Computational study of a column generation algorithm for bin packing and cutting stock problems

This paper reports on our attempt to design an efficient exact algorithm based on column generation for the cutting stock problem. The main focus of the research is to study the extend to which standard branch-and-bound enhancement features such as variable fixing, the tightening of the formulation with cutting planes, early branching, and rounding heuristics can be usefully incorporated in a branch-and-price algorithm. We review and compare lower bounds for the cutting stock problem. We propose a pseudo-polynomial heuristic. We discuss the implementation of the important features of the integer programming column generation algorithm and, in particular, the implementation of the branching scheme. Our computational results demonstrate the efficiency of the resulting algorithm for various classes of bin packing and cutting stock problems. Received October 18, 1996 / Revised version received May 14, 1998?Published online July 19, 1999     

Representations of unbounded optimization problems as integer programs

Any optimization problem in a finite structure can be represented as an integer or mixed-integer program in integral quantities.We show that, when an optimization problem on an unbounded structure has such a representation, it is very close to a linear programming problem, in the specific sense described in the following results. We also show that, if an optimization problem has such a representation, no more thann+2 equality constraints need be used, wheren is the number of variables of the problem.We obtain a necessary and sufficient condition for a functionf:SZ, withS Z ⁿ , to have a rational model in Meyer's sense, and show that Ibaraki models are a proper subset of Meyer models.This research was supported by NSF Grant No. GP-37510X1 and ONR Contract No. N00014-75-C0621, NR047-048.     

Approximating disjoint-path problems using packing integer programs

In a packing integer program, we are given a matrix $A$ and column vectors $b,c$ with nonnegative entries. We seek a vector $x$ of nonnegative integers, which maximizes $c^{T}x,$ subject to $Ax \leq b.$ The edge and vertex-disjoint path problems together with their unsplittable flow generalization are NP-hard problems with a multitude of applications in areas such as routing, scheduling and bin packing. These two categories of problems are known to be conceptually related, but this connection has largely been ignored in terms of approximation algorithms. We explore the topic of approximating disjoint-path problems using polynomial-size packing integer programs. Motivated by the disjoint paths applications, we introduce the study of a class of packing integer programs, called column-restricted. We develop improved approximation algorithms for column-restricted programs, a result that we believe is of independent interest. Additional approximation algorithms for disjoint-paths are presented that are simple to implement and achieve good performance when the input has a special structure.Work partially supported by NSERC OG 227809-00 and a CFI New Opportunities Award. Part of this work was done while at the Department of Computer Science, Dartmouth College and partially by NSF Award CCR-9308701 and NSF Career Award CCR-9624828.This work was done while at the Department of Computer Science, Dartmouth College and partially supported by NSF Award CCR-9308701 and NSF Career Award CCR-9624828.     

Polyhedral results for the edge capacity polytope

Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.?As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated.?The derived inequalities are tested on (i) the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems. Received: September 2000 / Accepted: October 2001?Published online March 27, 2002     

The mathematics of playing golf, or: a new class of difficult non-linear mixed integer programs

We consider a class of non-linear mixed integer programs with n integer variables and k continuous variables. Solving instances from this class to optimality is an NP-hard problem. We show that for the cases with k=1 and k=2, every optimal solution is integral. In contrast to this, for every k≥3 there exist instances where every optimal solution takes non-integral values. Received: August 2001 / Accepted: January 2002?Published online March 27, 2002     

Box-inequalities for quadratic assignment polytopes

Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. Their quality has turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started during the last decade [34, 31, 21, 22]. They have lead to basic knowledge on these polytopes concerning questions like their dimensions, affine hulls, and trivial facets. However, no large class of (facet-defining) inequalities that could be used in cutting plane procedures had been found. We present in this paper the first such class of inequalities, the box inequalities, which have an interesting origin in some well-known hypermetric inequalities for the cut polytope. Computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds. The most effective ones among the new inequalities turn out to be indeed facet-defining for both the non-symmetric as well as for the symmetric quadratic assignment polytope. Received: April 17, 2000 / Accepted: July 3, 2001?Published online September 3, 2001     

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.     

Binary clutter inequalities for integer programs

We introduce a new class of valid inequalities for general integer linear programs, called binary clutter (BC) inequalities. They include the -cuts of Caprara and Fischetti as a special case and have some interesting connections to binary matroids, binary clutters and Gomory corner polyhedra. We show that the separation problem for BC-cuts is strongly -hard in general, but polynomially solvable in certain special cases. As a by-product we also obtain new conditions under which -cuts can be separated in polynomial time. These ideas are then illustrated using the Traveling Salesman Problem (TSP) as an example. This leads to an interesting link between the TSP and two apparently unrelated problems, the T-join and max-cut problems.Mathematics Subject Classification: 90C10     

The packing property

A clutter (V, E) packs if the smallest number of vertices needed to intersect all the edges (i.e. a minimum transversal) is equal to the maximum number of pairwise disjoint edges (i.e. a maximum matching). This terminology is due to Seymour 1977. A clutter is minimally nonpacking if it does not pack but all its minors pack. An m×n 0,1 matrix is minimally nonpacking if it is the edge-vertex incidence matrix of a minimally nonpacking clutter. Minimally nonpacking matrices can be viewed as the counterpart for the set covering problem of minimally imperfect matrices for the set packing problem. This paper proves several properties of minimally nonpacking clutters and matrices. Received: December 1, 1997 / Accepted: April 6, 1999?Published online October 18, 2000     

Using continuous nonlinear relaxations to solve. constrained maximum-entropy sampling problems

Received January 9, 1997 / Revised version received January 26, 1998 Published online November 24, 1998     

A decomposition-based solution method for stochastic mixed integer nonlinear programs

This is a summary of the main results presented in the author’s PhD thesis, supervised by D. Conforti and P. Beraldi and defended on March 2005. The thesis, written in English, is available from the author upon request. It describes one of the very few existing implementations of a method for solving stochastic mixed integer nonlinear programming problems based on deterministic global optimization. In order to face the computational challenge involved in the solution of such multi-scenario nonconvex problems, a branch and bound approach is proposed that exploits the peculiar structure of stochastic programming problem.     

Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities

Let VIP(F,C) denote the variational inequality problem associated with the mapping F and the closed convex set C. In this paper we introduce weak conditions on the mapping F that allow the development of a convergent cutting-plane framework for solving VIP(F,C). In the process we introduce, in a natural way, new and useful notions of generalized monotonicity for which first order characterizations are presented. Received: September 25, 1997 / Accepted: March 2, 1999?Published online July 20, 2000     

Primal cutting plane algorithms revisited

Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are well-known and form the basis of the highly successful branch-and-cut method. It is rather less well-known that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutting planes is to enable a feasible solution to the original problem to be improved. Research on these algorithms has been almost non-existent.  In this paper we argue for a re-examination of these primal methods. We describe a new primal algorithm for pure 0-1 problems based on strong valid inequalities and give some encouraging computational results. Possible extensions to the case of general mixed-integer programs are also discussed.     

Mixing mixed-integer inequalities

Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure. Received: April 1998 / Accepted: January 2001?Published online April 12, 2001     

A Log-Barrier method with Benders decomposition for solving two-stage stochastic linear programs

An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving two-stage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run in polynomial-time. Received: August 1998 / Accepted: August 2000?Published online April 12, 2001     

Solving a class of semidefinite programs via nonlinear programming

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important SDP problems. We also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data. This transformation, together with the efficient gradient formulas, enables the solution of very large-scale SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a first-order log-barrier method designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global convergence of the algorithm is established under mild and reasonable assumptions. Received: January 5, 2000 / Accepted: October 2001?Published online February 14, 2002     

An integer linear programming approach for a class of bilinear integer programs

We propose an Integer Linear Programming (ILP) approach for solving integer programs with bilinear objectives and linear constraints. Our approach is based on finding upper and lower bounds for the integer ensembles in the bilinear objective function, and using the bounds to obtain a tight ILP reformulation of the original problem, which can then be solved efficiently. Numerical experiments suggest that the proposed approach outperforms a latest iterative ILP approach, with notable reductions in the average solution time.