Optimization Theory of Large Systems

Tabak, Daniel

doi:10.1109/tsmc.1971.4308301

Cited by 228 publications

(331 citation statements)

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“…sx es − y e ≤ 0, e ∈ E, s ∈ S. (12) Next, we have the formula of the Lagrangian function (Lasdon, 1970) obtained for the ILP formulation of RSA-UAM-DPP-AvgSpec and assigned dual variables, Note that the first term of the formula corresponds to the objective function while the rest of terms determine the constraints of the dual problem to the LP problem described by the formulas (8)-(12).…”

Section: 21mentioning

confidence: 99%

A column generation technique for routing and spectrum allocation in cloud–ready survivable elastic optical networks

Goścień

Walkowiak

2017

International Journal of Applied Mathematics and Computer Science

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Driven by increasing user requirements and expectations, the fast development of telecommunications networks brings new challenging optimization problems. One of them is routing and spectrum allocation (RSA) of three types of network flows (unicast, anycast, multicast) in elastic optical networks (EONs) implementing dedicated path protection (DPP). In the paper, we model this problem as integer linear programming (ILP) and we introduce two new optimization approaches-a dedicated heuristic algorithm and a column generation (CG)-based method. Then, relying on extensive simulations, we compare algorithm performance with reference methods and evaluate CG efficiency in detail. The results show that the proposed CG method significantly outperforms reference algorithms and achieves results very close to optimal ones (the average distance to optimal results was at most 2.1%).

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Section: 21mentioning

confidence: 99%

A column generation technique for routing and spectrum allocation in cloud–ready survivable elastic optical networks

Goścień

Walkowiak

2017

International Journal of Applied Mathematics and Computer Science

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“…The following outline of column generation is based on [8] and [9,Chapter 3.3], in addition to the textbook [4]. A more detailed account is found in Appendix 2 The problem (1) can be formulated in di erent ways.…”

Section: Column Generationmentioning

confidence: 99%

“…It may not necessarily be the case that Benders' algorithm terminates when (26) presented in Section 2.3, is that the algorithm is guaranteed to converge nitely since there is only a nite number of extreme points to the feasible set of the dual of Benders subproblem [9].…”

Section: Benders' Algorithmmentioning

confidence: 99%

Modeling and Solving Vehicle Routing Problems with Many Available Vehicle Types

Barman

Lindroth

Strömberg

2015

Springer Proceedings in Mathematics &Amp; Statistics

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In this thesis, models have been formulated and mathematical optimization methods developed for the heterogeneous vehicle routing problem with a very large set of available vehicle types, called many-hVRP. This is an extension of the standard heterogeneous vehicle routing problem (hVRP), in which typically fairly small sets of vehicle types are considered.Two mathematical models based on standard models for the hVRP have been formulated for the many-hVRP. Column generation and dynamic programming have been applied to both these models, following a successful algorithm for the hVRP. Benders' decomposition algorithm has also been applied to one of the models. In addition to the standard cost structure, where the cost of a pair of a vehicle and a route is determined by the length of the route and the vehicle type, we have studied costs that depend also on the load of the vehicle along the route. These load dependent costs were easily incorporated into the models, and other extensions could be similarly incorporated.By using a standard set of test instances (with between three and six vehicle types in each instance) we have been able to compare our implementation with published results for hVRP. For many-hVRP, we have extended these instances to include larger sets of vehicle types (with between 91 and 381 vehicle types in each instance). The results show that the algorithms implemented for the two models nd optimal solutions in a similar amount of time, but Benders' algorithm at times takes much longer to verify optimality. However, some other properties of Benders' algorithm suggests that it may constitute a good basis for a heuristic, when instances with even larger sets of vehicle types are used.

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“…Consider the primal-dual solution pair (f (π), u(π); π, λ(π)). It follows from the saddle point conditions (see for example [4]) that the primal point (f (π), u(π)) is an optimal solution of (9), and the dual point (π, λ(π)) is an optimal solution of (10), so λ(π)ĥ = πu(π). Since u(π) ∈ U(ĥ), by assumption we have that u(π) ∈ U(h) and hence λ(π)h ≤ πu(π) = λ(π)ĥ.…”

Section: Necessary and Sufficient Condition For Ordinary Dominationmentioning

confidence: 99%

On Traffic Domination in Communication Networks

Ben‐Ameur

Pavón‐Mariño

Pióro

2011

Performance Evaluation of Computer and Communication Systems. Milestones and Future Challenges

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Abstract. Input data for communication network design/optimization problems involving multi-hour or uncertain traffic can consist of a large set of traffic matrices. These matrices are explicitly considered in problem formulations for link dimensioning. However, many of these matrices are usually dominated by others so only a relatively small subset of matrices would be sufficient to obtain proper link capacity reservations, supporting all original traffic matrices. Thus, elimination of the dominated matrices leads to substantially smaller optimization problems, making them treatable by contemporary solvers. In the paper we discuss the issues behind detecting domination of one traffic matrix over another. We consider two basic cases of domination: (i) total domination when the same traffic routing must be used for both matrices, and (ii) ordinary domination when traffic dependent routing can be used. The paper is based on our original results and generalizes the domination results known for fully connected networks.

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Optimization Theory of Large Systems

Cited by 228 publications

References 0 publications

A column generation technique for routing and spectrum allocation in cloud–ready survivable elastic optical networks

A column generation technique for routing and spectrum allocation in cloud–ready survivable elastic optical networks

Modeling and Solving Vehicle Routing Problems with Many Available Vehicle Types

On Traffic Domination in Communication Networks

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