Comparison of Bundle and Classical Column Generation

An updated version of this paper has appeared in : Math. Program., Ser. A, 2006 DOI 10.1007/s10107-006-0079-zWhen a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley's method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparaisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman

Stabilized Benders methods for large-scale combinatorial optimization, with appllication to data privacy

The Cell Suppression Problem (CSP) is a challenging Mixed-Integer Linear Problem arising in statistical tabular data protection. Medium sized instances of CSP involve thousands of binary variables and million of continuous variables and constraints. However, CSP has the typical structure that allows application of the renowned Benders’ decomposition method: once the “complicating” binary variables are fixed, the problem decomposes into a large set of linear subproblems on the “easy” continuous ones. This allows to project away the easy variables, reducing to a master problem in the complicating ones where the value functions of the subproblems are approximated with the standard cutting-plane approach. Hence, Benders’ decomposition suffers from the same drawbacks of the cutting-plane method, i.e., oscillation and slow convergence, compounded with the fact that the master problem is combinatorial. To overcome this drawback we present a stabilized Benders decomposition whose master is restricted to a neighborhood of successful candidates by local branching constraints, which are dynamically adjusted, and even dropped, during the iterations. Our experiments with randomly generated and real-world CSP instances with up to 3600 binary variables, 90M continuous variables and 15M inequality constraints show that our approach is competitive with both the current state-of-the-art (cutting-plane-based) code for cell suppression, and the Benders implementation in CPLEX 12.7. In some instances, stabilized Benders is able to quickly provide a very good solution in less than one minute, while the other approaches were not able to find any feasible solution in one hour.Peer ReviewedPreprin

Solutions diversification in a column generation algorithm

International audienceColumn generation algorithms have been specially designed for solving mathematical programs with a huge number of variables. Unfortunately, this method suffers from slow convergence that limits its efficiency and usability. Several accelerating approaches are proposed in the literature such as stabilization-based techniques. A more classical approach, known as "intensification, consists in inserting a set of columns instead of only the best one. Unfortunately, this intensication typically overloads the master problem, and generates a huge number of useless variables. This article covers some characteristics of the generated columns from theoretical and experimental points of view. Two selection criteria are compared. The first one is based on column reduced cost and the second on column structure. We conclude our study with computational experiments on two kinds of problems: the acyclic vehicle routing problem with time windows and the one-dimensional cutting stock. problem

Quadratic stabilization of Benders decomposition

The foundational Benders decomposition, or variable decomposition, is known to have the inherent instability of cutting plane-based methods. Several techniques have been proposed to improve this method, which has become the state of the art for important problems in operations research. This paper presents a complementary improvement featuring quadratic stabilization of the Benders cutting-plane model. Inspired by the level-bundle methods of nonsmooth optimization, this algorithmic improvement is designed to reduce the number of iterations of the method. We illustrate the interest of the stabilization on two classical problems: network design problems and hub location problems. We also prove that the stabilized Benders method has the same theoretical convergence properties as the usual Benders method

Using the primal-dual interior point algorithm within the branch-price-and-cut method

AbstractBranch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the vehicle routing problem with time windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm

Use of Machine Learning Models to Warmstart Column Generation for Unit Commitment

The unit commitment problem is an important optimization problem in the energy industry used to compute the most economical operating schedules of power plants. Typically, this problem has to be solved repeatedly with different data but with the same problem structure. Machine learning techniques have been applied in this context to find primal feasible solutions. On the other hand, Dantzig-Wolfe decomposition with a column generation procedure has been shown to be successful in solving the unit commitment problem to tight tolerance. We propose the use of machine learning models not to find primal feasible solutions directly but to generate initial dual values for the column generation procedure. Our numerical experiments compare machine learning based methods for warmstarting the column generation procedure with three baselines: column pre-population, the linear programming relaxation and coldstart. The experiments reveal that the machine learning approaches are able to find both tight lower bounds and accurate primal feasible solutions in a shorter time compared to the baselines. Furthermore, these approaches scale well to handle large instances.Comment: 26 page

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

Incremental bundle methods using upper models

We propose a family of proximal bundle methods for minimizing sum-structured convex nondifferentiable functions which require two slightly uncommon assumptions, that are satisfied in many relevant applications: Lipschitz continuity of the functions and oracles which also produce upper estimates on the function values. In exchange, the methods: i) use upper models of the functions that allow to estimate function values at points where the oracle has not been called; ii) provide the oracles with more information about when the function computation can be interrupted, possibly diminishing their cost; iii) allow to skip oracle calls entirely for some of the component functions, not only at "null steps" but also at "serious steps"; iv) provide explicit and reliable a-posteriori estimates of the quality of the obtained solutions; v) work with all possible combinations of different assumptions on how the oracles deal with not being able to compute the function with arbitrary accuracy. We also discuss the introduction of constraints (or, more generally, of easy components) and use of (partly) aggregated models