. Such sets are of one of the following types (Lovász [1]): A split S: c ax 1 + bx c + 1, where a, b, c Z and gcd(a, b) 1; 1

2 A triangle with an integer point in the relative interior of each of the edges; these can be further classified in to one of the following three types (Dey and Wolsey [1]): 1. A type 1 triangle T 1 : a triangle with integer vertices and exactly one integer point in the relative interior of each edge.. A type triangle T : a triangle with more than one integer point on one edge and exactly one integer point in the relative interior of each of the other two edges. 3. A type 3 triangle T 3 : a triangle with exactly one integer point in the relative interior of each edge and non-integral vertices. A quadrilateral Q with exactly one integer point in the relative interior of each edge such that the four integer points form a parallelogram of area one. Inequalities of the form () corresponding to the above sets are called split, (type 1, or 3) triangle, and quadrilateral cuts, respectively. From the maximality of the above integer-free sets, it follows that any non-trivial facet of conv(x) is either a split, triangle or quadrilateral cut [1, 6]. Split cuts are the classical Gomory mixed integer (GMI) or mixed-integer rounding cuts [13]. Recently there has been a great deal of activity in comparing triangle and quadrilateral cuts to split cuts for two row MIPs. Basu et al. [4] compared the rank-1 closure (the convex set obtained by adding in a single round all possible cuts from the family) corresponding to the three cuts classes. They showed that the triangle closure (considering all three types of triangle cuts) and the quadrilateral closure are contained in the split closure, suggesting that triangle and quadrilateral cuts are in some sense stronger than split cuts. Dey [9] showed that type, type 3 triangle cuts and quadrilateral cuts have a finite split ranks (i.e. such a cut can be constructed via a finite sequence of split cuts) while only type 1 triangle cuts can have infinite split rank. However, empirical studies demonstrating the success of triangle and quadrilateral cuts in comparison to split (or GMI) cuts have been limited. Espinoza [11] reported some success with intersection cuts generated from some classes of integer-free triangles and quadrilaterals. Basu et al. [3] considered strengthened versions of a class of type triangle cuts and showed that combining these cuts with GMI cuts give somewhat better performance than GMI cuts alone. Dey et al. [8] presented computational results on randomly generated multi-knapsack instances and showed that a subclass of type triangle cuts can close more gap than GMI cuts. We present a probabilistic comparison of type 1 triangle cuts and split cuts. Specifically we address the question: what is the likelihood that a split cut will dominate with respect to cut coefficients or cut off more volume from the linear programming relaxation than a type 1 triangle cut for an arbitrary instance of the two-row MIP (1) given a specific probability distribution of the problem parameters? Our analysis reveals that, for the given distribution of the instances, such likelihood is high. The analysis also suggests some guidelines on when type 1 triangle cuts are likely to be more effective than split cuts and vice versa. Setup In this section, we discuss the distributional model for instances of the two-row MIP (1) and the two metrics used in our probabilistic comparison of type 1 triangle and split cuts. Without loss of generality, (by translating x by f and scaling y j by r j ) we can assume that f i < 1 for i 1, and r j 1 for all j in (1). Then r j 1 cos θ j and r j sin θ j where θ j is the angle between r j and the positive x 1 -axis.

3 The input model: We consider instances of (1) where f is a realization of a random vector f that is uniformly distributed with support : (, 1), i.e., the open unit square in the plane, and θ j is a realization of a random variable θ j that is uniformly distributed over [, π) for all j. (When f is on the boundary of cl(), the coefficients for some split and type 1 triangle cuts can be +, causing technical issues in their comparison.) Moreover, f, θ 1,..., θ k are independent random variables. nder this probabilistic input model, the cut corresponding to the integer-free body L is of the form ψ L (f, θ j )y j 1, (4) where the cut coefficient ψ L (f, θ j ) of variable y j is a random variable depending on f and θ j and is given by (3). Our analysis compares the random cut (4) when the set L is a split or a type 1 triangle. To guarantee that f int(l) with probability one, we only consider integer-free splits and type 1 triangles that contain. This ensures that the inequality (4) corresponding to L cuts off the infeasible point (f, ) for every realization f of f. There are only two splits containing (the valid inequality corresponds to the GMI cut for each row of system (1)) and there are only four type 1 triangles containing, with one of the four vertices of as its right-angle vertex (see Figure 1). There are various criteria for comparing cuts. We choose two criteria suitable for comparing two individual cuts rather than cut families. The first one is based on cut dominance. Definition 1. Suppose C 1 : k a jy j 1 and C : k b jy j 1 are two distinct valid inequalities for system (1), then C 1 dominates C if a j b j for j 1,, k with at least one of the inequalities being strict. We use C 1 D C to denote that C 1 dominates C. If C 1 D C, then C is implied by C 1. The second criteria is based on the volume cut off by the cuts from the linear relaxation. Definition. Suppose C 1 : k a jy j 1 and C : k b jy j 1 are two distinct valid inequalities for system (1). Let X LP be the linear relaxation of (1). Then C 1 V C if C 1 cuts off more volume than C from X LP, i.e. vol(x LP (x, y) : a j y j 1 ) > vol(x LP (x, y) : b j y j 1 ). We probabilistically compare split and type 1 triangle cuts with respect to these two metrics. 3 Conditional Probabilities with respect to f We first analyze the conditional probabilities of split cuts dominating and cutting off more volume than triangle cuts with respect to the fractional point f. This analysis helps with computing the total probabilities in Section 4, and also provides some insight into values of f for which type 1 triangle cuts are likely to be better than split cuts and vice versa. 3.1 Cut coefficient comparison Without loss of generality, we select one split from the two splits and one type 1 triangle from the four type 1 triangles in Figure 1. The analysis easily extends to the other splits and type 1 triangles by symmetry. The chosen split S 1 and type 1 triangle T 1 are shown in Figure. The split 3

4 S 1 x x T1 S (,1) (,1) T x 1 (,) (1,) (,) (1,) T4 x 1 (a) Two simple splits T3 (b) Four type 1 triangles Figure 1: The integer-free bodies selected for comparison S 1 is defined by AD and BC and the type 1 triangle T 1 is defined by AEF. Suppose that C S1 is the split cut for S 1 and C T1 is the triangle cut for T 1, and recall that ψ S1 (f, θ j ) and ψ T1 (f, θ j ) are the corresponding (random) cut coefficients for variable y j. We use Pr[ψ T1 (f, θ j ) < ψ S1 (f, θ j ) f] to denote the conditional probability of the event ψ T1 (f, θ j ) < ψ S1 (f, θ j ) when f f. Lemma 1. For each j 1,, k, Pr[ψ T1 (f, θ j ) < ψ S1 (f, θ j ) f] α(f), Pr[ψ S1 (f, θ j ) ψ T1 (f, θ j ) f] β(f) and Pr[ψ S1 (f, θ j ) < ψ T1 (f, θ j ) f] γ(f), where α(f) arccos f (f 1)+(1 f 1 ) [f +(1 f 1 ) ][(1 f ) +(1 f 1 ) ] π, β(f) arccos f1 +f f [f 1 +f ][f 1 +( f ) ] π, f1 arccos +f f 1 + arccos f1 +f f 1 3f + [f γ(f) 1 +f ][(1 f 1) +f ] [(1 f ) +(1 f 1 ) ][f1 +( f ) ] π Proof. Since θ j (j 1,, k) are i.i.d., we only need to prove ( the result ) for some j. For simplicity, cos θ we supress the index j here and prove it for some ray r. sin θ As shown in Figure, is the unit square with vertices A, B, C and D and O is the fractional point f. Let OR be the ray defined by f + λr. Let OM be the line parallel to the x 1 -axis that intersects S and T 1 at M and N respectively. Then θ is the angle between OM and OR in the counterclockwise direction. Let the symbol denote an angle less than π. Since the probability density function of θ is 1 π I(θ [, π)), by the law of total probability, Pr[ψ S1 (f, θ) < ψ T1 (f, θ) f] I(ψ S1 (f, θ) < ψ T1 (f, θ)) dθ, (5) π where I(A) is the indicator function of event ( A. ) By (3), ψ S1 (f, θ) 1 cos θ λ S1, where f + λ S1 boundary(s), and ψ sin θ T1 (f, θ) 1 λ T1 where ( ) ( ) cos θ cos θ f + λ T1 boundary(t sin θ 1 ). Therefore, ψ S1 (f, θ) < ψ T1 (f, θ) if the ray f + λ hits sin θ ( ) cos θ the boundary of T 1 first, and ψ T1 (f, θ) < ψ S1 (f, θ) if the ray f + λ hits the boundary of sin θ 4

5 F x The split S 1 The triangle T 1 D C R A O M B N E x 1 Figure : Computing Pr[ψ S (f, θ) < ψ T1 (f, θ)] S 1 first. When θ [, MOC) or θ (π MOB, π), OR is contained in the cone bounded by OB and OC, and hits the boundary of S first, so ψ T1 (f, θ) < ψ S1 (f, θ). Similarly, when θ ( MOC, MOF ) or θ (π MOA, π MOB), ψ S1 (f, θ) < ψ T1 (f, θ); when θ [ MOF, π MOA] or θ is equal to MOC or π MOB, ψ S1 (f, θ) ψ T1 (f, θ). Therefore, by (5), Pr[ψ S1 (f, θ) < ψ T1 (f, θ) f] AOB + COF π Pr[ψ T1 (f, θ) < ψ S1 (f, θ) f] BOC. π, Pr[ψ S1 (f, θ) ψ T1 (f, θ) f] AOF π, In BOC, OB (1 f 1 ) + f, OC (1 f 1 ) + (1 f ) and BC 1. By the law of cosines, cos BOC OB + OC BC OB OC f (f 1) + (1 f 1 ) [f + (1 f 1 ) ][(1 f ) + (1 f 1 ) ] πα(f). Therefore, Pr[ψ T1 (f, θ) < ψ S1 (f, θ) f] α(f). Similarly, AOF πβ(f) and AOB + COF πγ(f). Therefore, Pr[ψ S1 (f, θ) ψ T1 (f, θ) f] β(f), Pr[ψ S1 (f, θ) < ψ T1 (f, θ) f] γ(f). Lemma 1 provides the probabilities that a single coefficient of the split cut C S1 is smaller than, equal to, and larger than that of the triangle cut C T1 as a function of f. To compare the other split and type 1 triangles in Figure 1, we only need to change f 1 to 1 f 1 or f to 1 f in α(f), β(f) and γ(f) by symmetry. The following theorem gives the conditional probability that the split cut C S1 dominates the triangle cut C T1 with respect to f and the number of continuous variables k. Theorem 1. Pr[C S1 D C T1 f] [β(f) + γ(f)] k [β(f)] k. 5

6 Proof. Pr[C S1 D C T1 f] Pr[ψ S1 (f, θ j ) ψ T1 (f, θ j ), j f] Pr[ψ S1 (f, θ j ) ψ T1 (f, θ j ), j f] Pr[ψ S1 (f, θ j ) ψ T1 (f, θ j ) f] k Pr[ψ S1 (f, θ j ) ψ T1 (f, θ j ) f] k [β(f) + γ(f)] k [β(f)] k, where the second equality follows from the assumption that θ j (j 1,, k) are i.i.d.. Given integer free bodies L 1 and L, let R D (L 1, L ) {f : Pr[C L1 D C L f] > Pr[C L D C L1 f]}. The following corollary follows from Theorem 1. Corollary 1. R D (S 1, T 1 ) {f : γ(f) > α(f)} and R D (T 1, S 1 ) {f : α(f) > γ(f)}. By symmetry, after appropriately translating f, we can similarly describe the regions R D (S i, T j ) and R D (T j, S i ) for i 1, and j 1,, 3, 4 corresponding to any of the two splits and four type 1 triangles in Figure 1. Figures 3(a) and 3(b) show the regions 4 R D(S 1, T j ) and 4 R D(S, T j ), respectively shaded in black. The white regions in these figures indicate 4 R D(T j, S 1 ) and 4 R D(T j, S ), respectively. Since the union of the two black regions covers the unit square, there is no f for which a type 1 triangle cut C T satisfies that Pr[C T D C Si f] > Pr[C Si D C T f] (i 1, ). It follows from the discussion above that if we are only allowed to add one cut, when f 4 R D(S 1, T j ), we would select S 1, and when f 4 R D(T j, S 1 ), we would select S. (a) The region 4 R D(S 1, T j) (b) The region 4 R D(S, T j) Figure 3: The region. 3. Volume comparison In this section, we compare cuts based on the volume cut off from the linear relaxation of system (1). First we describe how the volume cut off is computed. 6

7 Suppose that C : k a jy j 1, with a j for all j, is a valid inequality for system (1). Consider the linear relaxation of (1) x f + r j y j x R, y j. { Let X LP be the set of feasible solutions of system (6) and S C X LP (x, y) : } k a jy j 1. Let vol(s C ) denote the volume of the polyhedron S C, which is also the volume cut off from S by the valid inequality C. The following lemma gives the volume of S C. Lemma. { + if j such that aj vol(s C ) otherwise (7) α n! k a j where α is a constant depending on the rays r 1,, r k. Proof. When a j for some j, S C is an unbounded polyhedron, and vol(s C ) +. When a j > for all j, S C is a k-dimensional polytope containing (f, ). Let } Proj y (S C ) {y R k : x R such that (x, y) S C be the projection of S C onto the y space. Proj y (S C ) is a k-dimensional simplex with, 1 a 1 e 1,, 1 a k e k as its (k + 1) vertices, where e j is the j-th unit vector. Therefore, vol(proj y (S C )) n! a 1 a k 1 n! k a. j Each point in S C is just an affine transformation of a point in the simplex Proj y (S C ), so vol(s C ) and vol(proj y (S C )) only differ by a factor α depending on the rays r 1,, r k. Thus vol(s C ). α n! k a j By Lemma, it suffices to compute the product of cut coefficients when we compare cuts based on the volume cut off from the linear relaxation. Now consider the split S 1 and type 1 triangle T 1 as in Section 3.1. As before, the analysis easily extends to another pair of split and type 1 triangle bodies by symmetry. Note that for fixed f (, 1), ψ T1 (f, θ j ) > with probability one. Moreover, since θ j is continuously distributed, Pr[ j s.t. ψ S1 (f, θ j ) ] Pr[ j s.t. θ j π or 3π ]. Theorem. Pr[C S1 V C T1 f] Pr[ ln ψ S 1 (f, θ j ) ψ T1 (f, θ j ) < ]. Proof. From Definition, Lemma and the fact that ψ S1 (f, θ j ) > and ψ T1 (f, θ j ) > with probability one, we have that Pr[C S1 V C T1 f] Pr[vol(S CS1 ) > vol(s CT1 ) f] α Pr[ n! k ψ S 1 (f, θ j ) > α n! k ψ T 1 (f, θ j ) ] Pr[ ln ψ S 1 (f, θ j ) ψ T1 (f, θ j ) < ]. (6) 7

8 Next we analyze the asymptotic behavior of the probability Pr[C S1 V C T1 f] as the number of continuous variables k increases. Before presenting further results, we give two technical lemmas. Lemma 3. Proof. See the appendix. ln cos xdx π ln and (ln cos x) dx <. To simplify the notation, let X j (f) ln ψ S 1 (f,θ j ) for every j 1,..., k. Note that for a fixed ψ T1 (f,θ j ) f (, 1), the random variable X j (f) is uniquely determined by θ j. The assumption that θ j, for j 1,..., k, are i.i.d. implies that X j (f), for j 1,..., k, are also i.i.d.. Let µ f E[X j (f)] and σf Var[X j(f)] for any j 1,..., k. Lemma 4. µ f < and σ f <. Proof. See the appendix. Now we present the asymptotic result on the probability that a split cut cuts off more volume than a type 1 triangle cut as the number of continuous variables increases. Theorem 3. lim Pr[C S 1 V C T1 f] 1 if µ f < 1/ if µ f if µ f >. Proof. From Theorem, we know Pr[C S1 V C T1 f] Pr[ k X j(f) < ]. Since X j (f) (j 1,..., k) are i.i.d., we can apply the Weak Law of Large Numbers and the Central Limit Theorem. k X j (f) Let X k (f) k. Since µ f is finite (Lemma 4), by the Weak Law of Large Numbers, lim Pr[ X k(f) µ f < ] 1 for any >. We consider three cases: µ f <, µ f > and µ f. (1) µ f <. Choose µ f. Then Pr[ X j (f) < ] Pr[X k (f) < ] Pr[X k (f) µ f < ] Pr[ X k (f) µ f < ]. Thus, lim inf Since lim sup Pr[ Pr[ X j (f) < ] lim inf X j (f) < ] 1, lim Pr[ X j (f) < ] 1. () µ f >. Choose µ f. Then Pr[ Pr[ X k(f) µ f < ] lim Pr[ X k(f) µ f < ] 1. X j (f) < ] Pr[X k (f) < ] Pr[X k (f) µ f < ] Pr[ X k (f) µ f > ]. 8

9 Thus, lim sup Pr[ Since lim inf X j (f) < ] lim sup Pr[ X k (f) µ f > ] lim Pr[ X k(f) µ f > ]. X j < ], Pr[ X j (f) < ]. Pr[ (3) µ f. From Lemma 4, σf is finite. By the Central Limit Theorem, X k(f) µ f σ f / converges to k the standard normal random variable N (, 1) in distribution. Thus lim Pr[ X j < ] lim Pr[X k(f) µ f σ f / k < ] 1. Define R V (S 1, T 1 ) {f : µ f < } and R V (T 1, S 1 ) {f : µ f > }. Then, R V (S 1, T 1 ) indicates the region where the split cut C S1 cuts off more volume than the type 1 triangle cut C T1 with probability close to 1 when k is large, and R V (T 1, S 1 ) indicates the region where the type 1 triangle cut C T1 cuts off more volume than the split cut C S1 with probability close to 1 when k is large. Even though θ j has a simple distribution, it is difficult to analytically compute µ f. However we can estimate µ f by Monte Carlo simulation for a given value of f, and identify the regions R V (S 1, T 1 ) and R V (T 1, S 1 ). The black and white regions in Figure 4 indicate R V (S 1, T 1 ) and R V (T 1, S 1 ), respectively. These have been identified as follows. First we randomly generate 1 5 fractional points f in ; then for each f, we independently generate 1 θ j uniformly from [, π) and check if the sample mean of ln ψ S 1 (f,θ j ) ψ T1 (f,θ j ) is less or greater than zero to identify if the corresponding f is in R V (S 1, T 1 ) or R V (T 1, S 1 ). The area of the black region is approximately.9. nless f 1 is close to 1, the split cut C S1 cuts off more volume than the type 1 triangle cut C T1 with probability close to 1 when k is large, and therefore C S1 is preferred. Figure 4: The shape of R V (S 1, T 1 ) and R V (T 1, S 1 ). 9

10 4 Total Probabilities In this section, we use the conditional probabilities from the previous section to compute coefficient dominance and volume cut off probabilities for split and type 1 triangle cuts when f is random. As before, we focus on the split cut C S1 and the type 1 triangle cut C T1 and note that the analysis and conclusions extend to another pair of split and type 1 triangle bodies by symmetry. The total probability analysis provides some insight on how these cuts are likely to perform when no information about the instance is available. 4.1 Cut coefficient comparison By the law of total probability Pr[C S1 D C T1 ] Pr[ψ S1 (f, θ j ) < ψ T1 (f, θ j ), j] Pr[ψ S1 (f, θ j ) < ψ T1 (f, θ j ), j f]dφ(f) {Pr[ψ S1 (f, θ j ) < ψ T1 (f, θ j ) f]} k dφ(f), where Φ(f) is the cumulative distribution function of f and the last equality follows from the fact that θ j are i.i.d. for j 1,..., k. Recall that the conditional probability Pr[ψ S1 (f, θ j ) < ψ T1 (f, θ j ) f] is given in Lemma 1. The following theorem describes the performance of the split cut C S1 and type 1 triangle cut C T1 when there is only one continuous variable. Theorem 4. If k 1 then Pr[C S1 D C T1 ].46 >.5 Pr[C T1 D C S1 ]. Proof. Note that BOC, AOB and COF are shown in Figure. Then BOC Pr[C T1 D C S1 ] Pr[ψ T1 (f, θ) < ψ S1 (f, θ)]dφ(f) dφ(f). π Similarly, Pr[C S1 D C T1 ] The proof then follows from Lemma 5. AOB + COF π dφ(f). Lemma 5. BOC dφ(f) π and Proof. See the appendix. COD dφ(f) π COF π DOA dφ(f) π dφ(f).176. AOB dφ(f).5, π Now we consider the case k > 1. Theorem 5. For any k, Pr[C S1 D C T1 ] > Pr[C T1 D C S1 ]. Proof. Pr[C S1 D C T1 ] > AOB + COF ( π ( AOB π )k dφ(f) Pr[C T1 D C S1 ]. ) k dφ(f) ( BOC ) k dφ(f) π The second equality follows from symmetry since f is uniformly distributed in (, 1). 1

11 Theorem 5 states that a single split cut is more likely to dominate a single type 1 triangle cut under our probabilistic model no matter how many continuous variables there are in system (1). We also use Monte Carlo simulation to estimate the magnitude of the probabilities that one cut dominates another. The result is shown in Figure $&%('*),+-$&.(' $&. ' ),+/$&% ' # "! Figure 5: Pr[C S1 D C T1 ] and Pr[C T1 D C S1 ] wrt the number of rays k. From Figure 5, although Pr[C S1 D C T1 ] > Pr[C T1 D C S1 ] for all k, both probabilities are very small when k 5 indicating that it is unlikely that one cut totally dominates another when there are many continuous variables. 4. Volume comparison In this section we estimate Pr[C S1 V C T1 ] with respect to the number of continuous variables k ψ S1 (f, θ j ) k. Recall that Pr[C S1 V C T1 ] Pr[ ψ T1 (f, θ j ) < 1]. We use Monte Carlo simulation to estimate the above probabilities as follows. For each k {1,..., 1}, we randomly generate N 1 5 samples of f 1, f, θ 1,, θ k according to our probabilistic input model. The probability k ψ S (f, θ j ) k Pr[ ψ T1 (f, θ j ) < 1] is then estimated by the proportion of the N samples with ψ S (f, θ j ) ψ T1 (f, θ j ) < 1. The estimated probabilities with respect to k are shown in Figure 6. The estimated probability that C S1 cuts off more volume from the linear relaxation than C T1 increases as the number of continuous variables increases, converging to approximately.9. To explain this, note that lim Pr[C S 1 V C T1 ] lim Pr[C S1 V C T1 f]dφ(f). 11

12 # # Since Pr[C S1 V C T1 f] is bounded, by interchanging limit and integral and applying Theorem 3 we have lim Pr[C S 1 V C T1 ] lim Pr[C S 1 V C T1 f]dφ(f) {I(µ f < ) + 1 I(µ f )}dφ(f) I(µ f < )dφ(f) Pr[f R V (S 1, T 1 )], where I(A) is the indicator function of event A and R V (S 1, T 1 ) is defined in Section 3.. Figure 6 presents Pr[C S1 V C T1 ] with respect to the number of continuous variables k (in two different scales). Recall that, as observed in Figure 4, the area of R V (S 1, T 1 ) is approximately.9, which coincides with the observation in Figure 6. We can conclude C S1 is more likely to cut off more volume than C T1 when k is not too small given any instance of (1) with parameters distributed according to our probabilistic input model ,- 1., ,- 1.,-.8 ()+* ()+* $%'& $/'&!" ()+* ()+* $%'& $/'&!" Figure 6: Estimated Pr[ k ψ S (f,θ j ) ψ T1 (f,θ j ) < 1] with respect to k. 5 Conclusions In this paper, we propose a probabilistic model to compare split cuts and type 1 triangle cuts. The analysis can be extended to other classes of facet defining intersection cuts where the corresponding integer-free body contains the unit square, such as type triangles and quadrilaterals containing. In particular, for the comparison of volume cut off, similar results as in Theorem, Lemma 4 and Theorem 3 can be derived, since the type triangles and quadrilaterals are all bounded and the corresponding cut coefficients are strictly greater than zero. Although it might be difficult to compute the associated probabilities analytically, we can still estimate the probability numerically and obtain regions of f where one cut dominates another or cuts off more volume. The analysis for type 3 triangles is much less obvious since such a triangle does not contain. Another interesting question is how to extend our probabilistic analysis on cut comparisons to the model with explicit bounds on the y variables. In this model, the region cut off from the LP relaxation by an individual cut is not always a simplex, and therefore the volume comparison becomes more complicated. It would also be interesting to study how to extend our analysis on volume comparison to multiple rounds of cuts. Finally we note that, recently, after the first submission of the current paper, two 1

13 groups, Del Pia et al. [7] and Basu et al. [5], have also conducted probabilistic analyses of the strength of various families of two-row cuts, using different probabilistic models and comparison criteria. 6 Acknowledgments This work is supported by NSF grant CMMI to the Georgia Institute of Technology. The authors would also like to thank an anonymous associate editor and three anonymous referees for their thoughtful comments. References [1] K. Andersen, Q. Louveaux, R. Weismantel, and L. A. Wolsey, Inequalities from two rows of a simplex tableau, in IPCO XII, M. Fischetti and D. P. Williamson, eds., vol of Lecture Notes in Computer Science, Springer, 7, pp [] E. Balas, Intersection cuts-a new type of cutting planes for integer programming, Operations Research, 19 (1971), pp [3] A. Basu, P. Bonami, G. Cornuéjols, and F. Margot, Experiments with two-row cuts from degenerate tableaux, to appear in INFORMS Journal on Computing, (9). [4], On the relative strength of split, triangle and quadrilateral cuts, in SODA, C. Mathieu, ed., SIAM, 9, pp [5] A. Basu, G. Cornuéjols, and M. Molinaro, A probabilistic analysis of the strength of the split and triangle closures, (October 1). strength6.pdf. [6] G. Cornuéjols and F. Margot, On the facets of mixed integer programs with two integer variables and two constraints, Mathematical Programming, 1 (9), pp [7] A. Del Pia, C. Wagner, and R. Weismantel, A probabilistic comparison of the strength of split, triangle, and quadrilateral cuts, (September 1). arxiv/pdf/19/19.553v1.pdf. [8] S. S. Dey, A. Lodi, A. Tramontani, and L. A. Wolsey, Experiments with two row tableau cuts, 1. to appear in IPCO XIV. [9] S. S. Dey and Q. Louveaux, Split rank of triangle and quadrilateral inequalities, CORE Discussion Paper, 55 (9). [1] S. S. Dey and L. A. Wolsey, Lifting integer variables in minimal inequalities corresponding to lattice-free triangles, in IPCO XIII, A. Lodi, A. Panconesi, and G. Rinaldi, eds., vol. 535 of Lecture Notes in Computer Science, Springer, 8, pp [11] D. Espinoza, Computing with multi-row Gomory cuts, in IPCO XIII, A. Lodi, A. Panconesi, and G. Rinaldi, eds., vol. 535 of Lecture Notes in Computer Science, Springer, 8, pp

14 [1] L. Lovász, Geometry of numbers and integer programming, in Mathematical programming: recent developments and applications, M. Iri and K. Tanabe, eds., Kluwer Academic Publishers, 1989, pp [13] G. L. Nemhauser and L. A. Wolsey, Integer and combinatorial optimization, Wiley- Interscience, New York, Appendices A Proof of Lemma 3 Proof. By substitution of variables, ln cos xdx ln sin xdx π ln π ln π ln ln( sin x cos x )dx ln dx ln sin xdx. Then, ln sin x dx + ln sin ydy + ln sin ydy + ln sin ydy 4 π 4 ln cos x dx ln cos zdz ln sin ydy Therefore, π ln ln sin xdx. By substitution of variables, (ln cos x) dx (ln sin x) dx. Since sin x x for x π, then (ln sin x) (ln x). Moreover, (ln x) dx x(ln x) x ln x + x + d, where d is a constant. Thus, (ln x) dx π (ln π ) π ln π + π <. Therefore, (ln sin x) dx is finite. B Proof of Lemma 4 k X j (f) Proof. To simplify the notation, let X k (f) k. µ f E[ln ψ S1 (f, θ j )] E[ln ψ T1 (f, θ j )]. By (3), ψ T1 (f, θ j ) is bounded and strictly positive for fixed f (, 1). Thus ( ln ψ T1 (f, ) θ j ) is bounded and E[ln ψ T1 (f, θ j )] is finite. By (3), ψ S1 (f, θ j ) 1 cos θj λ S1 where f + λ S1 hits sin θ j the boundary of the split S 1. Thus, f 1 + λ S1 cos θ j 1 when θ j [, π ) and θ j ( 3π, π), and f 1 + λ S1 cos θ j when θ j ( π, 3π ). Therefore, ψ S 1 (f, θ j ) cos θ j 1 f 1 when θ j [, π ) and θ j ( 3π, π), and ψ S 1 (f, θ j ) cos θ j f 1 when θ j ( π, 3π ). The probability density function of θ j is 14

15 1 π I(θ j [, π)). Therefore, E[ln ψ S1 (f, θ j )] 1 π [ ln cos θ j 1 π [ 1 π [4 3π π 1 f 1 dθ j + ln cos θ j dθ j ln ψ S1 (f, θ j ) 1 π dθ j 3π π ln cos θ j f 1 dθ j + ln(1 f 1 )dθ j + ln f 1 dθ j + ln cos θ j dθ j 3π ln cos θ j dθ j π ln f 1 (1 f 1 )] 3π 3π π 3π ln cos θ j 1 f 1 dθ j ] ln( cos θ j )dθ j ln(1 f 1 )dθ j ] By Lemma 3, ln cos θ jdθ j π ln. Therefore, E[ln ψ S1 (f, θ j )] is finite and µ f <. It only remains to verify that σ f is finite. Since σf E[(X j(f)) ] µ f, we need to verify that E[(X j (f)) ] is finite. E[(X j (f)) ] E[(ln ψ S 1 (f, θ j ) ψ T1 (f, θ j ) ) ] E[(ln ψ S1 (f, θ j )) ] E[ln ψ S1 (f, θ j ) ln ψ T1 (f, θ j )] + E[(ln ψ T1 (f, θ j )) ]. Since we have shown that ln ψ T1 (f, θ j ) is bounded and E[ln ψ S1 (f, θ j )] is finite for fixed f, the last two terms in the above equation are finite. For the first term E[(ln ψ S1 (f, θ j )) ], substitute the formula for ln ψ S1 (f, θ j ) and expand it as an integration, E[(lnψ S1 (f, θ j )) ] (ln cos θ j 1 f 1 ) 1 π dθ j + 3π 1 π [4 (ln cos θ j ) dθ j 4 ln f 1 (1 f 1 ) π (ln cos θ j ) 1 f 1 π dθ j + 3π (ln cos θ j 1 f 1 ) 1 π dθ j ln cos θ j dθ j + π(ln(1 f 1 )) + π(ln f 1 ) ] By Lemma 3, (ln cos θ j) dθ j and ln cos θ jdθ j are both finite. Thus, E[(ln ψ S1 (f, θ j )) ] <. Therefore, Var(X j ) is finite. C Proof of Lemma 5 Proof. Indeed, since Φ is uniformly distributed over, COD 1 1 dφ(f) lim π COD df 1 df π In COD, OC (1 f 1 ) + (1 f ), OD f 1 + (1 f ) and CD 1. By the law of cosines, cos COD OD + OC CD OD OC f 1 (f 1 1) + (1 f ) [f 1 + (1 f ) ][(1 f 1 ) + (1 f ) ] 15

16 f Therefore, COD arccos 1 (f 1 1)+(1 f ). Similarly, [f 1 +(1 f ) ][(1 f 1 ) +(1 f ) ] By substitution of variables, BOC arccos f 1 1 g,f g 1 g 1 f 1,g f Similarly, we can show Therefore, 1 1 BOC π f (f 1) + (1 f 1 ) [f + (1 f 1 ) ][(1 f ) + (1 f 1 ) ] DOA df 1 df π (1 ) Thus, dφ(f).5. Now we compute COF π dφ(f). COF dφ(f) lim π COD df 1 df π arccos arccos arccos arccos 1 1 f 1 (f 1 1)+(1 f ) [f 1 +(1 f ) ][(1 f 1 ) +(1 f ) ] π g (g 1)+(1 g 1 ) [(1 g ) +(1 g 1 ) ][g +(1 g 1) ] π g (g 1)+(1 g 1 ) [g +(1 g 1 ) ][(1 g ) +(1 g 1 ) ] π f (f 1)+(1 f 1 ) [f +(1 f 1 ) ][(1 f ) +(1 f 1 ) ] BOC df 1 df π BOC df 1 df π π AOB π df 1df 1 1 BOC + COD + DOA + AOB df 1 df π π π df 1df arccos lim COF π df 1df f1 +f f 1 3f + [(1 f ) +(1 f 1 ) ][f1 +( f ) ] π df 1 df dg dg 1 dg 1 dg df 1 df BOC df 1 df π df 1 df.176. In the final step, we used the Matlab function dblquad with 1 8 for the numerical calculation. 16