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Simultaneous columnandrow generation for largescale linear programs with columndependentrows. http://www.optimizationonline.org/DB_FILE/2010/11/2815.pdf
, 2010
"... Abstract: In this paper, we develop a simultaneous columnandrow generation algorithm that could be applied to a general class of largescale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining ..."
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Abstract: In this paper, we develop a simultaneous columnandrow generation algorithm that could be applied to a general class of largescale 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 columndependentrows. To solve these problems, we need to be able to generate both columns and rows onthefly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branchandcutandprice framework. We first characterize the underlying assumptions for the proposed columnandrow generation algorithm. These assumptions aregeneral enough and cover all problems with columndependentrows 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 columnandrow generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate our approach, the paper is concluded by applying the proposed framework to the multistage cutting stock and the quadratic set covering problems.
Automatic Decomposition and BranchandPrice  A Status Report
"... We provide an overview of our recent efforts to automatize DantzigWolfe reformulation and column generation/branchandprice for structured, largescale integer programs. We present the need for and the benefits from a generic implementation which does not need any user input or expert knowledge. A ..."
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We provide an overview of our recent efforts to automatize DantzigWolfe reformulation and column generation/branchandprice for structured, largescale integer programs. We present the need for and the benefits from a generic implementation which does not need any user input or expert knowledge. A focus is on detecting structures in integer programs which are amenable to a DantzigWolfe reformulation. We give computational results and discuss future research topics.
Separation of Generic Cutting Planes in BranchandPrice Using a Basis
, 2015
"... DantzigWolfe reformulation of a mixed integer program partially convexifies a subset of the constraints, i.e., it implicitly adds all valid inequalities for the associated integer hull. Projecting an optimal basic solution of the reformulation’s LP relaxation to the original space does is in gener ..."
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DantzigWolfe reformulation of a mixed integer program partially convexifies a subset of the constraints, i.e., it implicitly adds all valid inequalities for the associated integer hull. Projecting an optimal basic solution of the reformulation’s LP relaxation to the original space does is in general not yield a basic solution of the original LP relaxation. Cutting planes in the original problem that are separated using a basis like Gomory mixed integer cuts are therefore not directly applicable. Range [22] (and others) proposed as a remedy to heuristically compute a basic solution and separate this auxiliary solution also with cutting planes that stem from a basis. This might not only cut off the auxiliary solution, but also the solution we originally wanted to separate. We discuss and extend Range’s ideas to enhance the separation procedure. In particular, we present alternative heuristics and consider additional valid inequalities strengthening the original LP relaxation before separation. Our full implementation, which is the first of its kind, is done within the GCG framework. We evaluated the effects on several problem classes. Our experiments show that the separated cuts strengthen the formulation on instances where the integrality gap is not too small. This leads to a reduced number of nodes and reduced solution times.
A BranchPriceandCut Algorithm for Packing Cuts in Undirected Graphs
"... Abstract. The cut packing problem in an undirected graph is to find a largest cardinality collection of pairwise edgedisjoint cuts.We provide the first experimental study of this NPhard problem that interested theorists and practitioners alike. We propose a branchpriceandcut algorithm to optima ..."
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Abstract. The cut packing problem in an undirected graph is to find a largest cardinality collection of pairwise edgedisjoint cuts.We provide the first experimental study of this NPhard problem that interested theorists and practitioners alike. We propose a branchpriceandcut algorithm to optimally solve instances from various graph classes, random and from the literature, with up to several hundred vertices. In particular we investigate how complexity results match computational experience and how combinatorial properties help improving the algorithm’s performance. 1
On Stabilized BranchandPrice for Constrained Tree Problems
, 2011
"... We consider a rather generic class of network design problems in which a set or subset of given terminal nodes must be connected to a dedicated root node by simple paths and a variety of resource and/or quality of service constraints must be respected. These extensions of the classical Steiner tree ..."
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We consider a rather generic class of network design problems in which a set or subset of given terminal nodes must be connected to a dedicated root node by simple paths and a variety of resource and/or quality of service constraints must be respected. These extensions of the classical Steiner tree problem on a graph can be well modeled by a path formulation in which individual variables are used for all feasible paths. To solve this formulation in practice, branchandprice is used. It turns out, however, that a naive implementation of column generation suffers strongly from certain degeneracies of the pricing subproblem, leading to excessive running times. After analyzing these computational problems, we propose two methods for stabilizing column generation by using alternative dualoptimal solutions. This stabilized branchandprice is practically tested on the rooted delayconstrained Steiner tree problem and a quotaconstrained version of it. Results indicate that the new stabilization methods in general speed up the solution process dramatically, far more than a piecewise linear stabilization to which we compare. Furthermore, our stabilized branchandprice exhibits on most test instances a better performance than a so far leading mixed integer programming approach based on a layered graph model and branchandcut. As the new stabilization technique utilizing alternative dualoptimal solutions is generic in the sense that it easily adapts to the inclusion of a large variety of further constraints and different objective functions, the proposed method is highly promising for a large class of network design problems.
www.ac.tuwien.ac.at/tr On Optimal Design of Charging Stations for Electric Vehicles
, 2015
"... In this article we consider the Network Design Problem with Relays (NDPR) which gives answers to some important strategic design questions in context of emobility. Given a family of origindestination pairs electric vehicles need to travel, and given the existing links that can be traversed, these ..."
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In this article we consider the Network Design Problem with Relays (NDPR) which gives answers to some important strategic design questions in context of emobility. Given a family of origindestination pairs electric vehicles need to travel, and given the existing links that can be traversed, these questions are: (1) What are the optimal locations for placing the charging stations and how many of them are needed? (2) Could the available infrastructure be enhanced by including additional links (shortcuts), to reduce the travel distances? In contrast to previous work on the NDPR which mainly focused on heuristic approaches, we discuss exact methods based on different mixed integer linear programming formulations for the problem. We develop BranchandPrice and BranchPriceandCut algorithms that build upon models with an exponential number of variables (and constraints). In an extensive computational study, we analyze the performance of these approaches for instances that reflect different realworld settings.