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Cutting planes in integer and mixed integer programming
Discrete Applied Mathematics, 2002
This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for (i) general integer programs, (ii) problems with local structure such as knapsack constraints, and (iii) problems with 0-1 coe cient matrices, such as set packing, are examined in turn. Finally, the use of valid inequalities for classes of problems with structure, such as network design, is explored.
Projected Chvátal-Gomory cuts for mixed integer linear programs
Mathematical Programming, 2008
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.
Mod‐2 Cuts Generation Yields the Convex Hull of Bounded Integer Feasible Sets
SIAM Journal on Discrete Mathematics, 2006
This paper focuses on the outer description of the convex hull of all integer solutions to a given system of linear inequalities. It is shown that if the given system contains lower and upper bounds for the variables, then the convex hull can be produced by iteratively generating so-called mod-2 cuts only. This fact is surprising and might even be counterintuitive, since many integer rounding cuts exist that are not mod-2, i.e., representable as the zero-one-half combination of the given constraint system. The key, however, is that in general many more rounds of mod-2 cut generation are necessary to produce the final description compared to the traditional integer rounding procedure.
Conjunctive Cuts for Integer Programs
This paper deals with a family of conjunctive inequalities. Such inequalities are needed to describe the polyhedron associated with all the integer points that satisfy several knapsack constraints simultaneously. Here we demonstrate the strength and potential of conjunctive inequalities in connection with lifting from a computational point of view. 1
Mathematical Programming, 2014
We compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints, two unrestricted integer variables and any number of nonnegative continuous variables. In particular, we prove that the closure of Type 2 triangle (resp. Type 3 triangle; quadrilateral) inequalities, are all within a factor of 1.5 of the integer hull, and provide examples showing that the approximation factor is not less than 1.125. There is no fixed approximation ratio for split or Type 1 triangle inequalities however.
Integer Programming, Lattices, and Results in Fixed Dimension
Handbooks in Operations Research and Management Science, 2005
We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running time is polynomial in fixed dimension. We also briefly describe an algorithm, based on a different principle, to count integer points in an integer polytope. We then turn the attention to integer optimization. Again, we describe three algorithms: binary search, a linear algorithm for a fixed number of constraints, and a randomized algorithm for a varying number of constraints. The topic of the next part of our chapter is how to use lattice basis reduction in problem reformulation. Finally, we review cutting plane results when the dimension is fixed.
Cutting Planes for Mathematical Programming: Separation from Convex Hulls
2020
Cutting planes methods are of crucial importance to solve huge or nonconvex optimization problems. Minimizing a linear functional over general sets F may not be possible directly, however, optimizing on a wisely chosen relaxation R ⊇ F can be computationally tractable. Using the equivalence between optimization and separation, a popular approach consists in optimizing over a lightweight relaxation R 0 and separate the resulting incumbent from other cherry-picked relaxations R via cutting planes. Doing so, we implicitly optimize over R 0 intersected with cc R, the closed convex hull of R. For any type of relaxation R, we present a framework to efficiently separate an incumbent from cc R, granted that optimizing a linear functional over R can be done efficiently. In particular, our approach can generate in an efficient way 1) Balas' disjunctive cuts without having to solve a problem being multiple times the size of the original (for mixed integer linear conic problems) nor using perspective functions (for nonlinear problems), 2) Boyd's Fenchel cuts (for mixed integer linear problems) without enumerating all combinations of a small number of the integer constrained variables and 3) The hull generated by a Dantzig-Wolfe (DW) scheme via cutting planes only. In its classical use, DW is usable only when F exhibits a block-separable sub-structure. However, when facing a mixed integer problem with convex generalized nonlinear inequalities, given a partition of the variables, we show a generic way to construct a relaxation that is block-separable with respect to that partition.
Decomposition and Dynamic Cut Generation in Integer Programming: Theory and Algorithms
Abstract Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to compute bounds for integer programming problems. We discuss a framework for integrating dynamic cut generation with traditional decomposition methods in order to obtain improved bounds and present a new paradigm for separation called decompose and cut.
An optimality cut for mixed integer linear programs
European Journal of Operational Research, 1999
We derive the penalty cut, a simple optimality cut of general applicability in pure or mixed linear programs. This cut is tested on a number of examples and comparisons with the classical Gomory cut are provided.
Foundation-penalty cuts for mixed-integer programs
Operations Research Letters, 2003
We propose a new class of Foundation-Penalty (FP) cuts for GUB-constrained (and ordinary) mixed-integer programs, which are easy to generate by exploiting standard penalty calculations that are routinely employed in branch-and-bound contexts. The FP cuts are derived with reference to a selected integer variable or GUB set, and a foundation function that is typically a reduced cost function corresponding to an optimal linear programming basis. The concept behind the generation of these cuts generalizes the lifting process, and as we demonstrate, bears relationships with other classical cuts such as disjunctive cuts, lift-and-project cuts, convexity cuts, Gomory cuts, and mixedinteger rounding cuts. For example, the easily derived but often useful cutting planes at the level of Gomory cuts and mixed integer rounding cuts are subsumed by related FP cuts that simply 'plug in' penalty values from standard calculations (where the penalties are allowed to go beyond those from the first primitive mixed-integer programming codes). In general, the strength of these FP cuts can be varied according to the trade-offs between the strengths of alternative penalty calculations and the effort required to compute them, by virtue of the interactions between the foundation function and the branching disjunctions that underlie the penalty computations. By this means, FP cuts are especially convenient for use in branch-and-cut procedures, where penalty calculations are employed as a matter of course, and afford new strategies for generating cutting planes in this setting.