An extended model and a column generation algorithm for the planar multicommodity flow problem

This paper proposes a column generation approach for the Point-Feature Cartographic Label Placement problem (PFCLP). The column generation is based on a Lagrangean relaxation with clusters proposed for problems modeled by conflict graphs. The PFCLP can be represented by a conflict graph where vertices are positions for each label and edges are potential overlaps between labels (vertices). The conflict graph is decomposed into clusters forming a block diagonal matrix with coupling constraints that is known as a restricted master problem (RMP) in a Dantzig-Wolfe decomposition context. The clusters’ sub-problems are similar to the PFCLP and are used to generate new improved columns to RMP. This approach was tested on PFCLP instances presented in the literature providing in reasonable times better solutions than all those known and determining optimal solutions for some difficult large-scale instances.

A column generation approach to capacitated p-median problems

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO).

When 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 comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

Many transportation, communications, and logistics problems can be formulated as large multi-commodity flow problems, which are often solved using the traditional column generation method. During the application of the column generation, one needs to find initial solutions, so-called feasible columns in a first step. In order to determine these feasible columns and also solve dependent price problems within an acceptable time, this paper proposes several novel methods: Two methods by solving the global problem and two methods by finding the shortest and widest paths are proposed to compute the initial columns. Furthermore, we propose two methods by finding the shortest paths with commodities and sources (SPC and SPS) for solving price problems. Several well-known network instances are used as case studies in our evaluation to test the efficiency of each method. It is shown that the widest-path method is the best for finding initial columns. For solving the price problems, the preference of the two path-finding methods depend on the network structure.

A primal partitioning approach for single and non-simultaneous multicommodity flow problems

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm.

The wireless relay mesh network (WRMN) is designed to provide robust and fault tolerant communications between relay and user nodes in broadband wireless networks. In this paper, we investigate the benefits of decoupled optimization of routing and scheduling in WRMNs using the physical interference model and spatial reuse to maximize overall throughput. We model the routing optimization as a linear program using multicommodity flows (MCF). We refer to this problem as multicommodity flow routing optimization (MCF-ROPT). Using the flow per link determined from MCF-ROPT, we develop an optimization formulation to schedule the link traffic such that interference is minimized and time slots are reused appropriately based on spatial TDMA (STDMA). Furthermore, our scheduling approach incorporates the effect of reuse of multiple carriers on the transmission schedule. We refer to this problem as SM-TSS (STDMA multicarrier traffic sensitive scheduling). The SM-TSS is NP-hard and thus is solved using column generation. We compare our formulations with decoupled optimizations that use the protocol interference model and/or single carrier systems and show that our approach guarantees higher throughput by mitigating interference effectively.

One of the best approaches for some classes of multicommodity flow problems is a specialized interior‐point method that solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient. Its efficiency depends on the spectral radius—in [0,1)—of a certain matrix in the definition of the preconditioner. In a recent work, the authors improved this algorithm (i.e., reduced the spectral radius) for general block‐angular problems by adding a quadratic regularization to the logarithmic barrier. This barrier was shown to be self‐concordant, which guarantees the convergence and polynomial complexity of the algorithm. In this work, we focus on linear multicommodity problems, a particular case of primal block‐angular ones. General results are tailored for multicommodity flows, allowing a local sensitivity analysis on the effect of the regularization. Extensive computational results on some standard and some difficult instances, testing several regularization strategies, are also provided. These results show that the regularized interior‐point algorithm is more efficient than the nonregularized one. From this work it can be concluded that, if interior‐point methods based on conjugate gradients are used, linear multicommodity flow problems are most efficiently solved as a sequence of quadratic ones. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Optimization of demand-oriented train timetables under overtaking operations: A surrogate-dual-variable column generation for eliminating indivisibility

Branch-and-price-and-cut for large-scale multicommodity capacitated fixed-charge network design

The minimum sum-of-squares clustering problem (MSSC), also known as k-means clustering, refers to the problem of partitioning n data points into k clusters, with the objective of minimizing the total sum of squared Euclidean distances between each point and the center of its assigned cluster. We propose an efficient algorithm for solving large-scale MSSC instances, which combines column generation (CG) with dynamic constraint aggregation (DCA) to effectively reduce the number of constraints considered in the CG master problem. DCA was originally conceived to reduce degeneracy in set partitioning problems by utilizing an aggregated restricted master problem obtained from a partition of the set partitioning constraints into disjoint clusters. In this work, we explore the use of DCA within a CG algorithm for MSSC exact solution. Our method is fine-tuned by a series of ablation studies on DCA design choices, and is demonstrated to significantly outperform existing state-of-the-art exact approaches available in the literature. History: Accepted by Andrea Lodi, Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada [Grant 2023-04466]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0938 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0938 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Models and hybrid methods for the onshore wells maintenance problem

This article is concerned with robust stability of uncertain discrete-time linear systems. The matrix defining the linear system (system matrix) is assumed to depend affinely on a set of time-invariant unknown parameters lying on a known polytope. Robust stability is investigated by checking whether a certain integer power κ of the uncertain system matrix has spectral norm less than one. This peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by κ. A unique feature is that no extra variables need to be added to the problems being solved. Numerical experiments reveal that the value of κ needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle. By identifying the proposed stability test with a particular choice of a parameter-dependent Lyapunov function, extra variables can be introduced, yielding linear matrix inequalities optimisation problems of improved convergence.

The objective of the maximum weighted set of disjoint submatrices problem is to discover K disjoint submatrices that together cover the largest sum of entries of an input matrix. It has many practical data-mining applications, as the related biclustering problem, such as gene module discovery in bioinformatics. It differs from the maximum-weighted submatrix coverage problem introduced in [6] by the explicit formulation of disjunction constraints: submatrices must not overlap. In other words, all matrix entries must be covered by at most one submatrix. The particular case of \(K=1\), called the maximal-sum submatrix problem, was successfully tackled with constraint programming in [5]. Unfortunately, the case of \(K > 1\) is more challenging to solve as the selection of rows cannot be decided in polynomial time solely from the selection of K sets of columns. It can be proved to be \(\mathcal {NP}\)-hard. We introduce a hybrid column generation approach using constraint programming to generate columns. It is compared to a standard mixed integer linear programming (MILP) through experiments on synthetic datasets. Overall, fast and valuable solutions are found by column generation while the MILP approach cannot handle a large number of variables and constraints.