BCOL RESEARCH REPORT 07.04 Industrial Engineering & Operations Research University of California, Berkeley, CA 94720-1777 LIFTING FOR CONIC MIXED-INTEGER PROGRAMMING ALPER ATAMTÜRK AND VISHNU NARAYANAN Abstract. Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lower-dimensional restrictions. In order to simplify the computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect. Keywords: Valid inequalities, conic optimization, integer programming. October 2007 This research is supported, in part, by the National Science Foundation Grant 0700203: Conic Integer Programming. The first author is grateful to the hospitality of the Georgia Institute of Technology, where part of this research was conducted. Address: Industrial Engineering and Operations Research Department, 4141 Etcheverry Hall MC 1777, University of California, Berkeley CA 94720-1777 USA. Email: atamturk@berkeley.edu, vishnu@ieor.berkeley.edu.
LIFTING FOR CONIC MIXED-INTEGER PROGRAMMING 1 1. Introduction A conic mixed-integer program is an optimization problem of the form min c x : b Ax C, x Z n R p, where C R m is a closed, convex, pointed cone with nonempty interior. For instance, if C equals the linear cone R m +, then b Ax C reduces to the system of linear inequalities Ax b. Therefore, a conic integer program is a natural generalization of a linear integer program obtained by replacing R m + with a more general cone. A particularly interesting (nonlinear) conic constraint is the conic quadratic (or second-order conic) constraint Ax b 2 c x d, which includes a convex quadratic constraint as a special case. Many engineering and science problems, e.g., signal processing, portfolio optimization, support vector machines, are formulated with conic quadratic constraints [9]. We refer the reader to Ben-Tal and Nemirovski [5] for in-depth lecture notes on continuous conic optimization. Due to the growing demand for solving conic quadratic integer programs, commercial optimization packages, such as CPLEX and Mosek, already offer branch-and-bound solvers for conic quadratic integer programs. However, these solvers are currently far from being able to handle large-scale conic quadratic integer programs, and much work is needed to make them nearly as effective as their counterparts for linear integer programs. Although there is a wealth of literature on linear and quadratic integer programming, and more generally on global optimization, research on conic integer programming is so far limited. Çezik and Iyengar [7] give Chvátal-Gomory and disjunctive cuts for conic integer and conic 0-1 mixed programs. Atamtürk and Narayanan [4] describe mixed-integer rounding inequalities for conic mixed-integer programming. Vielma et al. [15] develop a branch-and-bound algorithm based on the polyhedral approximation of conic quadratic programs due to Ben-Tal and Nemirovski [6]. Aktürk et al. [1] give strong conic quadratic reformulations for mixed 0-1 problems with separable convex objective. Strong relaxations obtained by adding valid inequalities to formulations are crucial for solving linear integer programs with branch-and-bound methods. Lifting is an effective procedure for deriving such valid inequalities from simpler restrictions of the feasible set of solutions. Since 1970s lifting has been studied extensively and has become one of the most effective techniques for developing valid inequalities for linear integer programs. We list references [2, 3, 8, 11, 10, 12, 13, 16, 17] as a small sample of work on lifting for linear mixed-integer programming.
2 ALPER ATAMTÜRK AND VISHNU NARAYANAN Contributions. The success of lifting as an effective method for generating valid inequalities for linear integer programming is our motivation for generalizing it to conic mixed-integer programming in an effort to derive conic valid inequalities for this problem class. The generalization is achieved by working with generalized (conic) inequalities (Ben-Tal and Nemirovski [5]) and extending lifting functions for the linear case to lifting sets for conic lifting. When restricted to linear cones, conic lifting reduces to lifting for linear mixed-integer programming as one may expect. In a recent paper Richard and Tawarmalani [14] develop an interesting nonlinear lifting procedure for {x R n : f(x) 0}, where f is a function with affine minorant. Thus, their results do not apply for concave f. Our interest, however, is the set of integer points contained in a convex set. We describe a lifting approach for conic integer programming that lifts a valid conic constraint in a low-dimensional restriction to a valid conic constraint for the original conic mixed-integer set. Keeping the lifted constraint conic responds to a practical concern: If the continuous relaxations are solved with a conic solver, e.g., an SOCP solver, we would like to keep the valid inequalities added to the formulation conic. Moreover, maintaining the conic structure as an inequality is lifted, makes it easy to characterize the lifting sets and appropriate bounds for them. Outline. In Section 2 we develop the theory of conic lifting and describe how to derive lifted conic valid inequalities for conic mixed-integer programming by sequentially extending a low dimensional conic inequality. In Section 3 we discuss when lifting can be done independent of a lifting sequence so as to simplify the computations. In Section 4 we describe an application of sequence-independent conic lifting to a conic quadratic 0-1 set for illustration. In Section 5 we discuss the connections to the special case of linear mixed-integer programming. Finally, in Section 6 we conclude with a few final remarks. Throughout the paper for two sets A and B in R m, A + B denotes their Minkowski sum. 2. Conic Lifting Let us consider a conic mixed-integer set { S n (b) := (x 0,..., x n ) X 0 X n : b } n A i x i C, where A i is an m n i matrix, b R m, and C R m is a proper cone, i.e., closed, convex, pointed cone with nonempty interior, and each X i is a mixed-integer set in R n i. We assume that 0 X i for all i = 0,..., n. We are interested in deriving conic valid inequalities for S n (b), from conic inequalities valid for restrictions of S n (b). As indexing is arbitrary, consider the restriction S 0 (b) by fixing x i = 0 for all i = 1,..., n. We assume that S 0 (b) is nonempty. In principle, variables can be fixed to any value i=0
LIFTING FOR CONIC MIXED-INTEGER PROGRAMMING 3 as long as the restriction is nonempty. The choice of the zero value is for notational simplicity and without loss of generality by an appropriate change of variables; see, for instance, Atamtürk [3]. Let K R p be again a proper cone and consider a valid inequality h F 0 x 0 K (1) for S 0 (b), where h and F 0 are matrices of appropriate dimensions. Note that C and K may be different cones. For instance, K may be the cone of positive semidefinite matrices and C may be the linear cone, or K may be the linear cone and C may be the conic quadratic cone. Starting from valid inequality (1) for the restriction S 0 (b), our goal is to iteratively compute (matrices) F 1,..., F i such that the lifted inequality h i F j x j K (2) j=0 is valid for S i (b) for i = 1,..., n. Toward this end, for conic inequality (2) we define a lifting set below. We use (F i, h) to denote the intermediate lifted conic inequality (2). Definition 1. For v R m and i {0,..., n}, let the lifting set corresponding to inequality (2) be i Φ i (v) := d Rp : h F j x j d K for all (x 0,..., x i ) S i (b v). j=0 The lifting set Φ i (v) is a subset of R p parameterized by v R m. Observe that if S i (b v) =, then Φ i (v) = R p. The lifting set Φ i is used for computing F i+1 as shown in the sequel. Proposition 1. Given any v such that Φ i (v), Φ i (v) is a convex set with recession cone K. Proof. Convexity of Φ i (v) follows from the convexity of K. Also, for any t K, α R + and d Φ i (v), it is clear that d+αt Φ i (v). Conversely, let t be a ray of the recession cone. Then, for any d Φ i (v) and α R +, d+αt Φ i (v), i.e., h i j=0 F j x j (d + αt) K for all (x 0,..., x i ) S i (b v). Taking inner product with y K, we see that αt y (h i j=0 F j x j d) y for all α R +, which implies that t y 0. Hence, t K. Proposition 2. 0 Φ i (0) for all i = 0,..., n. Proof. Immediate from the validity of (2) for S i (b). Proposition 3. Φ i+1 (v) Φ i (v) for all v and i = 0,..., n 1. Proof. Follows from 0 X i and S i (b) S i+1 (b) for all i = 0,..., n 1.
4 ALPER ATAMTÜRK AND VISHNU NARAYANAN The next proposition describes the set of valid lifting matrices F i+1 based on the lifting set Φ i corresponding to the intermediate lifted inequality (F i, h). Proposition 4. Inequality (F i+1, h) is valid for S i+1 (b) if and only if F i+1 t Φ i (A i+1 t) for all t X i+1. Proof. Suppose that the condition is satisfied. Then, it immediately follows from the definition of Φ i that (F i+1, h) is valid for S i+1 (b). Conversely, suppose that there exists some x i+1 X i+1 such that F i+1 x i+1 Φ i (A i+1 x i+1 ). Then, there exists an ( x 0,..., x i ) S i (b A i+1 x i+1 ) such that h i j=0 F j x j F i+1 x i+1 K. However, ( x 1,..., x i+1 ) S i+1 (b), which implies that (F i+1, h) is not valid for S i+1 (b). A recursive relationship between lifting sets, which is used in the next section, follows from Definition 1. Proposition 5. Given Φ i and F i+1 for i = 0,..., n 1, Φ i+1 can be computed recursively as Φ i+1 (v) = ( Φi (v + A i+1 t) F i+1 t ). t X i+1 Proof. From the definition of Φ, we have i+1 Φ i+1 (v) = d : h F j x j d K, x S i+1 (b v) j=0 = i d : h F j x j (F i+1 t+d) K, x S i (b v A i+1 t) t X i+1 = t X i+1 j=1 ( Φi (v + A i+1 t) F i+1 t ). 3. Sequence-independent lifting Lifting, in general, is sequence-dependent; that is, the order in which variables x i, i = 1,..., n are introduced to the lifted inequality, can change Φ i and, consequently, the lifted inequality. In the previous section, we have seen that Φ i+1 (v) Φ i (v) holds in general. If the converse is also true for all i = 0,..., n 1, then Φ i (v) = Φ 0 (v) for all v and i = 1,..., n, and thus the order of lifting does not matter. If this condition is satisfied, then lifting is said to be sequence-independent. Recognizing sequence independence of lifting is computationally desirable, as in this case, it is sufficient to compute only the first lifting set Φ 0. Here we give a sufficient condition for sequence-independent lifting and show how to lift an inequality efficiently by a superadditive subset of the lifting set
LIFTING FOR CONIC MIXED-INTEGER PROGRAMMING 5 Φ 0. Sequence-independence of lifting by superadditive functions have been shown for linear 0-1 and mixed 0-1 programs by Wolsey [17] and Gu et al. [8]. The results below generalize of the ones in Atamtürk [3] for the linear mixed-integer programming case. Definition 2. The parameterized set Φ(v) is called superadditive on D R m, if for all u, v, u + v D, we have Φ(u) + Φ(v) Φ(u + v), i.e., α + β Φ(u + v) for all α Φ(u), β Φ(v). Theorem 1. The parameterized set Ψ(v) := v,w,v+w W π Φ 0 (w) where W = {w : Φ 0 (w) }, is superadditive. ( ) Φ 0 (v + w) π, (3) Proof. If either Ψ(u) = or Ψ(v) =, then Ψ(u) + Ψ(v) Ψ(u + v) holds trivially. Otherwise, suppose for contradiction that α Ψ(u) and β Ψ(v), but α + β Ψ(u + v). Then, there exist a w and π Φ 0 (w) such that α + β + π Φ 0 (u + v + w). However, because β Ψ(v), we must have β +π Φ 0 (v +w), which, together with α Φ 0 (u+v +w) (β +π), implies that α Ψ(u). A contradiction with the choice of α. The next theorem implies that Ψ is contained in any lifting set, independent of the lifting order. Theorem 2. The parameterized set Ψ is a subset of the last lifting set; that is, Ψ(v) Φ n (v) for all v. Proof. It follows from Proposition 5 that ( n ) Φ n (v) = {Φ 0 v + A i x i x X 1 X n i=1 } n F i x i Ψ(v), where the inclusion follows from n i=1 F i x i Φ 0 ( n i=1 Ai x i ) by validity of the lifted inequality (F n, h) and from the definition of Ψ by taking w = n i=1 Ai x i and π = n i=1 F i x i. Corollary 1. The parameterized set Ψ is a subset of all lifting sets; that is, Ψ(v) Φ n (v) Φ 0 (v) for all v. Proof. Follows from Proposition 3 and Theorem 2. i=1 Theorem 3. Ψ = Φ 0 if and only if Φ 0 is superadditive. Proof. If Ψ = Φ 0, from Theorem 2 Φ 0 is superadditive. Conversely, if Φ 0 is superadditive, then Φ 0 (v) Φ 0 (v + w) π for all π Φ 0 (w). Hence, Φ 0 (v) π Φ0 (w)(φ 0 (v + w) π). Since this is true for all w, we can take intersection over all w to see that Φ 0 (v) Ψ(v). But from Corollary 1 Φ 0 (v) Ψ(v) holds as well.
6 ALPER ATAMTÜRK AND VISHNU NARAYANAN Corollary 2. If Φ 0 is superadditive, then Ψ = Φ 0 = Φ 1 = = Φ n, i.e., lifting is sequence-independent. The next proposition states that even if Φ 0 is not superadditive, in general any superadditive subset of Φ 0 may be employed for sequence-independent lifting of (F 0, h). Proposition 6. Suppose that Ω(v) Φ 0 (v) for all v and Ω is superadditive, then (F n, h) is a lifted valid inequality for S n (b) whenever F i t Ω(A i t) for all t X i and i = 1,..., n. Proof. From the assumptions of the proposition, for any x S n (b), we have n i=1 ( n ) Ω(F i x i ) Ω F i x i i=1 Φ 0 ( n i=1 F i x i ). Then n i=1 F i x i Φ 0 ( n i=1 F i x i) and thus h n i=0 F i x i K. Corollary 3. If Φ 0 is superadditive, then (F n, h) is a lifted valid inequality for S n (b) whenever F i t Φ 0 (A i t) for all t X i and i = 1,..., n. 4. An illustrative application In this section we illustrate sequence-independent conic lifting for a conic quadratic mixed 0-1 set. Recall that an m + 1-dimensional second-order cone is defined as Q m+1 := {(t, t o ) R m R : t 2 t o }. Consider the conic quadratic mixed 0-1 set S = (x, y, t) {0, ( n 2 1}n+1 R 2 : a i x i β) + y 2 t. We should point out that given a high dimensional second order conic constraint Ax b t, where A R m n and b R m, we can construct a relaxation with Q 2+1 of the form S, by aggregating m 1 terms in the square root into the nonnegative term y 2 in S. Inequalities obtained from simpler relaxations of integer programs are common in linear integer programming. For conic 0-1 programming the set S plays a similar role to that of single row 0-1 knapsack relaxations in linear 0-1 programming. i=0
LIFTING FOR CONIC MIXED-INTEGER PROGRAMMING 7 If necessary, by complementing the binary variables, we assume that a > 0. Fixing x i = 0, i = 1,..., n, we arrive at the restriction { (x0 S 0 = (x 0, y, t) {0, 1} R 2 : β ) 2 + y 2 t}, where we assume without loss of generality that a 0 = 1. The continuous relaxation if S 0 has the unique extreme point (β, 0, 0), which is fractional for x when 0 < β < 1. This fractional extreme point is cut off by the conic inequality ((1 2β)x 0 + β) 2 + y 2 t. (4) Indeed, Atamtürk and Narayanan [4] show that adding inequality (4) to the continuous relaxation of S 0 is sufficient to describe conv(s 0 ). Our goal in this section is to lift the conic quadratic inequality (4) for S 0 to a conic quadratic inequality of the form ( 2 n (1 2β)x 0 + α i x i + β) + y 2 t, (5) i=1 that is valid for S. Now, letting b = ( β, 0, 0), let us write S in matrix form S n (b) = (x 0, y, t) {0, 1} R 2 : β 0 x 0 n y a i 0 x i Q 3 0 t i=1 0 as in Section 2. We will compute the lifting set Φ 0 (v) corresponding to (4), where v R 3 and v 1 0. Given v, this amounts to computing the set of all d R 3 such that (1 2β)x + β y d Q 3 (6) t for all (x, y, t) S 0 (b v). That is, Φ 0 (v) = { d R 3 : (6) holds for all (x, y, t) S 0 (b v) }. In order to derive the coefficients α i, i = 1,..., n, of (5), we will construct a superadditive subset Ω of the lifting set Φ 0 per Proposition 6. Recall that lifting coefficients F i must satisfy F i t Ω(A i t) for t {0, 1}. Because in this case, F i = α i 0 and A i = a i 0 for i = 1,..., n, (7) 0 0 it is sufficient to concentrate on the first component of the lifting set. Hence, the lifting set is then Φ 0 (v 1, 0, 0) = λ(v 1 ) Q 3, where λ(v 1 )= d 1 ((1 ) 0 2 : 2β)x + β + d1 + y 2 t, (x, y, t) S 0 β v 1 0. 0 0
8 ALPER ATAMTÜRK AND VISHNU NARAYANAN For computing λ(v 1 ), we consider the cases x 0 = 0 and x 0 = 1 below. Case x 0 = 0 : (d 1, 0, 0) λ(v 1 ) if and only if (β ) 2 (v1 + d1 + y 2 t for all (y, t) s.t. β ) 2 + y 2 t. This condition holds if and only if β + d 1 v 1 β. Case x 0 = 1 : (d 1, 0, 0) λ(v 1 ) if and only if (1 ) 2 (1 β + d1 + y 2 t for all (y, t) s.t. + v1 β ) 2 + y 2 t, which holds if and only if 1 β + d 1 1 + v 1 β. It follows from these conditions that λ(v 1 ) is a polyhedral set. We now consider the following cases: 1. v 1 β: In this case, the conditions reduce to β + d 1 v 1 β and 1 β + d 1 1+v 1 β; and they are satisfied if and only if v 1 d 1 v 1 2β. 2. 0 v 1 < β: In this case, the conditions become β + d 1 β v 1 and 1 β + d 1 1 + v 1 β; and they hold if and only if max {v 1 2β, v 1 2(1 β)} d 1 v 1. Therefore, we have λ(v 1 )= d 1 0 max {v : 1 2β, v 1 2(1 β)} d 1 v 1 if 0 v 1 < β v 0 1 d 1 v 1 2β if v 1 β. λ is superadditive for v 1 β; however, it is not so over the entire v 1 0 as the lower bound for 0 v 1 β is not subadditive. Therefore, in order to construct a valid lifting set, by only picking the upper bound for 0 v 1 β, we define { λ ( v 1, 0, 0) if 0 v 1 < β, (v) = λ(v 1 ) if v 1 β. Consider now the parameterized set Ω(v) := λ (v) Q 3. Because λ (v) λ(v), we have Ω(v) Φ 0 (v). On the other hand, superadditivity of Ω follows from superadditivity of λ, which follows from fact that ϕ 1 (v 1 ) = v 1 is both subadditive and superadditive, and that ϕ 2 (v 1 ) = v 1 2β is superadditive as β > 0. Hence, we have shown that Ω(v) is a superadditive lifting set for (4). Finally, we compute lifting matrices F i by letting F i Ω(A i ). Recalling (7), we find the strongest inequalities (5) based on Ω, by picking extreme values { { a i } if a i < β α i for i = 1,..., n. { a i, a i 2β} if a i β Observe that the superadditive conic lifting inequality (5) is not unique; for a i β, α i may be chosen as either a i or a i 2β. Therefore, the lifted inequalities form an exponential class. Unlike the conic MIR inequalities in
LIFTING FOR CONIC MIXED-INTEGER PROGRAMMING 9 Atamtürk and Narayanan [4], the lifted inequalities presented here make an explicit use of the binary variables. Example 1. Let S be defined by a = 1 and β = 1/2; thus the conic constraint is ( n 2 x i 1/2) + y 2 t. (8) i=0 Let us consider lifting the special case of inequality (4) (1/2) 2 + y 2 t, which is valid for the restriction S 0 = { x 0 {0, 1}, y R, t R : } (x 0 1/2) 2 + y 2 t. Then, by choosing α i = 1 or 0 for i = 1,..., n, we obtain exponentially many superadditive conic lifting inequalities ( 2 x i 1/2) + y 2 t for T {1,..., n}. (9) i T Observe that for each set T inequality (9) supports conv(s) at points (x, y, t): (0, 0, 1 2 ) and ( i Y e i, 0, Y 1 2 ) for any Y T. In contrast, the conic MIR inequality of Atamtürk and Narayanan [4] for this example is just the special case of inequality (9) with T =. This example illustrates the power of the conic lifting for describing the coefficients of valid inequalities for conic MIPs. On the other hand, the nonlinear lifting method of Richard and Tawarmalani [14] does not apply to this example since after writing the conic constraint (8) as f(x, t, y) := x j j N we see that f does not have an affine minorant. 2 + j N x j + t 2 1 4 y2 0, 5. Lifting for linear mixed-integer programming Here we show that lifting linear valid inequalities for linear mixed-integer programs is a direct consequence of conic lifting. Let the cone C = R m +, so that { } n S n (b) = (x 0,..., x n ) X 0 X n : A i x i b. With K = R +, we have the inequality F 0 x 0 h valid for S 0 (b), where h R, F 0 is a 1 n 0 row vector. In this case, the lifting set is given by i=0 Φ i (v) = {α R : α ϕ i (v)},
10 ALPER ATAMTÜRK AND VISHNU NARAYANAN where ϕ i (v) is the lifting function i ϕ i (v) = min h F j x j : (x 0,..., x i ) S i (b v) j=0 Note that β Φ i (v) if and only if β ϕ i (v). Consequently, Proposition 2 states that ϕ i (0) 0 and Proposition 3 implies that ϕ i (v) ϕ i+1 (v) for all i = 1,..., n 1. Suppose we have an intermediate lifted inequality (F i, h) for S i (b), then inequality (F i+1, h) is valid for S i+1 (b) if and only if F i+1 t ϕ i (A i+1 t) for all t X i+1, by Proposition 4. It is also seen that all results in Section 3 specialize to sequence-independent lifting in the case of linear mixed-integer programs [3, 8, 17]. The lifting set Φ 0 is superadditive if and only if the lifting function ϕ 0 is superadditive. Similarly, in the linear case, superadditive subsets of Φ 0 correspond to superadditive lower bounding functions of ϕ 0. Hence, we see that lifting in the linear case can be obtained by letting the cones C and K be nonnegative orthants of appropriate dimension. 6. Final remarks We have shown how to lift a conic inequality valid for lower-dimensional restrictions of a conic mixed-integer set into valid conic inequalities for the original set. As in the linear case, superadditive lifting sets lead to sequenceindependent lifting. The conic lifting approach is illustrated on a conic quadratic mixed 0-1 set. Because the coefficients are described by lifting sets rather than lifting functions, sequence independent lifting of the inequalities may lead to an exponential class of lifted inequalities as illustrated with an example. Based on the many successful applications of lifting in the linear case, it is reasonable to expect conic lifting to become an effective method for deriving conic inequalities for structured conic mixed-integer programs as well in due time. References [1] M. S. Aktürk, A. Atamtürk, and S. Gürel. A strong conic quadratic reformulation for machine-job assignment with controllable processing times. Research Report BCOL.07.01, IEOR, University of California- Berkeley, April 2007. [2] A. Atamtürk. On the facets of mixed integer knapsack polyhedron. Mathematical Programming, 98:145 175, 2003. [3] A. Atamtürk. Sequence independent lifting for mixed integer programming. Operations Research, 52:487 490, 2004. [4] A. Atamtürk and V. Narayanan. Cuts for conic mixed integer programming. In M. Fischetti and D. P. Williamson, editors, Proceedings of the 12th International IPCO Conference, pages 16 29, 2007.
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