Mixed-integer Rounding Research Articles

Computational aspects of infeasibility analysis in mixed integer programming

The analysis of infeasible subproblems plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems: conflict graph analysis and dual proof analysis. While conflict graph analysis detects sets of contradicting variable bounds in an implication graph, dual proof analysis derives valid linear constraints from the proof of the dual LP’s unboundedness. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide a comprehensive computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first combined approach that uses both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set Miplib 2017; the presented algorithms have been implemented within the non-commercial MIP solver SCIP and the commercial MIP solver FICO Xpress.

Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs

In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1–20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of \(l_p\)-norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. \(p=1\) and \(p =2\), respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts.

Lifting the knapsack cover inequalities for the knapsack polytope

Valid inequalities for the knapsack polytope have proven to be very useful in exact algorithms for mixed-integer linear programming. In this paper, we focus on the knapsack cover inequalities, introduced in 2000 by Carr and co-authors. In general, these inequalities can be rather weak. To strengthen them, we use lifting. Since exact lifting can be time-consuming, we present two fast approximate lifting procedures. The first procedure is based on mixed-integer rounding, whereas the second uses superadditivity.

Monoidal cut strengthening and generalized mixed-integer rounding for disjunctions and complementarity constraints

In the early 1980s, Balas and Jeroslow presented monoidal disjunctive cuts exploiting the integrality of variables. This article investigates the relation of monoidal cut strengthening to other classes of cutting planes for general two-term disjunctions. We introduce a generalization of mixed-integer rounding cuts and show equivalence to monoidal disjunctive cuts. Moreover, we demonstrate the effectiveness of these cuts via computational experiments on instances involving complementarity constraints. Finally, we present an adaptation of the mixed-integer rounding approach for mixed-complementarity problems.

Mixed-Integer Rounding Enhanced Benders Decomposition for Multiclass Service-System Staffing and Scheduling with Arrival Rate Uncertainty

We study server scheduling in multiclass service systems under uncertainty in the customer arrival volumes. Common practice in such systems is to first identify staffing levels and then determine schedules for the servers that cover these levels. We propose a new stochastic integer programming (SIP) model that integrates these two decisions, which can yield lower scheduling costs by exploiting the presence of alternative server configurations that yield similar quality of service. We find that a branch-and-cut algorithm based on Benders decomposition may fail due to the weakness of the relaxation bound. We propose a novel application of mixed-integer rounding to improve the Benders cuts used in this algorithm, a technique that is applicable to any SIP with integer first-stage decision variables. Numerical examples illustrate the computational efficiency of the proposed approach and the potential benefit of solving the integrated model compared to considering the staffing and scheduling problems separately. This paper was accepted by Yinyu Ye, optimization.

Another pedagogy for mixed-integer Gomory

We present a version of GMI (Gomory mixed-integer) cuts in a way so that they are derived with respect to a "dual form" mixed-integer optimization problem and applied on the standard-form primal side as columns, using the primal simplex algorithm. This follows the general scheme of He and Lee, who did the case of Gomory pure-integer cuts. Our input mixed-integer problem is not in standard form, and so our cuts are derived rather differently from how they are normally derived. A convenient way to develop GMI cuts is from MIR (mixed-integer rounding) cuts, which are developed from 2-dimensional BMI (basic mixed integer) cuts, which involve a nonnegative continuous variable and an integer variable. The nonnegativity of the continuous variable is not the right tool for us, as our starting point (the "dual form" mixed-integer optimization problem) has no nonnegativity. So we work out a different 2-dimensional starting point, a pair of somewhat arbitrary inequalities in one continuous and one integer variable. In the end, we follow the approach of He and Lee, getting now a finitely converging primal-simplex column-generation algorithm for mixed-integer optimization problems.

A Two-Stage Stochastic Integer Programming Approach to Integrated Staffing and Scheduling with Application to Nurse Management

We study the problem of integrated staffing and scheduling under demand uncertainty. This problem is formulated as a two-stage stochastic integer program with mixed-integer recourse. The here-and-now decision is to find initial staffing levels and schedules. The wait-and-see decision is to adjust these schedules at a time closer to the actual date of demand realization. We show that the mixed-integer rounding inequalities for the second-stage problem convexify the recourse function. As a result, we present a tight formulation that describes the convex hull of feasible solutions in the second stage. We develop a modified multicut approach in an integer L-shaped algorithm with a prioritized branching strategy. We generate 20 instances (each with more than 1.3 million integer and 4 billion continuous variables) of the staffing and scheduling problem using 3.5 years of patient volume data from Northwestern Memorial Hospital. Computational results show that the efficiency gained from the convexification of the recourse function is further enhanced by our modifications to the L-shaped method. The results also show that compared with a deterministic model, the two-stage stochastic model leads to a significant cost savings. The cost savings increase with mean absolute percentage errors in the patient volume forecast.

Split cuts and extended formulations for Mixed Integer Conic Quadratic Programming

We study split cuts and extended formulations for Mixed Integer Conic Quadratic Programming (MICQP) and their relation to Conic Mixed Integer Rounding (CMIR) cuts. We show that CMIR is a linear split cut for the polyhedral portion of an extended formulation of a quadratic set and it can be weaker than the nonlinear split cut of the same quadratic set. However, we also show that families of CMIRs can be significantly stronger than the associated family of nonlinear split cuts.

Mixed [formula omitted]-step MIR inequalities: Facets for the [formula omitted]-mixing set

Günlük and Pochet [O. Günlük, Y. Pochet, Mixing mixed integer inequalities. Mathematical Programming 90 (2001) 429–457] proposed a procedure to mix mixed integer rounding (MIR) inequalities. The mixed MIR inequalities define the convex hull of the mixing set {(y1,…,ym,v)∈Zm×R+:α1yi+v≥βi,i=1,…,m} and can also be used to generate valid inequalities for general as well as several special mixed integer programs (MIPs). In another direction, Kianfar and Fathi [K. Kianfar, Y. Fathi, Generalized mixed integer rounding inequalities: facets for infinite group polyhedra. Mathematical Programming 120 (2009) 313–346] introduced the n-step MIR inequalities for the mixed integer knapsack set through a generalization of MIR. In this paper, we generalize the mixing procedure to the n-step MIR inequalities and introduce the mixed n-step MIR inequalities. We prove that these inequalities define facets for a generalization of the mixing set with n integer variables in each row (which we refer to as the n-mixing set), i.e. {(y1,…,ym,v)∈(Z×Z+n−1)m×R+:∑j=1nαjyji+v≥βi,i=1,…,m}. The mixed MIR inequalities are simply the special case of n=1. We also show that mixed n-step MIR can generate valid inequalities based on multiple constraints for general MIPs. Moreover, we introduce generalizations of the capacitated lot-sizing and facility location problems, which we refer to as the multi-module problems, and show that mixed n-step MIR can be used to generate valid inequalities for these generalizations. Our computational results on small MIPLIB instances as well as a set of multi-module lot-sizing instances justify the effectiveness of the mixed n-step MIR inequalities.

On [formula omitted]-step MIR and partition inequalities for integer knapsack and single-node capacitated flow sets

Pochet and Wolsey [Y. Pochet, L.A. Wolsey, Integer knapsack and flow covers with divisible coefficients: polyhedra, optimization and separation, Discrete Applied Mathematics 59 (1995) 57–74] introduced partition inequalities for three substructures arising in various mixed integer programs, namely the integer knapsack set with nonnegative divisible/arbitrary coefficients and two forms of single-node capacitated flow set with divisible coefficients. They developed the partition inequalities by proving properties of the optimal solution in optimizing a linear function over these sets. More recently, the author and Fathi [K. Kianfar, Y. Fathi, Generalized mixed integer rounding inequalities: facets for infinite group polyhedra, Mathematical Programming 120 (2009) 313–346] have introduced the n-step mixed integer rounding (MIR) inequalities for the mixed-integer knapsack set with arbitrary coefficients through a generalization of MIR. In this paper, we show that the n-step MIR generates facet-defining inequalities not only for the three sets considered by Pochet and Wolsey but also for their generalization to the case where coefficients are not necessarily divisible. In the case of divisible coefficients, n-step MIR directly generates the partition inequalities for all three sets (and in some cases stronger inequalities for one of the sets). We show that n-step MIR gives facets for the integer knapsack set with arbitrary coefficients that either dominate or are not obtainable by the partition inequalities. Also, using the n-step MIR, we introduce families of facets for the two capacitated flow sets with arbitrary coefficients for the first time. Our results provide a new perspective based on n-step MIR into the polyhedral properties of these three substructures, extend them to the case of arbitrary coefficients, and underscore the power of n-step MIR to easily generate strong valid inequalities.

Computational Testing of a Separation Procedure for the Knapsack Set with a Single Continuous Variable

We study an exact separation procedure—SEP-MK—for the knapsack set with a single continuous variable XMK. Then, we address the question of whether SEP-MK can be of practical use in tightening mixed-integer programming (MIP) formulations when using standard (floating-point) MIP solvers. To this purpose, we present a separation procedure for MIP problems—SEP-MIPMK—where we derive knapsack sets of the form XMK by aggregating the continuous variables in the mixed knapsack inequalities of the formulation. Then, we use SEP-MK to generate cutting planes. Before the continuous variables are aggregated, the mixed knapsack inequalities are modified through the use of a bound substitution procedure to take into account fixed and variable bounds on the continuous variables. Bound substitution is made according to some heuristic rules, so even if its basic component SEP-MK is “exact,” the overall separation procedure for MIP problems, SEP-MIPMK, is heuristic. We perform a computational study on a wide set of mixed-integer programming instances from the MIPLIB 2003 [Achterberg, T., T. Koch, A. Martin. 2006. Mixed Integer Problem Library (MIPLIB) 2003. Konrad-Zuse-Zentrum für Informationstechnik Berlin, Berlin. http://miplib.zib.de] and Mittelmann [Mittelmann, H. 2010. MILP testcases. http://plato.asu.edu/ftp/milp] benchmark sets. Computational experiments confirm that lifted cover and mixed-integer rounding (MIR) inequalities are effective from a computational viewpoint. Nevertheless, there are several instances where SEP-MIPMK is able to significantly raise the lower bounds given by lifted cover and MIR inequalities.

A note on the MIR closure and basic relaxations of polyhedra

Anderson et al. (2005) [1] show that for a polyhedral mixed integer set defined by a constraint system A x ≥ b , along with integrality restrictions on some of the variables, any split cut is in fact a split cut for a basic relaxation, i.e., one defined by a subset of linearly independent constraints. This result implies that any split cut can be obtained as an intersection cut. Equivalence between split cuts obtained from simple disjunctions of the form x j ≤ 0 or x j ≥ 1 and intersection cuts was shown earlier for 0 / 1 -mixed integer sets by Balas and Perregaard (2002) [4]. We give a short proof of the result of Anderson, Cornuéjols and Li using the equivalence between mixed integer rounding (MIR) cuts and split cuts.

Generating facets for finite master cyclic group polyhedra using n-step mixed integer rounding functions

The n-step mixed integer rounding (MIR) functions generate n-step MIR inequalities for MIP problems and are facets for the infinite group problems. We show that the n-step MIR functions also directly generate facets for the finite master cyclic group polyhedra especially in many cases where the breakpoints of the n-step MIR function are not necessarily at the elements of the group (hence the linear interpolation of the facet coefficients obtained has more than two slopes).

N-step mingling inequalities: new facets for the mixed-integer knapsack set

The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313–346, 2009) are valid inequalities for the mixed-integer knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315–338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into n-step MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n = 2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that n-step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets.

Two-Step MIR Inequalities for Mixed Integer Programs

Two-step mixed integer rounding (MIR) inequalities are valid inequalities derived from a facet of a simple mixed integer set with three variables and one constraint. In this paper we investigate how to effectively use these inequalities as cutting planes for general mixed integer problems. We study the separation problem for single-constraint sets and show that it can be solved in polynomial time when the resulting inequality is required to be sufficiently different from the associated MIR inequalities. We discuss computational issues and present numerical results based on a number of data sets.

Numerically Safe Gomory Mixed-Integer Cuts

We describe a simple process for generating numerically safe cutting planes using floating-point arithmetic and the mixed-integer rounding procedure. Applying this method to the rows of the simplex tableau permits the generation of Gomory mixed-integer cuts that are guaranteed to be satisfied by all feasible solutions to a mixed-integer programming problem (MIP). We report on tests with the MIPLIB 3.0 and MIPLIB 2003 test collections as well as with MIP instances derived from the TSPLIB traveling salesman library.

On the exact separation of mixed integer knapsack cuts

During the last decades, much research has been conducted on deriving classes of valid inequalities for mixed integer knapsack sets, which we call knapsack cuts. Bixby et al. (The sharpest cut: the impact of Manfred Padberg and his work. MPS/SIAM Series on Optimization, pp. 309–326, 2004) empirically observe that, within the context of branch-and-cut algorithms to solve mixed integer programming problems, the most important inequalities are knapsack cuts derived by the mixed integer rounding (MIR) procedure. In this work we analyze this empirical observation by developing an algorithm to separate over the convex hull of a mixed integer knapsack set. The main feature of this algorithm is a specialized subroutine for optimizing over a mixed integer knapsack set which exploits dominance relationships. The exact separation of knapsack cuts allows us to establish natural benchmarks by which to evaluate specific classes of them. Using these benchmarks on MIPLIB 3.0 and MIPLIB 2003 instances we analyze the performance of MIR inequalities. Our computations, which are performed in exact arithmetic, are surprising: In the vast majority of the instances in which knapsack cuts yield bound improvements, MIR cuts alone achieve over 87% of the observed gain.

Mixing MIR inequalities with two divisible coefficients

This article is a polyhedral study of a generalization of the mixing set where two different, divisible coefficients are allowed for the integral variables. Our results generalize earlier work on mixed integer rounding, mixing, and extensions. They also directly apply to applications such as production planning problems involving lower bounds or start-ups on production, when these are modeled as mixed-integer linear programs. We define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed-integer set. We give a characterization of each of the maximal faces of the convex hull, as well as a closed form description of its extreme points and rays, and show how to separate over this set in $${\mathcal{O} (n\log n)}$$. Finally, we give several extended formulations of polynomial size, and study conditions under which adding certain simple constraints on the integer variables preserves our main result.

Mingling: mixed-integer rounding with bounds

Mixed-integer rounding (MIR) is a simple, yet powerful procedure for generating valid inequalities for mixed-integer programs. When used as cutting planes, MIR inequalities are very effective for mixed-integer programming problems with unbounded integer variables. For problems with bounded integer variables, however, cutting planes based on lifting techniques appear to be more effective. This is not surprising as lifting techniques make explicit use of the bounds on variables, whereas the MIR procedure does not. In this paper we describe a simple procedure, which we call mingling, for incorporating variable bound information into MIR. By explicitly using the variable bounds, the mingling procedure leads to strong inequalities for mixed-integer sets with bounded variables. We show that facets of mixed-integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure. In particular, the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of Marchand and Wolsey (Math Program 85:15–33, 1999) as well as the continuous integer knapsack cover and pack inequalities of Atamtürk (Math Program 98:145–175, 2003; Ann Oper Res 139:21–38, 2005). In addition, mingling inequalities give a generalization of the two-step MIR inequalities of Dash and Günlük (Math Program 105:29–53, 2006) under some conditions.

MIR closures of polyhedral sets

We study the mixed-integer rounding (MIR) closures of polyhedral sets. The MIR closure of a polyhedral set is equal to its split closure and the associated separation problem is NP-hard. We describe a mixed-integer programming (MIP) model with linear constraints and a non-linear objective for separating an arbitrary point from the MIR closure of a given mixed-integer set. We linearize the objective using additional variables to produce a linear MIP model that solves the separation problem exactly. Using a subset of these additional variables yields an MIP model which solves the separation problem approximately, with an accuracy that depends on the number of additional variables used. Our analysis yields an alternative proof of the result of Cook et al. (1990) that the split closure of a polyhedral set is again a polyhedron. We also discuss a heuristic to obtain MIR cuts based on our approximate separation model, and present some computational results.