Relations Between Facets of Low- and High-Dimensional Group Problems

Transcription

1 Mathematical Programming manuscript No. (will be inserted by the editor) Santanu S. Dey Jean-Philippe P. Richard Relations Between Facets of Low- and High-Dimensional Group Problems Received: date / Accepted: date Abstract One-dimensional infinite group problems have been extensively studied and have yielded strong cutting planes for mixed integer programs. Although numerical and theoretical studies suggest that group cuts can be significantly improved by considering higher-dimensional groups, there are no known facets for infinite group problems whose dimension is larger than two. In this paper, we introduce an operation that we call sequential-merge. We prove that the sequential-merge operator creates a very large family of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. Further, they exhibit two properties that reflect the benefits of using facets of highdimensional group problems: they have continuous variables coefficients that are not dominated by those of the constituent low-dimensional cuts and they can produce cutting planes that do not belong to the first split closure of MIPs. Further, we introduce a general scheme for generating valid inequalities for lower-dimensional group problems using valid inequalities of higher-dimensional group problems. We present conditions under which this construction generates facet-defining inequalities when applied to sequentialmerge inequalities. We show that this procedure yields some two-step MIR inequalities of Dash and Günlük. This research was supported by NSF Grant DMI S.S. Dey School of Industrial Engineering, Purdue University. santanu.dey@alumni.purdue.edu J.-P. P. Richard School of Industrial Engineering, Purdue University. jprichar@ecn.purdue.edu

2 2 Santanu S. Dey, Jean-Philippe P. Richard Keywords Mixed integer programming high-dimensional infinite group problem facet-defining inequalities cutting planes. Introduction Branch-and-cut algorithms are the cornerstone of solution methodologies for Mixed Integer Programming problems (MIPs); see Marchand et al. [2] and Johnson, Nemhauser, and Savelsbergh [20]. In the last decade, a vast amount of research has been invested into generating strong general purpose cutting planes that are easy to separate. One of the most common approaches is to generate cutting planes from single-constraint relaxations of the problem. This approach has proven to be successful in many cases. However it seems that stronger cutting planes can be obtained if information from multiple constraints of the problem is considered simultaneously as the interaction between multiple constraints can be better captured. In a series of papers, Gomory [4], Gomory and Johnson [5,6], Johnson [9], Gomory, Johnson, and Evans [8], and Gomory and Johnson [7] showed how group relaxations can be used to generate cutting planes for general mixed integer programs. Although the group-theoretic approach can be applied to problems with m constraints (m-dimensional group problems), most research have considered only one-dimensional group relaxations; see Gomory and Johnson [5,6,8,7], Aráoz et al. [3], Miller, Li and Richard [22], Richard, Li, and Miller [23] and Dash and Günlük [7] for descriptions and derivations of large families of facet-defining inequalities for one-dimensional group problems. In particular, Gomory s Mixed Integer Cut (GMIC), a facet of the one-dimensional group problem, has been empirically proven to be one of the most useful cutting plane for solving general mixed integer programs; see Bixby et al. [4] and Bixby and Rothberg [5]. In a recent review of non-traditional approaches to mixed integer programming Aardal, Weismantel, and Wolsey [] mentioned that: Given the recent computational interest in using Gomorys fractional cuts, mixed integer rounding inequalities and Gomorys mixed integer cuts, this reopens questions about the possible use of alternative subadditive functions to generate practically effective cutting planes. It is also natural to ask whether interesting higher dimensional functions can be found and put to use... Further, Fischetti and Saturni [3] and Dash and Günlük [8] conducted numerical studies to gauge the strength of general one-dimensional group cuts with respect to GMICs. These numerical experiments suggest that, deriving stronger group cuts through the simultaneous use of multiple tableau rows and through the strengthening of continuous variables coefficients is a research direction more promising than deriving new facets of one-dimensional group problems. Gomory and Johnson [7] also write about facets of twodimensional infinite group problems that: There are reasons to think that such inequalities would be stronger since they deal with the properties of two rows, not one. They can

3 Facets of Low- and High-Dimensional Group Problems 3 also much more accurately reflect the structure of the continuous variables. Therefore, discovering facet-defining inequalities for high-dimensional group problems is an important area of research towards deriving stronger cutting planes for general MIPs. Although, the theoretical foundation for the study of infinite group problems was laid in the 970 s, generating specific families of facet-defining inequalities for high-dimensional group problems has been a challenge ever since. This is because tools to prove that a function is facet-defining for one-dimensional infinite group problems do not have straightforward generalizations to high-dimensional problems; see Gomory and Johnson [7] for a discussion on the challenges in proving that functions are facet-defining for high-dimensional group problems. There are only a few papers that focus on group problems with multiple constraints. Johnson [9] presents general theoretical results for group relaxations of Mixed Integer Programs with multiple constraints. Recently, Dey and Richard [] introduced tools to study two-dimensional infinite group problems, developed techniques to prove functions are facet-defining for these problems and introduced two families of facet-defining inequalities for two-dimensional group relaxations. In particular, we note that only very few families of facet-defining inequalities are known for two-dimensional group problems, none are known for higher-dimensional group problems and tools to prove that functions are facet-defining are restricted to two dimensions. In this paper, we present a procedure that we call as sequential-merge because it consists of applying two group cuts one after the other in a process that depends on the order of the cuts applied. We introduce new theoretical tools to prove that this procedure generates facets of general highdimensional infinite group problems from facets of lower-dimensional infinite group problems. To the best of our knowledge, this is the first known family of facet-defining inequalities for general high-dimensional infinite group problems. We also show in this paper how facets of high-dimensional group problems can be used to generate facets of low-dimensional group problems. In Section 2 we give a brief review of the group approach and introduce some new results that are more suitable to prove functions are facet-defining in the case of sequential-merge inequalities. We also describe a relationship between valid inequalities of group problems and certain types of lifting functions. In Section 3, we describe the sequential-merge procedure. We show that this procedure shares some relationship with the two-step MIR procedure of Dash and Günlük [7] and can be used to explain the family of two-dimensional three-gradient facets obtained by Dey and Richard []. In Section 4, we prove that under mild conditions, the sequential-merge procedure generates facets for high-dimensional infinite group problems. In Section 5 we analyze the types of inequalities that can be generated for low-dimensional group problems using high-dimensional sequential-merge group cuts. Surprisingly, even though the sequential-merge procedure generates facet-defining inequalities for high-dimensional group problems under very general conditions, the conditions under which it generates facets for low-dimensional group problems are more restrictive. We conclude in Section 6 with directions of future research.

4 4 Santanu S. Dey, Jean-Philippe P. Richard An extended abstract of some of the results in Sections 2-4 specific to the case of two-dimensional group problems was presented at IPCO XII held in Ithaca, June 2007 [0]. 2 Group Approach and Lifting In this section, we present fundamental results about group problems and describe the notion of valid and facet-defining inequalities for these problems. These results were introduced and proven by Gomory and Johnson [5, 7] and Johnson [9]. Although some of these results were originally presented in the context of the one-dimensional group problem, most of the proofs are independent of the dimension of the group. We therefore present them in the more general setting. We then present some relations between valid inequalities of group problems and certain lifting functions that extend results from Richard, Li, and Miller [23]. These relations will allow us to give an intuitive interpretation of the sequential-merge operation as a two-stage cut generation procedure in Section 3. First, we denote by I m the group of real m-dimensional vectors where the group operation is addition modulo componentwise, i.e., I m = {(x, x 2...x m ) 0 x i < i m}. We refer to the vector (0, 0,..., 0) I m as o. Because it is clear from context, the symbol + is used to denote both the addition in R m and I m. Next we give a formal definition of the group problem. Definition (Gomory and Johnson [5], Johnson [9]) For r I m with r o, the group problem PI(r, m) is the set of functions t : I m R such that. t has a finite support, i.e., t(u) > 0 for a finite subset of I m. 2. t(u) is a non-negative integer for all u I m, 3. u Im ut(u) = r. Next we define the concept of a valid inequality for the group problem. Definition 2 (Gomory and Johnson [5], Johnson [9]) A function φ : I m R + is said to define a valid inequality for PI(r, m) if φ(o) = 0, φ(r) = and u Im φ(u)t(u), t PI(r, m). In the remainder of this paper, we will use the terms valid function and valid inequality interchangeably. It can be verified that given the simplex tableau n i= a ix i = b of an integer program P with m rows, the inequality n i= φ(p(a i))x i is valid for P if φ is valid for PI(r, m), P(a i ) = (a i (mod), a 2i (mod),...a mi (mod)) and P(b) = r; see Gomory and Johnson [7]. We next describe necessary conditions for valid inequalities φ to be strong. Definition 3 (Gomory and Johnson [5]) A valid inequality, φ, for PI(r, m) is said to be subadditive if φ(u) + φ(v) φ(u + v), u, v I m.

5 Facets of Low- and High-Dimensional Group Problems 5 Gomory and Johnson [5] prove that all valid functions of PI(r, m) that are not subadditive are dominated by valid subadditive functions of P I(r, m). Therefore it is sufficient to study the valid subadditive functions of P I(r, m). Next we introduce a definition to characterize strong inequalities. Definition 4 (Gomory and Johnson [5]) A valid inequality, φ, is said to be minimal for PI(r, m) if there does not exist a valid function φ for PI(r, m) different from φ such that φ (u) φ(u) u I m. We next present a result characterizing minimal functions. Theorem (Gomory and Johnson [5]) A valid function φ is minimal for PI(r, m) iff φ(u) + φ(r u) = u I m. Further, if φ is minimal, then φ is subadditive. Minimal inequalities for PI(r, m) are strong because they are not dominated by any single valid inequality. However, there is a stronger class of valid inequalities that Gomory and Johnson refer to as facet-defining inequalities. We present the definition of these inequalities next. Definition 5 (Facet, Gomory and Johnson [7]) Let P(φ) = {t PI(r, m) u I m, t(u)>0 φ(u)t(u) = }. We say that an inequality φ is facetdefining for PI(r, m) if there does not exist a valid function φ such that P(φ ) P(φ). Gomory and Johnson [7] proved that all facet-defining inequalities are minimal inequalities. To prove that a function is facet-defining, Gomory and Johnson [7] introduced a tool that they refer to as the Facet Theorem. We describe this result in Theorem 2 and introduce a necessary definition next. Definition 6 (Equality Set, Gomory and Johnson [7]) For each point u I m, we define g(u) to be the variable corresponding to the point u. We define the set of equalities of φ to be the system of equations g(u) + g(v) = g(u + v) for all u, v I m such that φ(u) + φ(v) = φ(u + v). We denote this set as E(φ). Theorem 2 (Facet Theorem, Gomory and Johnson [7]) If φ is minimal and subadditive, and if φ is the unique solution of E(φ) then φ is a facet. A related result that will be used to prove that functions are facets is presented next. Proposition (Dey [9]) Let φ be a valid, subadditive and minimal function for PI(r, m). The function φ is not facet-defining if and only if there exists a valid subadditive and minimal function φ such that E(φ ) E(φ). The following result of Aczél [2] is helpful in proving that E(φ) has a unique solution. Proposition 2 (Aczél [2]) Let K be the closed interval [0, ǫ] R for ǫ > 0. If g : K R is such that g(x) + g(y) = g(x + y) x, y K and g(x) 0 for arbitrarily small x K, then g(x) = cx x K, where c R +.

6 6 Santanu S. Dey, Jean-Philippe P. Richard The Facet Theorem is the only tool that has been used to date to prove that valid functions are facet-defining inequalities. Therefore, all known continuous facets of PI(r, ) satisfy the following property. Definition 7 Let φ be a valid continuous function for PI(r, k) where k Z + and r I k. We say that the solution of E(φ) is unique up to scaling if for any other continuous function φ : I k R +, E(φ ) E(φ) implies that φ = cφ for c R +. We observe that all facets for infinite group problems known to date are also piecewise linear. A function φ is defined to be piecewise linear if I m can be divided into polytopes such that the function φ is linear over each polytope; see Gomory and Johnson [7] and Dey and Richard []. Gomory and Johnson [7] conjectured that all facets of infinite group problems are piecewise linear. Therefore, when introducing tools to prove that inequalities are facet-defining, it is usual to assume that the inequality under study is piecewise linear. Next we present in Theorem 4 a result regarding the continuity of valid functions of P I(r, m) that is used in the proof of the Sequential-Merge Theorem of Section 4. Theorem 4 is proven using the following preliminary result. Theorem 3 (Dey at al. [2]) If a valid function φ for PI(r, m) satisfies the following conditions. φ(x) + φ(y) φ(x + y) x, y I m, 2. lim h 0 φ(p(hd)) h exists for all d R m, then φ is continuous. Theorem 4 Let φ be a minimal piecewise linear and continuous function for PI(r, m). If ψ is a valid function for PI(r, m) such that E(φ) E(ψ) then ψ is continuous. Proof Since φ is minimal φ(x) + φ(r x) = φ(r) x I m. By assumption E(φ) E(ψ), and therefore ψ satisfies these equalities. Since ψ is also valid, it follows from a Theorem that ψ is minimal. We conclude from Theorem that ψ is subadditive. Let d R m be any direction and denote the line segment between the origin and the point dǫ as [0, ǫ]. We choose ǫ sufficiently small so that φ is linear along the line segment [0, ǫ]. This can be done since φ is assumed to be piecewise linear and continuous. Then φ(x) + φ(y) = φ(x + y) for all x, y [0, ǫ 2 ]. Since ψ satisfies these equalities and ψ(x) 0 x Im, it follows from Proposition 2 that ψ(x) = cx for x [0, ǫ]. We conclude that ψ satisfies both conditions of Theorem 3 and so ψ is continuous. Generating strong inequalities for group problems is often difficult. However, Richard, Li and Miller [23] showed that lifting can be used to derive valid and facet-defining inequalities for one-dimensional group problems. The family of facet-defining inequalities we present here is also easier to derive using lifting functions. In the remainder of this section, given any x I m, we use x R m to denote the vector x = ( x, x 2,..., x m ) such that P( x) = x and 0 x i < i.

7 Facets of Low- and High-Dimensional Group Problems 7 Definition 8 (Lifting-Space Representation) Given a valid inequality φ for PI(r, m), we define the lifting-space representation of φ as [φ] r : R m R where m m [φ] r (x) = x i r i φ(p(x)). i= To illustrate the idea that motivates this definition, we discuss the case where m =. Consider a row of the simplex tableau n i= a ix i = a 0 of an integer program, where a i R i {,..., n}, the fractional part r of a 0 is nonzero, and x i s are nonnegative integer variables. If φ is a valid function for PI(r, ) we have that n i= φ(a i)x i is a valid cut for the original IP. Multiplying this cut with r and then subtracting it from the original row we obtain n i= [φ] r(a i )x i [φ] r (r). One well-known example of the relation between the group-space and the lifting-space representation of an inequality is that of the Gomory Mixed Integer Cut (GMIC) and the Mixed Integer Rounding (MIR) inequality. It can be easily verified that the form in which MIR is presented is [GMIC] r. Thus, intuitively, the construction of the lifting-space representation given in Definition 8 is a generalization of the relation that GMIC shares with MIR to other group cuts of one and higher dimensions. Propositions 3 and 4 are generalizations of results from Richard, Li, Miller [23]. i= Proposition 3 If φ is valid function for PI(r, m),. [φ] r (x + e i ) = [φ] r (x) +, where e i is the i th unit vector of R m. We say that [φ] r is pseudo-periodic. 2. [φ] r is superadditive iff φ is subadditive. Proof. [φ] r (x + e i ) = m i= x i + m i= r iφ(p(x)) = [φ] r (x) Assume first that φ is subadditive. For any x, y R m, we have m m m m [φ] r (x) + [φ] r (y) = x i r i φ(p(x)) + y i r i φ(p(y)) i= i= i= i= m m (x i + y i ) r i φ(p(x + y)) = [φ] r (x + y). Now assume that [φ] r is superadditive over R m. For any x, y I m we have m i= φ(x) + φ(y) = x m i [φ] r ( x) i= m i= r + ỹi [φ] r (ỹ) m i i= r i i= i= m i= ( x i + ỹ i ) [φ] r ( x + ỹ) m i= r i = φ(p( x + ỹ)) = φ(x + y).

8 8 Santanu S. Dey, Jean-Philippe P. Richard Motivated by Definition 8, we define next the inverse operation. Definition 9 (Group-Space Representation) Given a superadditive function ψ : R m R that is pseudo-periodic, we define the group-space representation of ψ as [ψ] r : I m R where [ψ] m i= r (x) = xi ψ(x) m. i= ri Proposition 4 A valid group-space function g : I m R is minimal iff [g] r is superadditive and [g] r (x) + [g] r (r x) = 0. Proof We know from Theorem that g is minimal iff g is subadditive and g(x) + g(r x) =. Note that by Definition, r o. Now, m i= x m i [g] r (x) i= m i= r + (r i x i ) [g] r (r x) m i i= r = i m i= r i [g] r (x) [g] r (r x) m i= r i = [g] r (x) + [g] r (r x) = 0. The result then follows from Proposition 3. 3 Sequential-Merge Inequalities for High-Dimensional Group Problems In this section, we introduce an operation that produces valid inequalities for PI(r, m+) from valid inequalities of PI(r, ) and PI(r, m). To simplify the notation, we denote x by x since the symbol is clear from the context. Definition 0 (Sequential-merge inequality) Let g and h be valid functions for PI(r, ) and PI(r 2, m) respectively. We define the sequential-merge of g and h as the function g h : I m+ R + where g h(x, x 2 ) = [[g] r (x + [h] r2 (x 2 ))] r (x, x 2 ) () and r = (r, r 2 ). In this construction, we refer to g as the outer function and to h as the inner function. Figure gives an example of facet-defining inequality for P I((0.4, 0.3), 2) that is obtained from the sequential-merge of a GMIC (Gomory and Johnson [5]) for PI(0.4, ) with a two-step MIR (Dash and Günlük [7]) for P I(0.3, ). There is an intuitive interpretation to the construction presented in Definition 0. Given m+ rows of a simplex tableau, we first generate a cutting plane in the lifting-space of the last m rows. This cutting plane is added to the first row of the tableau to generate a combined inequality. Finally, a one-dimensional cutting plane is generated from the combined inequality. Before presenting the analysis of sequential-merge inequalities, we mention that Gomory [4] also proposed a procedure to generate valid (and facetdefining) inequalities for group problems with multiple constraints. In particular, given two group problems based on finite groups G and G 2 respectively, Gomory described how to create facet-defining inequalities for P(G, r ) from

9 Facets of Low- and High-Dimensional Group Problems I Outer function I Inner function Sequential-Merge function (Facet of P I(0.4, )) (Facet of P I(0.3, )) (Facet of P I((0.4, 0.3), 2)) Fig. Example of sequential-merge operation facet-defining inequalities of P(G 2, r 2 ) provided that an homomorphism ψ is known from G onto G 2. We also mention that Gomory s construction depends on whether r 2 belongs to the kernel of ψ and we refer to [4] for details. A representative example of the use of this homomorphism result is the case where G is the direct product of a finite number of copies of a cyclic group G 2. The operation that aggregates the rows of the group problem defined on G can be verified to yield a homomorphism of G onto G 2. Therefore, Gomory s result shows that valid inequalities for the group problem P(G, r ) obtained by aggregating its rows and generating a facet-defining inequality for the corresponding group problem of the form P(G 2, r 2 ) are facet-defining for P(G, r ). The sequential-merge result that we will prove next is different in two respects. An obvious difference is that it applies to infinite group problems while the proof of Gomory s homomorphism result makes it applicable only for finite group problems. The more prominent difference, however, is that while Gomory s result uses only one inequality of a smaller group problem to generate an inequality of the larger group problem, our result uses two different source inequalities. Proposition 6 next states that the group-space representation of inequalities generated using the sequential-merge procedure is valid for P I(r, m+) when the outer function is nondecreasing in the lifting-space. Before we prove this result, we present a formula for the sequential-merge inequality in terms of the group representation of its inner and outer functions. For x I m, we use the notation X = m i= x i for x I m. Note that the summation defining X is performed in R, i.e., we add the numeric value of each component of x to obtain X. Proposition 5 Let g and h be valid inequalities for PI(r, ) and PI(r 2, m) respectively. Then g h(x, x 2 ) = R2h(x2)+rg(P(x+X2 R2h(x2))) r +R 2. Proof By Definition 8, [h] r2 (x 2 ) = X 2 R 2 h(x 2 ). Therefore [g] r (x +[h] r2 (x 2 )) = x +X 2 R 2 h(x 2 ) r g(p(x +X 2 R 2 h(x 2 ))). Let x +X 2 R 2 h(x 2 ) = p+q, where p = P(x +X 2 R 2 h(x 2 )) and q Z. Then [g] r (x +[h] r2 (x 2 )) =

10 0 Santanu S. Dey, Jean-Philippe P. Richard p+q r g(p). Finally using Definition 9, we obtain [[g] r (x +[h] r2 (x 2 ))] x +X 2 p q+r g(p) r +R 2 r = = R2h(x2)+rg(P(x+X2 R2h(x2))) r +R 2. Using Proposition 5 it is easy to verify that the sequential-merge operator is non-commutative in the case m = (h g is not defined for m > ). In the next proposition, we prove that sequential-merge inequalities are valid for the m+-dimensional group problems. Proposition 6 If g and h are valid subadditive functions for PI(r, ) and PI(r 2, m) respectively, and [g] r is nondecreasing, then g h is a valid subadditive function for PI(r, m+) where r (r, r 2 ). Proof To prove that g h is valid, it suffices to show that () g h(x, x 2 ) 0 (x, x 2 ) I m+, (2) g h(r, r 2 ) =, and (3) g h is subadditive.. Since g and h are valid inequalities, they are nonnegative by definition. Therefore it follows from Proposition 5 that g h(x, x 2 ) 0, (x, x 2 ) I m+. 2. Using substitution and the fact that g(r ) = h(r 2 ) =, we obtain that g h(r, r 2 ) =. 3. Because of Proposition 3, it is sufficient to verify that [g] r (x +[h] r2 (x 2 )) is superadditive. Since g and h are subadditive, it follows again from Proposition 3 that [g] r and [h] r2 are superadditive. For any (x, x 2 ) and (y, y 2 ) we obtain [g] r (x + [h] r2 (x 2 )) + [g] r (y + [h] r2 (y 2 )) [g] r (x + [h] r2 (x 2 ) + y + [h] r2 (y 2 )) [g] r (x + y + [h] r2 (x 2 + y 2 )) where the last inequality holds because [g] r is nondecreasing and [h] r2 is superadditive. We next show that the sequential-merge inequalities obtained using minimal inequalities are minimal. Proposition 7 If g and h are valid, minimal functions for PI(r, ) and PI(r 2, m), and [g] r is nondecreasing, then g h is a minimal function for PI(r, m+), where r (r, r 2 ). Proof It follows from Proposition 6, that [g h] (r,r 2) is superadditive. It therefore follows from Proposition 4 that we only need to show that [g h] r (x, x 2 ) + [g h] r (r x, r 2 x 2 ) = 0. We have [g h] r (x, x 2 ) + [g h] r (r x, r 2 x 2 ) = [g] r (x + [h] r2 (x 2 )) + [g] r (r x + [h] r2 (r 2 x 2 )) = [g] r (x + [h] r2 (x 2 )) + [g] r (r x [h] r2 (x 2 )) = [g] r (x + [h] r2 (x 2 )) [g] r (x + [h] r2 (x 2 )) = 0 where the second equality holds because h is minimal and the third equality holds because g is minimal. We next give two examples of known valid inequalities for group problems that can be obtained using the sequential-merge procedure.

11 Facets of Low- and High-Dimensional Group Problems Proposition 8 Consider κ(x) = [[ξ] r (x + [ξ] r (x))] (r,r)(x, x), where ξ is the GMIC, i.e., κ(x) is the sequential-merge inequality obtained using the same constraint twice and using GMIC as both the inner and outer function. Then κ(x) is a two-step MIR function of Dash and Günlük [7]. Proof For a right-hand-side of b, the two-step MIR of Dash and Günlük [7] is represented as g b,α (v) = { v( ρτ) k(v)(α ρ) ρτ( b) k(v)+ τv τ( b) if v k(v)α < ρ if v k(v)α ρ (2) where ρ = b α b/α, τ = b/α, k(v) = min{ v/α, τ}. For α = b +b, τ = + b = 2, ρ = b2 +b and the function is g b, b +b (v) = k(v) = { 0 v < α v α, +b 2b 2 2b 2 ( b) v 2v 2( b) v(+b 2b 2 ) b( b) 2b 2 ( b) v v < b2 b 2 +b +b v < b +b v < b b b v <. b +b (3) (4) Using Proposition 5, κ(x) = r2ξ(x)+rξ(p(2x r2ξ(x))) r +r 2. It is easy to verify that this function is the reflection automorphism of g b, b +b with r = b, i.e., g b, b +b (x) = κ( x); see Gomory and Johnson [7] for the Automorphism Theorem. We observe that sequential-merge procedure shares some relations with the two-step MIR procedure of Dash and Günlük [7]. An important difference however is that the sequential-merge procedure uses, in general, different rows of a simplex tableau. Also the two-step MIR procedure only uses MIR inequalities as constituent functions. We will define, compare and discuss further the properties of the sequential-merge procedure for one-dimensional group problems in Section 5. We describe in the next proposition another family of facets for the twodimensional group problem that can be obtained using the sequential-merge procedure. The proof involves simple algebraic manipulations and is omitted. Proposition 9 Consider ρ(x, y) = [[ξ] r (x + [ξ] r2 (y))] (r,r 2) (x, y), where ξ is the GMIC, i.e. ρ(x, y) is the sequential-merge inequality obtained using GMIC as both the inner and outer function. This inequality is the function ψ ϕ where ψ is three-gradient facet-defining inequality for PI((r, r 2 ), 2) of Dey and Richard [] and ϕ is the automorphism of I 2, ϕ(x, y) = ( y, x).

12 2 Santanu S. Dey, Jean-Philippe P. Richard 4 Facet-defining Sequential-Merge Inequalities for PI(r, m+) In this section, we derive conditions under which sequential-merge inequalities are facet-defining for the m+-dimensional group problem PI(r, m+). We begin by studying some geometric properties of g h. Definition (Spine) We define the set of points {(x, y) (I I m ) P(x+ Y R 2 h(y)) = 0} as the spine of the function g h. We denote the spine of g h by S(g h). To prove that g h is facet-defining sufficiently many points u, v I m+ must be found that satisfy the equality g h(u) + g h(v) = g h(u + v). The spine of g h simplifies this search for equalities as it satisfies the following important property: for every equality that h satisfies, there exists an equality that g h satisfies involving points of its spine. Thus, the spine of g h plays an important role in proving that g h is facet-defining, when h is facet-defining. Proposition 0 Let g and h be valid subadditive inequalities for PI(r, ) and PI(r 2, m) respectively. If v, v 2 I m are such that h(v ) + h(v 2 ) = h(v + v 2 ) and (u, v ), (u 2, v 2 ) S(g h) then. (u + u 2, v + v 2 ) S(g h). 2. g h(u, v ) + g h(u 2, v 2 ) = g h(u + u 2, v + v 2 ) Proof Assume that h(v )+h(v 2 ) = h(v +v 2 ). Let (u, v ), (u 2, v 2 ) S(g h). From Definition we know that u = P( V + R 2 h(v )) and u 2 = P( V 2 + R 2 h(v 2 )).. u + u 2 = P( V + R 2 h(v )) + P( V 2 + R 2 h(v 2 )) = P( (V + V 2 ) + R 2 (h(v )+h(v 2 ))) = P( (V +V 2 )+R 2 h(v +v 2 )) where the last equality holds because h(v ) + h(v 2 ) = h(v + v 2 ). Therefore, (u + u 2, v + v 2 ) S(g h). 2. For any point (u, v) S(g h) we have P(u+V R 2 h(v)) = 0. Therefore R2h(v)+rg(P(u+V R2h(v))) g h(u, v) = r +R 2 = R2 r +R 2 h(v). Now g h(u, v ) + g h(u 2, v 2 ) = u 2, v + v 2 ). R2 r +R 2 h(v ) + R2 r +R 2 h(v 2 ) = R2 r +R 2 h(v + v 2 ) = g h(u + Intuitively, because the function g h has the equalities of h on its spine, E(g h) will have an unique solution on its spine up to scaling whenever E(h) has a unique solution up to scaling. We next give a formal proof of this observation that will be used in the proof of Theorem 5. Proposition Let g and h be continuous, piecewise linear valid inequalities for PI(r, ) and PI(r 2, m) respectively. Assume that [g] r be nondecreasing, and that E(h) has an unique solution up to scaling. If ψ is a valid function for PI(r, m+) such that E(ψ) E(g h), then ψ(u, u 2 ) = cg h(u, u 2 ) = cr2 r +R 2 h(u 2 ) (u, u 2 ) S(g h) where c is a nonnegative real number. Proof Note first that Theorem 4 implies that ψ is continuous. Let η : I m I m+ be the function, η(u) = (P( U + R 2 h(u)), u), i.e., the image of η is

13 Facets of Low- and High-Dimensional Group Problems 3 the spine of g h. Since h is continuous, η is a continuous function. Define now the function h : I m R + as h(u) = r+r2 R 2 ψ η(u), i.e., h(u 2 ) = r +R 2 R 2 ψ(u, u 2 ) where (u, u 2 ) S(g h). We conclude h is a continuous function since ψ and η are continuous functions. Now let h(y ) + h(y 2 ) = h(y + y 2 ) be any equality that h satisfies. Then using Proposition 0 we obtain that g h(x, y ) + g h(x 2, y 2 ) = g h(x + x 2, y +y 2 ) where (x, y ), (x 2, y 2 ), (x +x 2, y +y 2 ) S(g h). Since E(ψ) E(g h), we obtain that ψ(x, y )+ψ(x 2, y 2 ) = ψ(x +x 2, y +y 2 ). Finally by the construction of h we obtain h(y )+ h(y 2 ) = h(y +y 2 ). Thus E( h) E(h). However E(h) has an unique solution up to scaling. Therefore h(u) = ch(u), or equivalently, ψ(u, u 2 ) = cr2 r +R 2 h(u 2 ) = cg h(u, u 2 ) (u, u 2 ) S(g h). Moreover since ψ is a nonnegative function, c is nonnegative. Although Proposition establishes that E(g h) has an unique solution up to scaling on its spine, it falls short of proving that E(g h) has an unique solution over I m+. Therefore, we identify in Propositions 2 and 4 some equalities that g h satisfies that help in extending the result of Proposition to I m+. Proposition 2 If g and h are valid inequalities for PI(r, ) and PI(r 2, m) respectively and if [g] r is nondecreasing, then g h(x, y ) + g h(δ, 0) = g h(x + δ, y ) δ I and (x, y ) S(g h). Proof Because (x, y ) S(g h), we know from Definition that P(x + Y R 2 h(y )) = 0. Therefore it can be verified that g h(x, y ) = R2h(y) r +R 2 and g h(x +δ, y ) = R2h(y)+rg(δ) r +R 2. Since g h(δ, 0) = rg(δ) r +R 2 we obtain that g h(x, y ) + g h(δ, 0) = g h(x + δ, y ). Next we present a simple result characterizing valid functions that are nondecreasing in their lifting-space representation. Proposition 3 Let f be a continuous and subadditive valid inequality for PI(r, k). If the function [f] r is nondecreasing then f(x) = X R for {x I k 0 x i r i i {,..., m}}. Formally, consider the topology on I induced by the metric function d : I I R + defined as d (u, v) = min { ũ ṽ, ũ ṽ } where ũ R such that 0 ũ < and P(ũ) = u. For I m, we consider the topology induced by the metric function d : I m I m R + defined as d(u, v) = m i= (d (u i, v i)) 2 ; see Dey et al. [2] for details. Using this metric, it can be verified that the function, P : R I is a continuous function. We illustrate the proof of continuity of P at 0 R: Let ǫ > 0. We need to show that there exists a δ > 0 such that if x < δ, then d (P(0), P(x)) < ǫ. If ǫ > 0.5, then δ can be any arbitrary positive number since d (P(0), P(x)) 0.5 x R. If ǫ 0.5, then let δ = ǫ. Note now for any x ( δ,0] d (P(0), P(x)) = min{ + x, ( + x)} = x < δ = ǫ. Similarly if x [0, δ), then d (P(0), P(x)) = x < δ = ǫ. Similarly, it can be verified that the function P( U) that maps I m to I is continuous. Since P : R I, P( U) : I m I, h : I m R are continuous, P( U+R 2h(u)) = P( U) + P(R 2h(u)) can be verified to be a continuous function and, in turn, η is a continuous function.

14 4 Santanu S. Dey, Jean-Philippe P. Richard Proof Since f is valid, we have that f(r) =. Therefore, [f] r (r) = R Rf(r) = 0. Now consider x R m such that 0 x i r i i {,..., m}. Let y R m be such that y i = r i x i i {,..., m}. Since f is subadditive, [f] r is superadditive. Then we have [f] r (x) + [f] r (y) [f] r (x + y) = [f] r (r) = 0. Finally, since [f] r is nondecreasing we have that 0 = [f] r (0) [f] r (x) and 0 = [f] r (0) [f] r (y). Therefore [f] r (x) = 0 or f(x) = X R. Henceforth, for x, y I m, we say x y if x i y i i m. Proposition 4 Assume that g and h are valid subadditive inequalities for PI(r, ) and PI(r, m) respectively. Assume also that [g] r and [h] r2 are nondecreasing functions. Then g h(x, x 2 ) = x+x2 r +R 2 (x, x 2 ) {(x, x 2 ) 0 x r, 0 x 2 r 2 }. Furthermore g h(u, v ) + g h(u 2, v 2 ) = g h(u + u 2, v + v 2 ) whenever u, u 2, u + u 2 r and v, v 2, v + v 2 r 2. Proof Since g is a valid subadditive function and [g] r is nondecreasing, we have from Proposition 3 that g(x ) = x r 0 x r. Similarly h(x 2 ) = X 2 R 2. The result then follows from Proposition 5. Theorem 5 (Sequential-Merge Theorem) Assume that g and h are continuous, piecewise linear, facet-defining inequalities of PI(r, ) and PI(r 2, m) respectively. Assume also that E(g) and E(h) are unique up to scaling and that [g] r and [h] r2 are nondecreasing. Then g h is a facet-defining inequality for PI((r, r 2 ), m+). Proof Assume by contradiction that g h is not facet-defining. Therefore we conclude from Theorem 2 that there exists a valid inequality ψ g h such that E(ψ) E(g h). To obtain a contradiction, we now prove that ψ(u) = g h(u) u I m+. The proof is in two steps. First we prove that g h(u) = ψ(u) u S(g h). Then we prove that g h(u) = ψ(u) u I m+. This will provide the required contradiction. We know from Proposition 4 that g h(x, x 2 ) = x+x2 r +R 2 0 x r and 0 (x 2 ) i (r 2 ) i i m. Because ψ satisfies all the equalities of g h, we have that for all 0 x r, 0 x 2 r 2 ψ(x, 0) + ψ(0, x 2 ) = ψ(x, x 2 ). (5) In particular, because g h(u, 0)+g h(u 2, 0) = g h(u +u 2, 0) (u, u 2 ) in the line segment between (0, 0) and ( r 2, 0), ψ must satisfy these equalities. Therefore by Proposition 2, Similarly it can be shown that ψ(kr, 0) = kψ(r, 0) k [0, ]. (6) ψ(0, kr 2 ) = kψ(0, r 2 ) k [0, ]. (7) Further, ψ must be minimal as g h is, and therefore ψ(r, 0) + ψ(0, r 2 ) =. (8)

15 Facets of Low- and High-Dimensional Group Problems 5 Let be the vector (,,..., ). Because h is a piecewise linear continuous function and h() = 0, there exists 0 < < such that h( δr 2 ) = δγ 0 δ for some γ 0. Select δ > 0 such that δ < min{,, }. Now r R 2+R 2γ g h(0, r 2 ) = R2 r +R 2, g h(δr 2 + R 2 γδ, δr 2 ) = R2 r +R 2 h( δr 2 ) = R2γδ r +R 2. Since r 2 δr 2 < r 2, we have from Proposition 3 that h(r 2 δr 2 ) = R2 δr2 R 2. Also δγr 2 + δr 2 < r, implies that g(γr 2 δ + δr 2 ) = γr2δ+δr2 r. Using these two relations, we obtain that g h(δr 2 + R 2 γδ, r 2 δr 2 ) = R2+R2γδ r +R 2. Thus, g h(0, r 2 )+g h(δr 2 +R 2 γδ, δr 2 ) = g h(δr 2 +R 2 γδ, r 2 δr 2 ). Therefore ψ satisfies the same equation, i.e., ψ(0, r 2 ) + ψ(δr 2 + R 2 γδ, δr 2 ) = ψ(δr 2 + R 2 γδ, r 2 δr 2 ). (9) r Since δ < R we have that δr 2+R 2γ 2 + R 2 γδ < r. Also r 2 δr 2 < r 2. Therefore using (5), (6), (7) we obtain ψ(δr 2 + R 2 γδ, r 2 δr 2 ) = δr 2 + R 2 γδ r ψ(r, 0) + R 2 δr 2 R 2 ψ(0, r 2 ). (0) Using (9) and (0) we obtain ψ(0, r 2 ) + ψ(δr 2 + R 2 γδ, δr 2 ) = δr 2 + R 2 γδ r ψ(r, 0) + R 2 δr 2 R 2 ψ(0, r 2 ). () From Proposition, we obtain that ψ(u) = cg h(u) u S(g h). In particular, we have c = ψ(0, r 2) g h(0, r 2 ) (2) since (0, r 2 ) belongs to S(g h) and g h(0, r 2 ) 0. It is easy to verify that (δr 2 + R 2 γδ, δr 2 ) S(g h) and (0, r 2 ) S(g h). Therefore by using Proposition again, we obtain ψ(δr 2 + R 2 γδ, δr 2 ) = cg h(δr 2 + R 2 γδ, δr 2 ) Substituting (3) in () we obtain = g h(δr 2 + R 2 γδ, δr 2 ) ψ(0, r 2 ) g h(0, r 2 ) = γδψ(0, r 2 ). (3) r ψ(0, r 2 ) = R 2 ψ(r, 0). (4) Thus using (8) and (4), we conclude that ψ(r, 0) = r r +R 2 = g h(r, 0) and ψ(0, r 2 ) = R2 r +R 2 = g h(0, r 2 ). Along with (2), this implies that c = and ψ(u) = g h(u) u S(g h). Next note that since ψ satisfies all the equalities of g h, ψ satisfies the equalities satisfied by g h along the x -axis. It is easy to verify that the equalities of g h along the x -axis are the same as those of g since g h(x, 0) =

16 6 Santanu S. Dey, Jean-Philippe P. Richard r r +R 2 g(x ). Since E(g) is unique up to scaling ψ(u) = c g h(u) for all u along the x -axis. Also ψ(r, 0) = g h(r, 0). Therefore ψ(u) = g h(u) for all u along the x -axis. Finally consider any point (u, v) I m+. There exists δ such that (u, v)+ (δ, 0) = (u, v), where (u, v) S(g h). Since ψ satisfies all the inequalities of g h, it follows from Proposition 2 that ψ(u, v) + ψ(δ, 0) = ψ(u, v). But ψ(u, v) = g h(u, v) since (u, v) belongs to S(g h) and ψ(δ, 0) = g h(δ, 0) as (δ, 0) belongs to the x -axis. Therefore, ψ(u, v) = g h(u, v). Theorem 5 presents some sufficient conditions under which the sequentialmerge operation generates facet-defining inequalities for high-dimensional group problems from facet-defining inequalities of lower-dimensional group problems. An interesting question is to determine which of these conditions are necessary for the sequential-merge operator to generate facet-defining inequalities. In Theorem 5, we assumed the technical conditions that g and h are facet-defining and E(g) and E(h) are unique up to scaling. Using Proposition it can be verified that if g is not facet-defining, then g h is not facet-defining, thus showing that the condition that g is facet-defining is necessary for g h to be facet-defining. The condition that E(g) is unique up to scaling is therefore not very restrictive as all known facet-defining inequalities for PI(r, ) satisfy this condition; see Gomory and Johnson [7]. On the other hand, the conditions that h is facet-defining and E(h) is unique up to scaling in Theorem 5 seem more restrictive. We conjecture that there may exist functions g h that are facet-defining but do not satisfy these conditions. We now extend the above results to the mixed integer case. To this end, we use a result from Johnson [9] which states that the coefficient of a continuous variable µ φ in a minimal group cut φ can be found as µ φ (u) = lim φ(p(hu)) h 0 + h where u R m+ is the column vector of coefficients of the continuous variable. The following proposition describes how the sequential-merge facets obtained for P I(r, m+) can be generalized to m+-dimensional mixed integer group cuts. Proposition 5 Let x R, y R m and let c + g = lim g(ǫ) ǫ 0 + ǫ = r, c g = lim g( ǫ) ǫ 0 + ǫ, c h (y) = lim h(ǫy) ǫ 0 + ǫ. The coefficients of the continuous variables for g h are given by R 2c h (y)+r c + g (x+y R2c h(y)) µ g h (x, y) = r +R 2 if (x + Y R 2 c h (y)) 0 R 2c h (y) r c g (x+y R2c h(y)) r +R 2 if (x + Y R 2 c h (y)) 0. (5) Proof The proofs for the two cases are similar. Therefore, we give the proof only for the first case. From Johnson [9] we obtain µ g h (u) = lim ǫ 0 + g h(p(ǫu)). (6) ǫ Since h is piecewise linear, h(ǫy) = ǫc h (y) for sufficiently small ǫ. Therefore P(ǫx + ǫy R 2 h(ǫy)) = P(ǫx + ǫy ǫr 2 c h (y)). Again, for sufficiently small

17 Facets of Low- and High-Dimensional Group Problems 7 ǫ, we have that g(p(ǫx + ǫy ǫr 2 c h (y))) = ǫc + g (x + Y R 2 c h (y)) since x + Y R 2 c h (y) 0. Therefore we obtain g h(p(ǫ(x, y))) ǫ = R 2h(ǫy) + r g(p(ǫx + ǫy R 2 h(ǫy))) (r + R 2 )ǫ = R 2ǫc h (y) + ǫr c + g (x + Y R 2 c h (y)) (r + R 2 )ǫ = R 2c h (y) + r c + g (x + Y R 2 c h (y)). (r + R 2 ) This completes the proof. Next we illustrate on an example how the sequential-merge procedure can be applied to mixed integer programs. Example Consider the following mixed integer set 3 x 3 y (7) 3 x y 3 2 (8) x, y Z + (9) whose feasible region is represented in Figure 2. We introduce continuous non-negative slack variables s and s 2 and perform simplex iterations to obtain the tableau x + 2s + s 2 = 7 2 (20) y s + s 2 = 2. (2) Using Proposition 5 and using GMIC as both the inner and outer functions, we obtain the sequential-merge cut s + 2s 2. This cut is illustrated in Figure 2 and is equivalent to x + y 3. It can easily be verified that this inequality is facet-defining for the convex hull of solutions to (7), (8) and (9). It is also easily shown that the two GMICs generated from the individual rows of the tableau are 4s +2s 2 and 2s +2s 2. They are equivalent to x 3 and 2x + y 6 respectively. Interestingly, it can be shown that these inequalities are not facet-defining for the convex hull of solutions to (7), (8) and (9). In this example, it can be proven that there exists no split disjunction that can be used to derive the cut x + x 2 3, therefore it cannot be obtained as a GMIC from any combination of tableau rows. To prove that no such disjunction exists, we show that the point ( 8 3, 2 3 ) belongs to the rank- split closure of (7), (8) and (9). This provides the desired proof as ( 8 3, 2 3 ) is cut off by x +x 2 3. Consider the integer points A (2, ), B (3, ), C (3, 0), and D (4, 0).

18 8 Santanu S. Dey, Jean-Philippe P. Richard 3 2 Sequential-Merge Cut ( 8 3, 2 3 ) A B 0 C D Fig. 2 Sequential-Merge cut with split rank 2. and Consider now any split disjunction P {(x, y) R 2 π x + π 2 y π 0 } (22) P 2 {(x, y) R 2 π x + π 2 y π 0 + } (23) where π 0, π, π 2 Z. Denote the LP relaxation of (7), (8) and (9) as LPR. Denote also the intersection of P with LPR as V and the intersection of P 2 with LPR as V 2. A point p is not cut off by the disjunctive cut if it is a convex combination of points in V and V 2. The following cases are sufficient to verify that ( 8 3, 2 3 ) belongs to the first split closure.. A, B, C P : It can be easily verified that the point ( 8 3, 2 3 ) LPR. Also ( 8 3, 2 3 ) P since ( 8 3, 2 3 ) = 3 (A + B + C). Therefore (8 3, 2 3 ) V and the disjunction does not cut off the point ( 8 3, 2 3 ). 2. A, D P : Note that ( 8 3, 2 3 ) P since ( 8 3, 2 3 ) = 2 3 A + 3D. Therefore ( 8 3, 2 3 ) V and the disjunction does not cut off the point ( 8 3, 2 3 ). 3. B, C, D P, A P 2 : It can be verified that ( 0 3, 3 ) LPR. Also ( 0 3, 3 ) P since ( 0 3, 3 ) = 3 (B + C + D). Therefore (0 3, 3 ) V. Also A LPR and A P 2, i.e., A V 2. Finally, note that ( 8 3, 2 3 ) = 2 (0 3, 3 ) + 2 A. Therefore (8 3, 2 3 ) is a convex combination of points in V and V 2 and the disjunction does not cut off the point ( 8 3, 2 3 ). 4. A, B P, C, D P 2 : It can be verified that the point ( 5 2, ) LPR. Also ( 5 2, ) = 2 (A + B). Therefore, (5 2, ) V. Also C V 2 since C LPR. Finally, note that ( 8 3, 2 3 ) = 2 3 (5 2, ) + 3 C. Therefore (8 3, 2 3 ) is a convex combination of points in V and V 2 and the disjunction does not cut off the point ( 8 3, 2 3 ). 5. A, C P, B, D P 2 : It can be verified that ( 9 4, 3 4 ) LPR. Also (9 4, 3 4 ) P since ( 9 4, 3 4 ) = 3 4 A + 4 C. Therefore, (9 4, 3 4 ) V. It can be verified that the point ( 7 2, 2 ) LPR. Also (7 2, 2 ) P 2 since ( 7 2, 2 ) = 2 (B + D).

19 Facets of Low- and High-Dimensional Group Problems 9 Finally, note that ( 8 3, 2 3 ) = 2 3 (9 4, 3 4 )+ 3 (7 2, 2 ). Therefore (8 3, 2 3 ) is a convex combination of points in V and V 2 and the disjunction does not cut off the point ( 8 3, 2 3 ). It can be seen from Proposition 5 that sequential-merge inequalities have very diverse continuous variable coefficients. To understand the strength of the continuous coefficients in sequential-merge inequalities we consider the following example. Example 2 Consider a continuous variable with coefficient (u, u 2 ) R 2 with u > 0, u 2 < 0, u + u 2 + r 2 c h u 2 = 0. The coefficient of this continuous r variable in g h is 2 r +r 2 ( u 2 c h ). If the group cut h was used to generate a cut from the second constraint alone, the coefficient of the continuous variable would be u 2 c h > r2 r +r 2 ( u 2 c h ). Similarly, if the group cut g was derived using the first constraint alone, the coefficient of the continuous variable would be r u. Since u + u 2 + r 2 c h u 2 = 0 the coefficient of the continuous r variable using g h, 2 r +r 2 ( u 2 c h ) = u+u2 r +r 2 < r u as u 2 < 0. Therefore in this case the continuous coefficients generated using the two different cuts g and h individually will be strictly weaker than those generated using g h. We conclude from Example 2 that if both the inner and outer functions used in the sequential-merge procedure are GMICs then the coefficient generated for the continuous variable is stronger than the coefficient generated using each of the individual group cuts for a continuous variable with coefficient (u, u 2 ) where u > 0, u 2 < 0, u + u 2 + r 2 c h u 2 = 0 (i.e., the coefficients of the sequential-merge inequalities are not dominated by those of the GMICs). This result is significant because it can be proven that GMIC generates the strongest possible coefficients for continuous variables among all facets of one-dimensional group problems. Note also that although the above discussion was based on the specific case where u > 0, u 2 < 0 and u + u 2 + r 2 c h u 2 = 0, there exists a larger range of continuous variables coefficients for which the sequential-merge procedure yields inequalities whose coefficients are not dominated by the continuous coefficients of the one-dimensional group cuts derived from the individual rows. 5 Projected Sequential-Merge Inequality In this section, we study methods to construct sequential-merge inequalities for low-dimensional group problems from sequential-merge inequalities for higher-dimensional group problems. Further, we derive conditions for the resulting inequalities to be strong. Consider first a valid subadditive function φ : I m+ R + for a m+- dimensional group problem. One intuitive procedure to generate a cut for the m-dimensional group problem from φ is to first duplicate one row of the m-dimensional system of constraints, scale the two identical rows with integer multipliers and then generate a m + -dimensional cut for the expanded system. Algebraically, this construction corresponds to defining the

20 20 Santanu S. Dey, Jean-Philippe P. Richard m-dimensional function f(x, x 2,..., x m ) = φ(x, x 2,..., n x m, n 2 x m ), (24) where n, n 2 are arbitrary integers. The procedure presented in (24) is one of many possible ways of deriving inequalities of m-dimensional group problems using inequalities of m+-dimensional group problems. We leave the investigation of different constructions for future research. Moreover, while the procedure presented in (24) can be applied to any m+-dimensional inequality, we study it only with respect to inequalities that are constructed by iteratively applying sequential-merge operators to one-dimensional facet-defining inequalities, i.e., we analyze m+-dimensional facet-defining inequalities φ of the form (g (g 2...(g m g m+ ))) where each g i is facet-defining for PI(r i, ) and [g i ] ri is nondecreasing. Based on the procedure presented in (24), we are interested in deriving conditions under which m-dimensional functions of the form (g (g 2...(g m g m+ ))) (x, x 2,.., n x m, n 2 x m ) are facet-defining. Observe first that if we prove that the function (g m g m+ ) (n x m, n 2 x m ) is a facet-defining inequality for the one-dimensional group problem and is non-decreasing in its lifting-space representation, it will follow from the sequential-merge theorem that the projected m-dimensional function f(x, x 2,..., x m ) is facet-defining. Therefore, we begin this section by deriving conditions for which one-dimensional infinite group cuts projected from twodimensional sequential-merge inequalities are facet-defining. This result is presented in Lemma and is then used to prove Theorem 6, a more general result that characterizes the projection of m+-dimensional sequential-merge inequalities. Geometrically, when m =, and n = n 2 =, the procedure in (24) corresponds to creating a one-dimensional cut by computing the value of φ along the diagonal of I 2. The construction of two-step MIR inequalities using sequential-merge inequalities that was presented in Proposition 8 is an example of this idea. It was proven in Section 4 that under very general conditions, the sequential-merge procedure creates facet-defining inequalities for high-dimensional group problems from facet-defining inequalities for low-dimensional group problems. Surprisingly, the one-dimensional projection does not always produce facet-defining inequalities for P I(r, ) even when the constituent functions are facet-defining for the one-dimensional group problem. In fact, the result does not even necessarily hold when the constituent inequalities are GMICs. Figure 3 shows an example of this observation where the outer function is GMIC, the inner function is a three-slope facet-defining inequality, and n = n 2 =. Figure 3 also shows an example where n 2 = 2 and both the inner and outer functions are GMICs. It can be verified that this function (denoted as φ) is not facet-defining by showing that it is not extreme, i.e., φ = 2 φ + 2 φ 2. Interestingly, when the inner function is GMIC and n 2 =, the sequential-merge procedure generates facet-defining inequalities for P I(r, ). We introduce some notation to represent this subfamily of one-dimensional projected sequential-merge inequalities. Definition 2 Let n Z + be such that n, r < /n. Given a valid inequality g for PI(nr, ) that is such that [g] nr is a nondecreasing func-