Lattice based extended formulations for integer linear equality systems

We study different extended formulations for the set $X = \{x\in\mathbb{Z}^n \mid Ax = Ax^0\}$ in order to tackle the feasibility problem for the set $X_+=X \cap \mathbb{Z}^n_+$. Here the goal is not to find an improved polyhedral relaxation of conv$(X_+)$, but rather to reformulate in such a way that the new variables introduced provide good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. For the case that $A$ has one row $a$ we analyze the reformulations in more detail. In particular, we determine the integer width of the extended formulations in the direction of the last coordinate, and derive a lower bound on the Frobenius number of $a$. We also suggest how a decomposition of the vector $a$ can be obtained that will provide a useful extended formulation. Our theoretical results are accompanied by a small computational study.Comment: uses packages amsmath and amssym

Two row mixed integer cuts via lifting

Recently, Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] characterized extreme inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible nonnegative integer variables (giving the two-row mixed integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We begin by observing that functions giving valid coefficients for the nonnegative integer variables can be constructed by lifting a subset of the integer variables and then applying the fill-in procedure presented in Johnson [23]. We present conditions for these 'general fill-in functions" to be extreme for the two-row mixed integer infinite-group problem. We then show that there exists a unique 'trivial' lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In all other cases (maximal lattice-free triangle with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the case of a triangle with one integer point in the relative interior of each side and non-integral vertices, we present sufficient conditions to yield an extreme inequality for the two-row mixed integer infinite-group problem.

Constrained infinite group relaxations of MIPs

Recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized. In this paper, we consider a stronger relaxation of general mixed integer sets by allowing constraints, such as bounds, on the free integer variables in the continuous group relaxation. We generalize a number of results for the continuous infinite group relaxation to this stronger relaxation and characterize the extreme inequalities when there are two integer variables.

Lot-sizing with stock upper bounds and fixed charges

Here we study the discrete lot-sizing problem with an initial stock variable and an associated variable upper bound constraint. This problem is of interest in its own right, and is also a natural relaxation of the constant capacity lot-sizing problem with upper bounds and fixed charges on the stock variables. We show that the convex hull of solutions of the discrete lot-sizing problem is obtained as the intersection of two simpler sets, one involving just 0-1 variables and the second a mixing set with a variable upper bound constraint. For these two sets we derive both inequality descriptions and polynomial-size extended formulations of their respective convex hulls. Finally we carry out some limited computational tests on single-item constant capacity lot-sizing problems with upper bounds and fixed charges on the stock variables in which we use the extended formulations derived above to strengthen the initial mixed integer programming formulations.mixed integer programming, discrete lot-sizing, stock fixed costs, mixing sets

Optimal Trees

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Optimizing Constrained Subtrees of Trees

Given a tree G = (V, E) and a weight function defined on subsets of its nodes, we consider two associated problems. The first, called the "rooted subtree problem", is to find a maximum weight subtree, with a specified root, from a given set of subtrees. The second problem, called "the subtree packing problem", is to find a maximum weight packing of node disjoint subtrees chosen from a given set of subtrees, where the value of each subtree may depend on its root. We show that the complexity status of both problems is related, and that the subtree packing problem is polynomial if and only if each rooted subtree problem is polynomial. In addition we show that the convex hulls of the feasible solutions to both problems are related: the convex hull of solutions to the packing problem is given by "pasting together" the convex hulls of the rooted subtree problems. We examine in detail the case where the set of feasible subtrees rooted at node i consists of all subtrees with at most k nodes. For this case we derive valid inequalities, and specify the convex hull when k < 4

Economic Lot-Sizing with Start-up Costs: The Convex Hull

A partial description of the convex hull of solutions to the economic lot-sizing problem with start-up costs (ELSS) has been derived recently. Here a larger class of valid inequalities is given and it is shown that these inequalities describe the convex hull of ELSS. This in turn proves that a plant location formulation as a linear program solves ELSS. Finally a separation algorithm is given

Lattice reformulation cuts

Here we consider the question whether the lattice reformulation of a linear integer program can be used to produce effective cutting planes. In particular, we aim at deriving split cuts that cut off more of the integrality gap than Gomory mixed-integer (GMI) inequalities generated from LP-tableaus, while being less computationally demanding than generating the split closure. We consider integer programs (IPs) in the form max{ Ax=b x =Zn+}, where the reformulation takes the form max\{cx +cQ> -xu u =Zn-m Z n - m\}, where Q is an n (n - m) integer matrix. Working on an optimal LP-tableau in the u -space allows us to generate n - m GMIs in addition to the m GMIs associated with the optimal tableau in the x space. These provide new cuts that can be seen as GMIs associated to n - m nonelementary split directions associated with the reformulation matrix \Q . On the other hand it turns out that the corner polyhedra associated to an LP basis and the GMI or split closures are the same whether working in the x or u spaces. Our theoretical derivations are accompanied by an illustrative computational study. The computations show that the effectiveness of the cuts generated by this approach depends on the quality of the reformulation obtained by the reduced basis algorithm used to generate Q and that it is worthwhile to generate several rounds of such cuts. However, the effectiveness of the cuts deteriorates as the number of constraints is increased