A Stabilized Structured Dantzig-Wolfe Decomposition Method

We discuss an algorithmic scheme, which we call the stabilized structured Dantzig-Wolfe decomposition method, for solving large-scale structured linear programs. It can be applied when the subproblem of the standard Dantzig-Wolfe approach admits an alternative master model amenable to column generation, other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron. Stabilization is achieved by the same techniques developed for the standard Dantzig-Wolfe approach and it is equally useful to improve the performance, as shown by computational results obtained on an application to the multicommodity capacitated network design problem

A note on "A LP-based heuristic for a time-constrained routing problem"

In their paper, Avella et al. (2006) investigate a time-constrained routing problem. The core of the proposed solution approach is a large-scale linear program that grows both row- and column-wise when new variables are introduced. Thus, a column-and-row generation algorithm is proposed to solve this linear program optimally, and an optimality condition is presented to terminate the column-and-row generation algorithm. We demonstrate by using Lagrangian duality that this optimality condition is incorrect and may lead to a suboptimal solution at termination

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm for a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints. These constraints are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on the fly within an efficient solution method. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm and then introduce the associated set of pricing subproblems in detail. The proposed methodology is demonstrated on numerical examples for the multi-stage cutting stock and the quadratic set covering problems

On the Computational Efficiency of Subgradient Methods: a Case Study with Lagrangian Bounds

Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications. We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables. We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different Lagrangian duals of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices. Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems

Dynamic smoothness parameter for fast gradient methods

We present and computationally evaluate a variant of the fast gradient method by Nesterov that is capable of exploiting information, even if approximate, about the optimal value of the problem. This information is available in some applications, among which the computation of bounds for hard integer programs. We show that dynamically changing the smoothness parameter of the algorithm using this information results in a better convergence profile of the algorithm in practice

Dynamic Smoothness Parameter for Fast Gradient Methods

We present and computationally evaluate a variant of the fast gradient method by Nesterov that is capable of exploiting information, even if approximate, about the optimal value of the problem. This information is available in some applications, among which the computation of bounds for hard integer programs. We show that dynamically changing the smoothness parameter of the algorithm using this information results in a better convergence profile of the algorithm in practice

On Exploiting Flow Allocation with Rate Adaptation for Green Networking

Network power consumption can be reduced considerably by adapting link data rates to their offered traffic loads. In this paper, we exploit how to leverage rate adaptation for green networking by studying the following flow allocation problem in wired networks: Given a set of candidate paths for each end-to-end communication session, determine how to allocate flow (data traffic) along these paths such that power consumption is minimized, subject to the constraint that the traffic demand of each session is satisfied. According to recent measurement studies, we consider a discrete step increasing function for link power consumption. We address both the single and multiple communication session cases and formulate them as two optimization problems, namely, the Single-session Flow allocation with Rate Adaptation Problem (SF-RAP), and the Multisession Flow Allocation with Rate Adaptation Problem (MFRAP). We first show that both problems are NP-hard and present a Mixed Integer Linear Programming (MILP) formulation for the MF-RAP to provide optimal solutions. Then we present a 2-approximation algorithm for the SF-RAP, and a general flow allocation framework as well as an LP-based heuristic algorithm for the MF-RAP. Simulation results show that the algorithm proposed for the SF-RAP consistently outperforms a shortest path based baseline solution and the algorithms proposed for the MF-RAP provide close-to-optimal solutions