Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use previously generated GMI cuts. We demonstrate that for problems in MIPLIB 3.0 and MIPLIB 2003, the cuts we generate form an important subclass of all rank-1 mixed-integer rounding cuts. Further, we use our heuristic to generate globally valid rank-1 GMI cuts at nodes of a branch-and-cut tree and use these cuts to solve a difficult problem from MIPLIB 2003, namely timtab2, without using problem-specific cuts. 1

We describe a simple process for generating numerically safe cutting planes using floating-point arithmetic and the mixed-integer rounding (MIR) 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. We report on tests with the MIPLIB 3.0 and MI-PLIB 2003 test collections, and with MIP instances derived from the TSPLIB traveling salesman library. 1

Split cuts represent the most widely used class of cutting planes currently employed by state-of-the-art branch-and-cut solvers for mixed integer linear programming. Rank-1 cuts often have better numerical properties than higher rank cuts. In this paper, we study several heuristics to generate new families of strong rank-1 split cuts, by considering integer linear combinations of the rows of the simplex tableau, and deriving the corresponding mixedinteger Gomory cuts. In particular, we propose several cut generation algorithms that share the following aims: reducing the number of nonzeroes, obtaining small coefficients, generating orthogonal cuts. A key idea is that of selecting a subset of the variables, and trying to generate a cut which cuts deeply on those variables. We show that variables with small reduced cost are good candidates for this purpose, yielding cuts that close a larger integrality gap. On a set of test instances where standard split cut generators close on average 28.8 % of the integrality gap in the first pass, we can close 35.3 % by investing 10 times as much cut generation time. Incorporating our new split cuts into a branch-and-cut algorithm yields an improvement in the overall performance: except on very easy instances, where our procedure is too slow to bring advantage, on average we can save 23 % computing time on instances which are solved, and close more integrality gap on unsolved instances in a fixed amount of time. 1 INTRODUCTION 2 1

Gomory mixed-integer cuts are one of the key components in Branch-and-Cut solvers for mixed-integer linear programs. The textbook formula for generating these cuts is not used directly in open-source and commercial software due to the limited numerical precision of the computations: Additional steps are performed to avoid the generation of invalid cuts. This paper studies the impact of some of these steps on the safety of Gomory mixed-integer cut generators. As the generation of invalid cuts is a relatively rare event, the experimental design for this study is particularly important. We propose an experimental setup that allows statistically significant comparisons of generators. We also propose a parameter optimization algorithm and use it to find a Gomory mixed-integer cut generator that is as safe as a benchmark cut generator from a commercial solver even though it generates many more cuts. 1