Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

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

Single item lot-sizing with non-decreasing capacities

We consider the single item lot-sizing problem with capacities that are non-decreasing over time. When the cost function is i) non-speculative or Wagner-Whitin (for instance, constant unit production costs and non-negative unit holding costs), and ii) the production set-up costs are non-increasing over time, it is known that the minimum cost lot-sizing problem is polynomially solvable using dynamic programming. When the capacities are non-decreasing, we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot-sizing problem to optimality when the objective function satisfies i) and ii). The formulation is based on mixing set relaxations and reduces to the (known) convex hull of solutions when the capacities are constant over time. We illustrate the use and effectiveness of this improved LP formulation on a new test instances, including instances with and without Wagner-Whitin costs, and with both non-decreasing and arbitrary capacities over time.lot-sizing, mixing set relaxation, compact reformulation, production planning, mixed integer programming

Uncapacitated Lot-Sizing with Stock Upper Bounds, Stock Fixed Costs, Stock Overloads and Backlogging: A Tight Formulation

For an n-period uncapacitated lot-sizing problem with stock upper bounds, stock fixed costs, stock overload and backlogging, we present a tight extended shortest path formulation of the convex hull of solutions with O(n^2) variables and constraints, also giving an O(n^2) algorithm for the problem. This corrects and extends a formulation in [11] for the problem with just stock upper bounds

Mixing sets linked by bidirected paths

Recently there has been considerable research on simple mixed-integer sets, called mixing sets, and closely related sets arising in uncapacitated and constant capacity lot- sizing. This in turn has led to study of more general sets, called network-dual sets, for which it is possible to derive extended formulations whose projection gives the convex hull of the network-dual set. Unfortunately this formulation cannot be used (in general) to optimize in polynomial time. Furthermore the inequalities definining the convex hull of a network-dual set in the original space of variables are known only for some special cases. Here we study two new cases, in which the continuous variables of the network-dual set are linked by a bi- directed path. In the first case, which is motivated by lot-sizing problems with (lost) sales, we provide a description of the convex hull as the intersection of the convex hulls of 2^n mixing sets, where n is the number of continuous variables of the set. However optimization is polynomial as only n + 1 of the sets are required for any given objective function. In the second case, generalizing single arc flow sets, we describe again the convex hull as an intersection of an exponential number of mixing sets and also give a combinatorial polynomial-time separation algorithm.mixing sets, extended formulations, mixed integer programming, lot-sizing with sales

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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