Cutting planes for integer programs with general integer variables

We investigate the use of cutting planes for integer programs with general integer variables. We show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0–1 variables. We also explore the use of Gomory's mixed-integer cuts. We address both theoretical and computational issues and show how to embed these cutting planes in a branch-and-bound framework. We compare results obtained by using our cut generation routines in two existing systems with a commercially available branch-and-bound code on a range of test problems arising from practical applications. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.This research was partly performed when the author was affiliated with CORE, Université Catholique de Louvain.     

Test sets for integer programs

In this paper I discuss various properties of the simplicial complex of maximal lattice free bodies associated with a matrixA. If the matrix satisfies some mild conditions, and isgeneric, the edges of the complex form the minimal test set for the family of integer programs obtained by selecting a particular row ofA as the objective function, and using the remaining rows to impose constraints on the integer variables.     

An improved algorithm for solving biobjective integer programs

A parametric algorithm for identifying the Pareto set of a biobjective integer program is proposed. The algorithm is based on the weighted Chebyshev (Tchebycheff) scalarization, and its running time is asymptotically optimal. A number of extensions are described, including: a technique for handling weakly dominated outcomes, a Pareto set approximation scheme, and an interactive version that provides access to all Pareto outcomes. Extensive computational tests on instances of the biobjective knapsack problem and a capacitated network routing problem are presented.     

Test sets of integer programs

This article is a survey about recent developments in the area of test sets of families of linear integer programs. Test sets are finite subsets of the integer lattice that allow to improve any given feasible non-optimal point of an integer program by one element in the set. There are various possible ways of defining test sets depending on the view that one takes: theGraver test set is naturally derived from a study of the integral vectors in cones; theScarf test set (neighbors of the origin) is strongly connected to the study of lattice point free convex bodies; the so-calledreduced Gröbner basis of an integer program is obtained from a study of generators of polynomial ideals. This explains why the study of test sets connects various branches of mathematics. We introduce in this paper these three kinds of test sets and discuss relations between them. We also illustrate on various examples such as the minimum cost flow problem, the knapsack problem and the matroid optimization problem how these test sets may be interpreted combinatorially. From the viewpoint of integer programming a major interest in test sets is their relation to the augmentation problem. This is discussed here in detail. In particular, we derive a complexity result of the augmentation problem, we discuss an algorithm for solving the augmentation problem by computing the Graver test set and show that, in the special case of an integer knapsack problem with 3 coefficients, the augmentation problem can be solved in polynomial time.Supported by a Gerhard-Hess-Forschungsförderpreis of the German Science Foundation (DFG).     

On mixed-integer sets with two integer variables

We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables if the coefficients of the integer variables form a matrix of rank 2. We also present an alternative characterization of the crooked cross cut closure of mixed-integer sets similar to the one on the equivalence of different definitions of split cuts presented in Cook et al. (1990) [4]. This characterization implies that crooked cross cuts dominate the 2-branch split cuts defined by Li and Richard (2008) [8]. Finally, we extend our results to mixed-integer sets that are defined as the set of points (with some components being integral) inside a closed, bounded and convex set.     

Enhancing CP-based column generation for integer programs

This is a summary of the author’s Ph.D. thesis supervised by Federico Malucelli and defended on 15 May 2008 at the Politecnico di Milano. The thesis is written in English and is available from the author upon request. This work presents new methods for enhancing the Column Generation approach based on Constraint Programming when it is used for solving combinatorial optimization problems. The methods proposed focus on the interactions between the linear programming solver and the constraint programming solver, and on how they impact on both a single iteration and the overall execution of the Column Generation procedure. The result of this work is the design and implementation of general-purpose optimization algorithms, whose efficiency is proven by solving two very different problems: the Minimum Graph Coloring Problem and a resource allocation problem arising in Wireless Ad Hoc Networks.     

An algorithm for a separable integer programming problem with cumulatively bounded variables

An algorithm is presented for obtaining the optimal solution of an integer programming problem in which the nested constraints represent the cumulative bounding of the variables and the objective function is a sum of concave functions of one variables. The algorithm requires O(m log(m)log²(b_m/m)) time, where m is the number of variables and b_m is an upper bound on the sum of the m variables, b_m ⩾m⩾1. It is also demonstrated that a special case of identical concave functions is solvable in O(m). Both results significantly improve the previous bounds for these problems.     

Resolving sets and integer programs for recommender systems

Recommender systems make use of different sources of information for providing users with recommendations of items. Such systems are often based on either collaborative filtering methods which make automatic predictions about the interests of a user, using preferences of similar users, or content based filtering that matches the user’s personal preferences with item characteristics. We adopt the content-based approach and propose to use the concept of resolving set that allows to determine the preferences of the users with a very limited number of ratings. We also show how to make recommendations when user ratings are imprecise or inconsistent, and we indicate how to take into account situations where users possibly don’t care about the attribute values of some items. All recommendations are obtained in a few seconds by solving integer programs.

     

An interior-point Benders based branch-and-cut algorithm for mixed integer programs

We present an interior-point branch-and-cut algorithm for structured integer programs based on Benders decomposition and the analytic center cutting plane method (ACCPM). We show that the ACCPM based Benders cuts are both pareto-optimal and valid for any node of the branch-and-bound tree. The valid cuts are added to a pool of cuts that is used to warm-start the solution of the nodes after branching. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances.     

Binary integer programs with two variables per inequality

Several recent papers have shown that some properties of the maximum weight stable set problem hold true in the more general setting of binary integer programs with two variables per inequality. In this paper, we show that in fact the two problems are equivalent by using the transitive closure of the binary integer program and (possibly) reducing the number of variables by fixing, complementing, or identifying them. We use this equivalence to prove two conjectures made by Johnson and Padberg regarding the perfection of bidirected graphs.     

A rounding algorithm for integer programs

There exist general purpose algorithms to solve the integer linear programming problem but none of them are polynomial. Polynomially bounded rounding algorithms have been studied, but most of them are problem specific. In this paper we study a generalized rounding algorithm that is polynomial, characterize matrices that may be used in this scheme and identify a class of integer programs that it solves.     

An analysis of the greedy algorithm for partially ordered sets

The worst-case analysis of the greedy algorithm for a combinatorial problem of linear maximization on a partially ordered set (introduced by U. Faigle [3]) is given.     

S3 sets. An extension of the Beale-Tomlin special ordered sets

In this work an extension of the Beale-Tomlin special ordered sets is introduced that has proved to be efficient for solving certain types of open shop scheduling problems. Besides their usual characteristics, exclusivity constraints in the jobs are allowed, more general than tree-like precedence structures are considered, and semi-active schedules that cannot be labeled as non-optimal solutions may occur. The problem is formulated as a large-scale 0–1 model. Computational experience on some real-life problems is reported.     

Strong duality for a special class of integer programs

A pair of primal-dual integer programs is constructed for a class of problems motivated by a generalization of the concept of greatest common divisor. The primal-dual formulation is based on a number-theoretic, rather than a Lagrangian, duality property; consequently, it avoids the dualitygap common to Lagrangian duals in integer programming.This research was partially supported by the National Science Foundation, Grant No. DCR-74-20584     

On the facets of mixed integer programs with two integer variables and two constraints

In this paper we consider an infinite relaxation of the mixed integer linear program with two integer variables, nonnegative continuous variables and two equality constraints, and we give a complete characterization of its facets. We also derive an analogous characterization of the facets of the underlying finite integer program.     

The greedy algorithm for partially ordered sets

Those independence systems on finite partially ordered sets are characterized for which the greedy algorithm always works.     

Global optimization using special ordered sets

The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick [4] points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland [2] or Soland [6], or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin [1], have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin [3] show how they have benefitted from the vast improvements in branch and bound integer programming capabilities over the last few years, as a result of being incorporated in a general mathematical programming system.Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis.This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two by-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval.The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.     

Generalized Chvátal-Gomory closures for integer programs with bounds on variables

Mathematical Programming - Integer programming problems that arise in practice often involve decision variables with one or two sided bounds. In this paper, we consider a generalization of...     

Parameterisation algorithms for the integer linear programs in binary variables

In this paper we consider the integer programmiing problem: minimize z(x) = c · x subject to Ax?b,x binary.Roodman appended the objective function z(x) to the body of the constraints and presented a modified version of the Balas additive algorithm by which each fathomed partial solution is attributed to the constraint which caused the fathoming. Exploiting this information, he conducted (a) ranging analysis, i.e. calculating bounds on the values of the parameters which leave the original optimal solution unchanged, and (b) parameter change analysis, i.e. determining new optimal solutions (if any) for revised values of the parameters outside the ranging bounds.We extend Roodman's results and construct parametric functions of the following form. Let Σ be any parameter of c or b or A, and replace Σ by Σ(λ) = Σ + λ. Then, holding every other parameter of the program fixed, and varying λ in the set of real numbers we construct the parametric function $\text{z}{}^1\text{(Σ(λ))}$ which matches non-overlapping intervals of λ with optimal solutions. This replaces by exact bounds in the linear programming sense, the bounds underestimated by Roodman's ranging analysis. It also determines optimal solutions for any values of λ, rather than for a revised set of values. Finally some results of computational experience are presented.