Sanjeeb Dash

Valid inequalities based on simple mixed-integer sets

In this paper we use facets of simple mixed-integer sets with three variables to derive a parametric family of valid inequalities for general mixed-integer sets. We call these inequalities two-step MIR inequalities as they can be derived by applying the simple mixed-integer rounding (MIR) principle of Wolsey (1998) twice. The two-step MIR inequalities define facets of the master cyclic group polyhedron of Gomory (1969). In addition, they dominate the strong fractional cuts of Letchford and Lodi (2002).

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.

Projected Chvátal–Gomory cuts for mixed integer linear programs

Recent experiments by Fischetti and Lodi show that the first Chvátal closure of a pure integer linear program (ILP) often gives a surprisingly tight approximation of the integer hull. They optimize over the first Chvátal closure by modeling the Chvátal–Gomory (CG) separation problem as a mixed integer linear program (MILP) which is then solved by a general- purpose MILP solver. Unfortunately, this approach does not extend immediately to the Gomory mixed integer (GMI) closure of an MILP, since the GMI separation problem involves the solution of a nonlinear mixed integer program or a parametric MILP. In this paper we introduce a projected version of the CG cuts, and study their practical effectiveness for MILP problems. The idea is to project first the linear programming relaxation of the MILP at hand onto the space of the integer variables, and then to derive Chvátal–Gomory cuts for the projected polyhedron. Though theoretically dominated by GMI cuts, projected CG cuts have the advantage of producing a separation model very similar to the one introduced by Fischetti and Lodi, which can typically be solved in a reasonable amount of computing time.

Exponential Lower Bounds on the Lengths of Some Classes of Branch-and-Cut Proofs

We examine the complexity of branch-and-cut proofs in the context of 0-1 integer programs. We establish an exponential lower bound on the length of branch-and-cut proofs that use 0-1 branching and lift-and-project cuts (called simple disjunctive cuts by some authors), Gomory-Chvatal cuts, and cuts arising from the N0 matrix-cut operator of Lovasz and Schrijver. A consequence of the lower-bound result in this paper is that branch-and-cut methods of the type described above have exponential running time in the worst case.

A model for fusion and code motion in an automatic parallelizing compiler

Loop fusion has been studied extensively, but in a manner isolated from other transformations. This was mainly due to the lack of a powerful intermediate representation for application of compositions of high-level transformations. Fusion presents strong interactions with parallelism and locality. Currently, there exist no models to determine good fusion structures integrated with all components of an auto-parallelizing compiler. This is also one of the reasons why all the benefits of optimization and automatic parallelization of long sequences of loop nests spanning hundreds of lines of code have never been explored. We present a fusion model in an integrated automatic parallelization framework that simultaneously optimizes for hardware prefetch stream buffer utilization, locality, and parallelism. Characterizing the legal space of fusion structures in the polyhedral compiler framework is not difficult. However, incorporating useful optimization criteria into such a legal space to pick good fusion structures is very hard. The model we propose captures utilization of hardware prefetch streams, loss of parallelism, as well as constraints imposed by privatization and code expansion into a single convex optimization space. The model scales very well to program sections spanning hundreds of lines of code. It has been implemented into the polyhedral pass of the IBM XL optimizing compiler. Experimental results demonstrate its effectiveness in finding good fusion structures for codes including SPEC benchmarks and large applications. An improvement ranging from 5% to nearly a factor of 2.75× is obtained over the current production compiler optimizer on these benchmarks.

On the MIR Closure of Polyhedra

We study the mixed-integer rounding (MIR) closure of polyhedra. The MIR closure of a polyhedron 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 approximately, with an accuracy that depends on the number of additional variables used. Our analysis yields a short proof of the result of Cook, Kannan and Schrijver (1990) that the split closure of a polyhedron is again a polyhedron. We also present some computational results with our approximate separation model.

Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra

In this paper, we study the relationship between 2D lattice-free cuts, the family of cuts obtained by taking two-row relaxations of a mixed-integer program (MIP) and applying intersection cuts based on maximal lattice-free sets in

Valid inequalities based on the interpolation procedure

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