15,831 research outputs found

    Relaxation methods for minimum cost ordinary and generalized network flow problems

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    Title from cover.Bibliography: p. 55-57.National Science Foundation Grant NSF-ECS-8217668.by Dimitri P. Bertsekas and Paul Tseng

    An E-relaxation method for separable convex cost generalized network flow problems

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    Cover title. "The extended abstract of this article appeared in the proceedings of the 5th International IPCO Conference, Vancouver, June 1996."--P. 1.Includes bibliographical references (p. 18-21).Supported by the National Science Foundation. CCR-9311621, DMI-9300494by Paul Tseng, Dimitri P. Bertsekas

    Auction algorithms for generalized nonlinear network flow problems

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    Thesis (Ph.D.)--Boston UniversityNetwork flow is an area of optimization theory concerned with optimization over networks with a range of applicability in fields such as computer networks, manufacturing, finance, scheduling and routing, telecommunications, and transportation. In both linear and nonlinear networks, a family of primal-dual algorithms based on "approximate" Complementary Slackness (ε-CS) is among the fastest in centralized and distributed environments. These include the auction algorithm for the linear assignment/transportation problems, ε-relaxation and Auction/Sequential Shortest Path (ASSP) for the min-cost flow and max-flow problems. Within this family, the auction algorithm is particularly fast, as it uses "second best" information, as compared to using the more generic ε-relaxation for linear assignment/transportation. Inspired by the success of auction algorithms, we extend them to two important classes of nonlinear network flow problems. We start with the nonlinear Resource Allocation Problem (RAP). This problem consists of optimally assigning N divisible resources to M competing missions/tasks each with its own utility function. This simple yet powerful framework has found applications in diverse fields such as finance, economics, logistics, sensor and wireless networks. RAP is an instance of generalized network (networks with arc gains) flow problem but it has significant special structure analogous to the assignment/transportation problem. We develop a class of auction algorithms for RAP: a finite-time auction algorithm for both synchronous and asynchronous environments followed by a combination of forward and reverse auction with ε-scaling to achieve pseudo polynomial complexity for any non-increasing generalized convex utilities including non-continuous and/ or non-differentiable functions. These techniques are then generalized to handle shipping costs on allocations. Lastly, we demonstrate how these techniques can be used for solving a dynamic RAP where nodes may appear or disappear over time. In later part of the thesis, we consider the convex nonlinear min-cost flow problem. Although E-relaxation and ASSP are among the fastest available techniques here, we illustrate how nonlinear costs, as opposed to linear, introduce a significant bottleneck on the progress that these algorithms make per iteration. We then extend the core idea of the auction algorithm, use of second best to make aggressive steps, to overcome this bottleneck and hence develop a faster version of ε-relaxation. This new algorithm shares the same theoretical complexity as the original but outperforms it in our numerical experiments based on random test problem suites

    Discrete Convex Functions on Graphs and Their Algorithmic Applications

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    The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

    The auction algorithm for assignment and other network flow problems

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    Cover title.Includes bibliographical references (p. 15-17).Research supported by the Army Research Office. DAAL 03-86-K-0171by Dimitri P. Bertsekas

    Relaxation methods for linear programs

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    Bibliography: p. 44-45.National Science Foundation grant NSF-ECS-3217668by Paul Tseng, Dimitri P. Bertsekas
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