José Manuel Valério de Carvalho

Using Extra Dual Cuts to Accelerate Column Generation

Column generation is often used to solve models with stronger linear-programming relaxations. From the dual standpoint, column-generation processes can be viewed as cutting plane algorithms. In this paper, we present conditions under which it is possible to restrict the dual space, but still preserve the strength of the primal model and recover an optimal primal solution. We derive a family of dual cuts that are valid for the space of dual optimal solutions of the one-dimensional cutting-stock problem. These cuts correspond to extra columns in the primal model. Inserting a polynomial number of cuts of this family in the problem formulation at initialization time restricts the dual space during the entire column-generation process. Computational experiments show that this idea helps, reducing substantially the number of columns generated, the number of degenerate iterations, and the computational time.

A survey of dual-feasible and superadditive functions

Dual-feasible functions are valuable tools that can be used to compute both lower bounds for different combinatorial problems and valid inequalities for integer programs. Several families of functions have been used in the literature. Some of them were defined explicitly, and others not. One of the main objectives of this paper is to survey these functions, and to state results concerning their quality. We clearly identify dominant subsets of functions, i.e. those which may lead to better bounds or stronger cuts. We also describe different frameworks that can be used to create dual-feasible functions. With these frameworks, one can get a dominant function based on other ones. Two new families of dual-feasible functions obtained by applying these methods are proposed in this paper.We also performed a computational comparison on the relative strength of the functions presented in this paper for deriving lower bounds for the bin-packing problem and valid cutting planes for the pattern minimization problem. Extensive experiments on instances generated using methods described in the literature are reported. In many cases, the lower bounds are improved, and the linear relaxations are strengthened.

Dual-Optimal Inequalities for Stabilized Column Generation

Column generation is one of the most successful approaches for solving large-scale linear programming problems. However, degeneracy difficulties and long-tail effects are known to occur as problems become larger. In recent years, several stabilization techniques of the dual variables have proven to be effective. We study the use of two types of dual-optimal inequalities to accelerate and stabilize the whole convergence process. Added to the dual formulation, these constraints are satisfied by all or a subset of the dual-optimal solutions. Therefore, the optimal objective function value of the augmented dual problem is identical to the original one. Adding constraints to the dual problem leads to adding columns to the primal problem, and feasibility of the solution may be lost. We propose two methods for recovering primal feasibility and optimality, depending on the type of inequalities that are used. Our computational experiments on the binary and the classical cutting-stock problems, and more specifically on the so-called triplet instances, show that the use of relevant dual information has a tremendous effect on the reduction of the number of column generation iterations.

Cutting Stock Problems

Column generation has been proposed by Gilmore and Gomory to solve cutting stock problem, independently of Dantzig-Wolfe decomposition. We survey the basic models proposed for cutting stock and the corresponding solution approaches. Extended Dantzig-Wolfe decomposition is surveyed and applied to these models in order to show the links to Gilmore-Gomory model. Branching schemes discussion is based on the subproblem formulation corresponding to each model. Integer solutions are obtained by combining heuristics and branch-and-price schemes. Linear relaxations are solved by column generation. Stabilization techniques such as dual-optimal inequalities and stabilized column generation algorithms that have been proposed to improve the efficiency of this process are briefly discussed.

Comparing Dantzig-Wolfe decompositions and branch-and-price algorithms for the multi-item capacitated lot sizing problem

In this article, we consider the multi-item capacitated lot sizing problem with setup times. Starting from an original mixed integer programming model, we apply the standard Dantzig–Wolfe decomposition (DWD) in two different ways: defining the subproblems by items and defining the subproblems by periods. A third decomposition is developed in which the subproblems of both types are integrated in the same model. The linear relaxation of this last approach, which we denote as multiple DWD, provides lower bounds (equal to or) better than the bounds obtained by the other decompositions, which in turn, provide lower bounds (equal to or) better than the ones given by the original model. For solving the three decomposition models, we implemented three branch-and-price algorithms. We describe their main aspects and report on their computational results in instances from the literature.

Multidimensional dual-feasible functions and fast lower bounds for the vector packing problem

In this paper, we address the 2-dimensional vector packing problem where an optimal layout for a set of items with two independent dimensions has to be found within the boundaries of a rectangle. Many practical applications in areas such as the telecommunications, transportation and production planning lead to this combinatorial problem. Here, we focus on the computation of fast lower bounds using original approaches based on the concept of dual-feasible functions.

New Stabilization Procedures for the Cutting Stock Problem

In this paper, we deal with a column generation-based algorithm for the classical cutting stock problem. This algorithm is known to have convergence issues, which are addressed in this paper. Our methods are based on the fact that there are interesting characterizations of the structure of the dual problem, and that a large number of dual solutions are known. First, we describe methods based on the concept of dual cuts, proposed by Valerio de Carvalho [Valerio de Carvalho, J. M. 2005. Using extra dual cuts to accelerate column generation. INFORMS J. Comput.17(2) 175--182]. We introduce a general framework for deriving cuts, and we describe a new type of dual cut that excludes solutions that are linear combinations of some other known solutions. We also explore new lower and upper bounds for the dual variables. Then we show how the prior knowledge of a good dual solution helps improve the results. It tightens the bounds around the dual values and makes the search converge faster if a solution is sought in its neighborhood first. A set of computational experiments on very hard instances is reported at the end of the paper; the results confirm the effectiveness of the methods proposed.

New lower bounds based on column generation and constraint programming for the pattern minimization problem

The pattern minimization problem is a cutting and packing problem that consists in finding a cutting plan with the minimum number of different patterns. This objective may be relevant when changing from one pattern to another involves a cost for setting up the cutting machine. When the minimization of the number of different patterns is done by assuming that no more than the minimum number of rolls can be used, the problem is also referred to as the cutting stock problem with setup costs. Most of the approaches described in the literature are based on heuristics. Solving the problem exactly has been a real challenge, and only very few exact solution methods have been reported so far in the literature. In this paper, we intend to contribute to the resolution of the pattern minimization problem with new results. We explore a different integer programming model that can be solved using column generation, and we describe different strategies to strengthen it, among which are constraint programming and new families of valid inequalities. Lower bounds for the pattern minimization problem are derived from the new integer programming model, and also from a constraint programming model. Our approaches were tested on a set of real instances, and on a set of random instances from the literature. For these instances, the computational experiments show a clear improvement on the quality of the lower bounds.

A framework for dependability evaluation of PROFIBUS networks

Fieldbus networks have been assuming a high acceptance in the industrial environment, replacing the old centralized control architectures. Due to time critical nature of the tasks involved in these environments, the fulfillment of dependability attributes is usually required. Therefore, the dependability is an important parameter on system design, which should be evaluated. Several factors can affect system dependability. The environmental ones are the most common and due to the particularity of the industrial environment this susceptibility is increased. In this paper, we propose a framework based on fault injection techniques, supported by a hardware platform, which emulates a fault set, representative of industrial environment scenarios, intending to disturb data communications on a PROFIBUS network. From these fault injection experiments, relevant data is gathered and a further analysis is carried out in order to evaluate dependability attributes.

On the Properties of General Dual-Feasible Functions

Dual-feasible functions have been used to compute fast lower bounds and valid inequalities for integer linear optimization problems. However, almost all the functions proposed in the literature are defined only for positive arguments, which restricts considerably their applicability. The characteristics and properties of dual-feasible functions with general domains remain mostly unknown. In this paper, we show that extending these functions to negative arguments raises many issues. We explore these functions in depth with a focus on maximal functions, i.e. the family of non-dominated functions. The knowledge of these properties is fundamental to derive good families of general maximal dual-feasible functions that might lead to strong cuts for integer linear optimization problems and strong lower bounds for combinatorial optimization problems with knapsack constraints.