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  • 51.
    Westerlund, Andreas
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Göthe-Lundgren, Maud
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    A stabilized column generation scheme for the traveling salesman subtour problem2006In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 154, no 15, p. 2212-2238Article in journal (Refereed)
    Abstract [en]

    Given an undirected graph with edge costs and both revenues and weights on the vertices, the traveling salesman subtour problem is to find a subtour that includes a depot vertex, satisfies a knapsack constraint on the vertex weights, and that minimizes edge costs minus vertex revenues along the subtour.

    We propose a decomposition scheme for this problem. It is inspired by the classic side-constrained 1-tree formulation of the traveling salesman problem, and uses stabilized column generation for the solution of the linear programming relaxation. Further, this decomposition procedure is combined with the addition of variable upper bound (VUB) constraints, which improves the linear programming bound. Furthermore, we present a heuristic procedure for finding feasible subtours from solutions to the column generation problems. An extensive experimental analysis of the behavior of the computational scheme is presented.

  • 52.
    Westerlund, Andreas
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
    Göthe-Lundgren, Maud
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
    Larsson, Torbjörn
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
    Mathematical Formulations of the Heterogeneous Fleet Vehicle Routing Problem2003In: ROUTE 2003, International Workshop on Vehicle Routing, Intermodal Transport and related areas,2003, 2003Conference paper (Other academic)
  • 53.
    Westerlund, Andreas
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Göthe-Lundgren, Maud
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Yuan, Di
    Linköping University, Department of Science and Technology. Linköping University, The Institute of Technology.
    Subgradient optimization prior to column generation: a means for predicting optimal columns (and non-binding restrictions)2005Report (Other academic)
    Abstract [en]

    Many classes of combinatorial and mixed integer optimization problems are attacked with decomposition methods. One technique is to perform subgradient optimization on a Lagrangean dual problem: another is to perform column generation on a Dantzig-Wolfe problem, or equivalently, cut generation on its linear programming dual. These techniques both have advantages and disadvantages. In this paper these techniques are combined into a two-phase method, with the purpose of overcoming drawbacks of the techniques.

    The two-phase method consists of a prediction phase and a solution phase. In the prediction phase, subgradient optimization is performed: subgradients found are stored and used as starting columns for the solution phase. (Optionally, non-binding restricitions can be predicted based on information from the prediction phase.) The columns found are used to construct a Dantzig/Wolfe master problem. In the solution phase, column generation is performed if needed. A solid theoretical bases for the two-phase method is presented which includes strong asymptotical results. ln practise, truncated usage must be performed: practical guidelines are given in the paper.

    The two-phase method is tested on two applications: a multicommodity network flow problem and a convexified version of the traveling salesman subtour problem. Two categories of numerical experiments are presented. ln the first category, various aspects of truncated usage of the theory are illustrated. In the second category, the two-phase method is tested on a relatively large number of test problems.

    An overall conclusion of our work is that the two-phase method can perform significantly better, in terms of CPU-times, compared to a (stabilized) Dantzig-Wolfe algorithm. ln general, whenever the subproblems are computationaly inexpensive, compared to the Dantzig-Wolfe master programs, the two-phase method might be an interesting alternative to use instead of pure Dantzig-Wolfe.

  • 54.
    Zhao, Hongmei
    et al.
    Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
    Lei, Lei
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, The Institute of Technology.
    Yuan, Di
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, The Institute of Technology.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
    Rönnberg, Elina
    Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
    Power efficient uplink scheduling in SC-FDMA: Bounding global optimality by column generation2013In: 2013 IEEE 18th International Workshop on Computer Aided Modeling and Design of Communication Links and Networks (CAMAD, IEEE , 2013, p. 119-123Conference paper (Refereed)
    Abstract [en]

    We study resource allocation in cellular systems and consider the problem of finding a power efficient scheduling in an uplink single carrier frequency division multiple access (SC-FDMA) system with localized allocation of subcarriers, that is, the subcarriers allocated to a user equipment have to be consecutive in the frequency domain in each time slot. This problem is discrete and nonconvex, thus the use of suboptimal algorithms has been a common practice. We leverage the power of mathematical programming in order to approach global optimality or a tight bounding interval confining global optimum, to arrive at an effective scheme for gauging the performance of suboptimal algorithms. Toward this end, we first provide a straightforward integer linear programming formulation, and then an alternative and less trivial, so-called column-oriented, formulation. The latter is solved by column generation, which is a solution technique for large-scale optimization problems with certain characteristics. The computational evaluation demonstrates that the column generation method produces very highquality subcarrier allocations that either coincide with the global optimum or enable an extremely sharp bounding interval. Hence the approach serves well for the purpose of benchmarking results for large-scale instances of power efficient SC-FDMA scheduling.

  • 55.
    Zhao, Yixin
    et al.
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Rönnberg, Elina
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    A Large Neighbourhood Search Principle for Column-Oriented Models: Theoretical Derivation and Example Applications2016In: Matheuristics 2016: Proceedings of the Sixth International Workshop on Model-based Metaheuristics / [ed] V. Maniezzo and T. Stutzle, Bruxelles, Belgium: IRIDIA , 2016Conference paper (Refereed)
  • 56.
    Zhao, Yixin
    et al.
    Nanjing University of Science and Technology, School of Automation.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Rönnberg, Elina
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    An integer programming column generation principlefor heuristic search methods2019In: International Transactions in Operational Research, ISSN 0969-6016, E-ISSN 1475-3995, Vol. 27, no 1, p. 665-695Article in journal (Refereed)
    Abstract [en]

    There is an increasing interest in integrating column generation and heuristic approaches to efficiently solve large-scale discrete optimisation problems. We contribute in this direction. Based on the insights from Lagrangian duality theory, we present an auxiliary problem that can be used for finding near-optimal solutions to a discrete column-oriented model. The structure of this auxiliary problem makes it suitable for being addressed with a heuristic search method involving column generation. To this end, we suggest a large neighbourhood search strategy where the repair step is to solve a column generation type subproblem. The suggested search strategy and mathematical models involved need to be tailored to the problem structure. To illustrate important design options and computational behaviour, four applications are studied: bin packing, generalised assignment, a resource allocation problem and the fixed-charge transportation problem.

  • 57.
    Zhao, Yixin
    et al.
    Nanjing University of Science and Technology, School of Automation.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Rönnberg, Elina
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Pardalos, Panos
    University of Florida, Department of Industrial and System Engineering.
    The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation2018In: Journal of Global Optimization, ISSN 0925-5001, E-ISSN 1573-2916, Vol. 72, no 3, p. 517-538Article in journal (Refereed)
    Abstract [en]

    A new and strong convexified formulation of the fixed charge transportation problem is provided. This formulation is obtained by integrating the concepts of Lagrangian decomposition and column generation. The decomposition is made by splitting the shipping variables into supply and demand side copies, while the columns correspond to extreme flow patterns for single sources or sinks. It is shown that the combination of Lagrangian decomposition and column generation provides a formulation whose strength dominates those of three other convexified formulations of the problem. Numerical results illustrate that our integrated approach has the ability to provide strong lower bounds. The Lagrangian decomposition yields a dual problem with an unbounded set of optimal solutions. We propose a regularized column generation scheme which prioritizes an optimal dual solution with a small 1-norm. We further demonstrate numerically that information gained from the strong formulation can also be used for constructing a small-sized core problem which yields high-quality upper bounds.

  • 58.
    Zhao, Yixin
    et al.
    Nanjing Univ Sci and Technol, Peoples R China.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Yuan, Di
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, Faculty of Science & Engineering.
    Rönnberg, Elina
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Lei, Lei
    Linköping University, Department of Science and Technology. Linköping University, Faculty of Science & Engineering.
    Correction: Power efficient uplink scheduling in SC-FDMA: benchmarking by column generation (vol 17, pg 695, 2016)2019In: Optimization and Engineering, ISSN 1389-4420, E-ISSN 1573-2924, Vol. 20, no 3, p. 959-959Article in journal (Other academic)
    Abstract [en]

    At the time of the final publication of the paper, in December 2016, Yixin Zhaos affiliation had changed.

  • 59.
    Zhao, Yixin
    et al.
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Larsson, Torbjörn
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Yuan, Di
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, Faculty of Science & Engineering.
    Rönnberg, Elina
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Lei, Lei
    Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, Faculty of Science & Engineering.
    Power efficient uplink scheduling in SC-FDMA: benchmarking by column generation2016In: Optimization and Engineering, ISSN 1389-4420, E-ISSN 1573-2924, Vol. 17, no 4, p. 695-725Article in journal (Refereed)
    Abstract [en]

    We study resource allocation in cellular systems and consider the problem of finding a power efficient scheduling in an uplink single carrier frequency division multiple access system. Due to the discrete nature of this problem and its computational difficulty, particularly in a real-time setting, the use of suboptimal algorithms is common practice. We aim at an effective way of gauging the performance of suboptimal algorithms by finding tight bounds on the global optimum. Toward this end, we first provide a basic integer linear programming formulation. Then we propose a significantly stronger column-oriented formulation and a corresponding column generation method, as well as an enhanced column generation scheme. The latter extends the first scheme through the inclusion of a stabilization technique, an approximate column generation principle, and a tailored heuristic that is embedded in the column generation scheme to find high-quality though not necessarily global optimal solutions. The computational evaluation demonstrates that compared with a poor performance by the integer linear programming formulation, the column generation method can produce near-optimal schedules that enable a sharp bounding interval. The enhanced column generation method significantly sharpens the bounding interval. Hence the column generation approach serves well for the purpose of benchmarking results for large-scale instances.

  • 60.
    Larsson, Torbjörn (Editor)
    Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
    Ranjbar, Zohreh (Editor)
    Linköping University, Department of Mathematics, Scientific Computing. Linköping University, Faculty of Science & Engineering.
    Skoglund, Ingegerd (Editor)
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Turesson, Bengt-Ove (Editor)
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Lürén, Oskar (Photographer)
    Fotograf Oskar, Linköping.
    Verksamhetsberättelse 2010: Matematiska institutionen2010Report (Other academic)
12 51 - 60 of 60
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