On the exact separation of mixed integer knapsack cuts - Marcos ...

32 R. Fukasawa, M. GoycooleaLikewise, if in some node of the branch and bound tree we have that x i ≥ l i + 1 issatisfied by all solutions in that node, then we may also impose that:and,x j = u j ∀ j ∈ C 1,g−1 , (8)x g ≥ u g − a i − ∑ j∈C g,ia j (u j − l j )a g. (9)In either case we will only be cutting off dominated solutions to the original MIP.Proof We will prove that if e f < e i , then any solution x ∈ K that satisfies x i ≤ u i −1and does not satisfy 6 or 7 is cost-dominated.First, let L f = l f + a i − ∑ j∈C a i, f −1 j (u j −l j )a fand L j = l j , ∀ j ∈ C f +1,n . Assume,for the sake of contradiction, that x t > L t for some t ∈ C f,n .Ifx j < u j for any j ∈C 1,t−1 , then we can define ¯x so that ¯x k = x k , ∀k ̸= j, t, ¯x t = x t −ɛ, ¯x j = x j +ɛa t /a jand have that for some ɛ>0 small enough, ¯x ∈ K . Since the variables are sortedby decreasing efficiency, and observing that no two continuous variables can have thesame efficiency, ¯x cost-dominates x.Therefore we may assume that x j = u j for all j ∈ C 1,t−1 . But then since t ≥ f ,wehave that ∑ j∈C i,na j (x j −l j ) = ∑ j∈C i,t−1a j (u j −l j )+ ∑ j∈C t,na j (x j −l j )>a i andthat e t ≤ e f < e i . Therefore, by Proposition 7 we will have that x is cost-dominated.The second part of the Proposition is analogous.⊓⊔4 Computational experimentsIn this section we describe the results of our computational experiments. All implementationswere developed in the “C” and “C++” programming languages, using theLinux operating system (v2.4.27) on Intel Xeon dual-processor computers (2GB ofRAM, at 2.66GHz). Since generating cuts which are invalid is a real point of concern,we use exact arithmetic rather than the floating point arithmetic typically usedin computer codes. Specifically, we used the exact LP solver of Applegate et al. [4]for solving LP 1 , and the GNU Multiple Precision (GMP) Arithmetic library [33] forimplementing all other computations.In order to test our results we use the MIPLIB 3.0 [13] and MIPLIB 2003 [1]data sets. Results are organized as follows. In Sect. 4.1 we utilize the methodologydeveloped in Sect. 2 to benchmark the MIR inequalities. In Sect. 4.2 we analyze theperformance of the mixed integer knapsack algorithm developed in Sect. 3.4.1 Benchmarking the MIR inequalitiesWe now describe our experiments computing the values z ∗ (C K ) and z ∗ (M K ) and ouruse of these values to benchmark the performance of MIR inequalities, as detailed123