Comparison of bundle and classical column generation

Nowadays, large-scale failures in elastic optical inter-data center networks (EO-DCNs) are critical issues for network survivability. Once the original connection or the primary DC fails, an alternative data center (DC) can serve the cloud services with replicated content. In this paper, we study disaster-resilient cloud services provisioning with distance-adaptive modulation formats, considering content placement, routing with modulation format adaption, protection of path and content, and spectrum allocation simultaneously. Both dedicated end-to-content backup path protection (DEBPP) and shared end-to-content backup path protection (SEBPP) are studied. First, integer linear program (ILP) models are formulated with the objective of minimum spectrum usage. For the sake of scalability, a column generation (CG) approach is developed to decompose the ILP model into a master problem and a pricing problem. The CG approach is shown to achieve comparable results to ILP models. Numerical simulations are conducted to investigate the spectrum usage of DEBPP and SEBPP with different traffic parameters and different network topologies.

A column generation (CG) approach for the solution of timetabling problems is presented. This methodology could be used for various instances of the timetabling problem, although in this paper the solution of the high-school situation in Greece is presented. The results obtained show clearly that the CG approach that has been extremely successful in recent years in the solution of airline crew scheduling problems could also be very efficient and robust for the solution of timetabling problems. Several large timetabling problems corresponding to real problems have been successfully solved, with the solutions obtained feasible and of very high quality in accordance with the problem definition. In addition, none of the solutions contained any idle hour for any of the teachers, which was one of the main goals of this optimization effort.

We consider two approaches for solving the classical minimum vertex coloring problem—that is, the problem of coloring the vertices of a graph so that adjacent vertices have different colors and minimizing the number of used colors—namely, constraint programming and column generation. Constraint programming is able to solve very efficiently many of the benchmarks but suffers from a lack of effective bounding methods. On the contrary, column generation provides tight lower bounds by solving the fractional vertex coloring problem exploited in a branch-and-price algorithm, as already proposed in the literature. The column generation approach is here enhanced by using constraint programming to solve the pricing subproblem and to compute heuristic solutions. Moreover, new techniques are introduced to improve the performance of the column generation approach in solving both the linear relaxation and the integer problem. We report extensive computational results applied to the benchmark instances: we are able to prove optimality of 11 new instances and to improve the best-known lower bounds on 17 other instances. Moreover, we extend the solution approaches to a generalization of the problem known as the minimum vertex graph multicoloring problem, where a given number of colors has to be assigned to each vertex.

A column generation approach to capacitated p-median problems

The problem of scheduling and tree routing in WiMAX/802.16 based mesh networks were not defined in the standard and are thus subject to extensive research. In this paper, we consider the problem of joint routing and scheduling in 802.16-based wireless mesh network, with the objective of determining a minimum length schedule that satisfies a given (uplink/downlink) end-to-end traffic demand. Minimizing the schedule length amounts to maximizing the spectrum spatial reuse by concurrently transmitting on as many links as possible, which we refer to as a transmission configuration (a group of links that can simultaneously transmit without violating the signal-to-interferenceplus- noise ratio (SINR) requirement). Our model is referred to as maximum spatial reuse (MSR). Since there is an overwhelming number of possible transmission configurations to be assigned to time slots, we adopt the column generation technique to construct our MSR model. We present two formulations for modeling MSR, namely the link-based column generation (CGLink) formulation and the path-based column generation (CGPath) formulation. These two formulations differ mainly in the number of routing decision variables. Our experimental results indicate that the path-based formulation needs much less computational (CPU) time than the link-based formulation in order to determine the (same) optimized solution with the same spatial reuse gain.

One of the major tasks of telecommunications network planners is deciding where and how much spare capacity to build so that disrupted traffic may be rerouted in the event of a network failure. This paper presents an experimental study comparing two techniques for restoration capacity planning. The first, linear programming using column generation, produces optimal fractional solutions that can be integerized in practice for little extra cost. This approach is time-consuming, however, and for some networks of interest the linear programs are too large to be practically solvable. The second algorithm is a fast heuristic called LOCAL, which can be practically applied to much larger problem sizes than column generation. A fast linear-programming lower bound is used to measure the efficiency of the solutions. The purpose of the study was threefold: to determine how much benefit is obtained by using column generation, when the problem size is small enough that column generation is practical; to determine the quality of solutions produced by LOCAL on both small and large problems; and to investigate the utility of the lower-bound LP.We find that column generation produces networks whose restoration capacity cost is 10% to 16% less than those produced by LOCAL. Sometimes the total cost of the network is of primary interest, including both the cost of service and restoration capacity, and in this case the difference between column generation and LOCAL is 4% to 6%. Column generation is the method of choice for making final decisions about purchasing spare capacity. When millions are to be spent, running time is not a consideration, unless the computation simply will not complete in the available time. In such cases, the differential between column generation and LOCAL is small enough and consistent enough that LOCAL can safely be used. LOCAL is the method of choice for network design spreadsheet applications, where different architectures can be quickly compared, modified, and then compared again, and where a consistent approximation to the optimum is sufficient. The bound produced by the lower-bound LP is consistently extremely close to the true optimum, so the lower-bound LP can be effectively used to monitor the performance of both methods.

We present a new model and a column generation algorithm for the linear minimum cost multicommodity flow problem. The new model has two differences when compared to the classic formulation based on flows on paths: it has an additional set of variables and an additional set of inequalities. The extra variables are associated with flows on circuits where some arcs are reversed. The extra inequalities assure that an optimal solution to the extended model can be converted to an optimal solution to the path formulation. In order to obtain an optimal solution to the extended model, we use column generation. The extra variables are explicitly considered in the restricted master problem, from the beginning of the column generation process. The subproblem remains a set of shortest path problems. Our present work is related to the use of dual–optimal inequalities in the sense that we extend a model with additional primal variables/dual inequalities. However, our extended model also has additional primal inequalities/dual variables. We present computational results of our method in planar instances and compare them with standard column generation, a bundle method, and a general‐purpose solver. For the tested instances, there is an effective improvement in computational time of the column generation method when the extended model is used. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 3–16 2007

In intensity-modulated radiation therapy (IMRT), a network flow is adopted to solve the pricing problem of the generic column generation approach in order to obtain a deliverable aperture. However, excessive computation results from the direct use of a network flow. In addition, a decline in plan quality may result from the direct determination of the leaf position using the gradient information. To overcome these problems, a column generation approach based on region growth is proposed. The proposed method is designed to reduce the computational cost of solving the pricing problem and improve the IMRT plan quality. First, the gradients of the beamlets are obtained by an objective function constructed under the constraint conditions of the organs. Second, the gradients are transformed nonlinearly. Third, the positions of the continuous negative gradient regions in each row of the aperture are determined and stored. Fourth, these gradients are taken as a whole and added to the aperture network flow, which is solved as a shortest-path problem. Finally, the deliverable aperture is obtained and added to the treatment plan. To verify the effectiveness of the proposed method, experiments involving five five-field prostate cancer cases and five nine-field head and neck cancer cases were conducted. Compared with the generic column generation method, the dose distribution of the target is ensured by the proposed method, which also effectively protects organs at risk and reduces the running time. Specifically, in ten groups of comparative experiments, the normal tissue complication probability of the proposed method is reduced by up to 3.37%, and the maximum acceleration rate is 20.44%. According to the experimental results, the proposed method is more consistent with clinical requirements compared with the generic column generation method.

We present a complementary column generation feature that produces tight upper bounds, thereby enhancing heuristic and exact column generation approaches for (minimization) set partitioning formulations that possess dense column structures. We also introduce a duality-based lower bound that prompts a useful termination criterion, which can be utilized to mitigate the tailing-off effect induced by column generation approaches. Computations are presented for the one-dimensional bin packing problem and a joint vehicle assembly-routing problem.

A column generation approach for the unconstrained binary quadratic programming problem

To develop a novel aperture-based algorithm for volumetric modulated are therapy (VMAT) treatment plan optimization with high quality and high efficiency. The VMAT optimization problem is formulated as a large-scale convex programming problem solved by a column generation approach. The authors consider a cost function consisting two terms, the first enforcing a desired dose distribution and the second guaranteeing a smooth dose rate variation between successive gantry angles. A gantry rotation is discretized into 180 beam angles and for each beam angle, only one MLC aperture is allowed. The apertures are generated one by one in a sequential way. At each iteration of the column generation method, a deliverable MLC aperture is generated for one of the unoccupied beam angles by solving a subproblem with the consideration of MLC mechanic constraints. A subsequent master problem is then solved to determine the dose rate at all currently generated apertures by minimizing the cost function. When all 180 beam angles are occupied, the optimization completes, yielding a set of deliverable apertures and associated dose rates that produce a high quality plan. The algorithm was preliminarily tested on five prostate and five head-and-neck clinical cases, each with one full gantry rotation without any couch/collimator rotations. High quality VMAT plans have been generated for all ten cases with extremely high efficiency. It takes only 5-8 min on CPU (MATLAB code on an Intel Xeon 2.27 GHz CPU) and 18-31 s on GPU (CUDA code on an NVIDIA Tesla C1060 GPU card) to generate such plans. The authors have developed an aperture-based VMAT optimization algorithm which can generate clinically deliverable high quality treatment plans at very high efficiency.

Column generation is a fundamental technique for solving large-scale combinatorial optimization problems such as unit commitment and vehicle routing, yet its performance is often limited by dual oscillation. This study explores the intrinsic cause of this phenomenon from the perspective of shadow price theory and demonstrates that dual oscillation arises from the lack of marginal interpretability of Lagrange multipliers when multiple dual solutions coexist. To address this issue, an improved column generation framework is proposed in which traditional multipliers are replaced with minimum-norm multipliers that possess clear economic meaning and act as directional shadow prices. A generalized pricing subproblem is formulated, and partial minimum-norm multipliers are obtained through convex quadratic optimization to guide column generation. Numerical experiments on a simplified single-period unit commitment case and large-scale cutting stock problems showed that the proposed approach eliminated invalid column generation and achieved speedy convergence to the optimal solution within only two iterations for the unit commitment case, and the classical column generation exhibited slow convergence with dual oscillation in large-scale scenarios while the improved algorithm achieved fast and stable convergence. The results indicate that the stabilization method enhances the consistency of dual variables and provides a more robust foundation for the theoretical and practical development of column generation algorithms.

A recurrent task in mathematical programming consists of optimizing polytopes with prohibitively many constraints, e.g., the primal polytope in cutting-planes methods or the dual polytope in column generation (CG). We present a method to optimize such polytopes using the following intersection sub-problem: given ray $${\mathbf r}\in \mathbb {Z}^n$$r?Zn, find the maximum $${t^*}\ge 0$$t??0 such that $${t^*}{\mathbf r}$$t?r is feasible. We interpret $${\mathbf 0_n}\rightarrow {\mathbf r}$$0n?r as a direction of growth from the origin $${\mathbf 0_n}$$0n; $${t^*}{\mathbf r}$$t?r is the intersection point between $${\mathbf 0_n}\rightarrow {\mathbf r}$$0n?r and the polytope boundary. To ease the exposition, the first part of the paper is devoted to the general context of a (primal or dual) polytope with rational data such that: (i) the intersection sub-problem can be effectively solved in practice for input rays $${\mathbf r}\in \mathbb {Z}^n$$r?Zn and (ii) $${\mathbf 0_n}$$0n is feasible. By iterating over such rays $${\mathbf r}$$r, our method produces a sequence of feasible solutions $${t^*}{\mathbf r}$$t?r that we prove to be finitely convergent. From Section 3 on, we focus on dual polytopes in CG formulations. Typically, if the CG (separation) sub-problem can be solved by dynamic programming (DP), so can be the intersection sub-problem. The proposed integer ray method (IRM) only uses integer rays, and so, the intersection sub-problem can be solved using a DP scheme based on states indexed by integer ray values. We show that under such conditions, the intersection sub-problem can be even easier than the CG sub-problem, especially when no other integer data is available to index states in DP, i.e., if the CG sub-problem input consists of fractional (or large-range) values. As such, the IRM can tackle scaled instances (with large-range weights) of capacitated problems that seem prohibitively hard for classical CG. This is confirmed by numerical experiments on various capacitated Set-Covering problems: Capacitated Arc-Routing, Cutting-Stock and other three versions of Elastic Cutting-Stock (i.e., a problem class that includes Variable Size Bin Packing).

We apply column generation with branch‐and‐price optimization to a multi‐target, multi‐task assignment problem, with precedence constraints. Column generation transforms the nonlinear program with separable costs and constraints into a linear program. This reformulation divides the original problem into a number of smaller problems, where one of these smaller problems accounts for the coupling constraints between agents and must be known by every agent. All other divisions consider local constraints affecting only one agent; these smaller problems are known by only one corresponding agent. Because of this reformulation, the assignment problem can be solved in a distributed manner. A theorem is proven which details the central analytical result of the paper, allowing a nonlinear program to be reformulated as a linear program. Simulation results for a multi‐target, single‐task assignment problem, as well as a multi‐target, multi‐task assignment problem with precedence constraints are presented. Published 2011. This article is a US Government work and is in the public domain in the USA.

This paper extends product mix/linear programming based problem with an infinite number of product types for decisions. The corresponding model becomes a class of semi-infinite linear programming model. With the problem assumptions, the optimal solution can still be theoretically solved using the Simplex based method. To handle the infinite number of decision variables, a set of finite number of variables can be initially generated and then solved and improve sequentially using the classical column generation approach. The proposed procedure was programmed as a MATLAB code and was tested. The computational results are reported.