Asymptotic analysis of the flow deviation method for the maximum concurrent flow problem

Bienstock, Daniel; Raskina, Olga

doi:10.1007/s101070100254

Cited by 40 publications

(30 citation statements)

References 21 publications

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“…Two primal block-angular problems have been considered: multicommodity network flows, and the minimum congestion (or maximum concurrent flow) problem [7]. They were solved with the specialized interior-point method for primal block-angular problems updated with a regularized function.…”

Section: Computational Results For Primal Block-angular Problemsmentioning

confidence: 99%

“…. , k. Let Θ be the symmetric diagonal matrix defined in (6), and B ∈ Rm ×m (m = ∑ k i=1 m i ), C ∈ Rm ×l and D ∈ R l×l the submatrices of AΘ A T defined in (7). Then, the spectral radius…”

Section: Outline Of the Interior-point Algorithm For Primal Block-angmentioning

confidence: 99%

“…In the literature, both problems are usually seen as one, and denoted as the maximum concurrent flow problem [7]. These problems arise in practical applications on telecommunications networks.…”

Section: Minimum Congestion Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Quadratic regularizations in an interior-point method for primal block-angular problems

Castro

Cuesta

2010

Math. Program.

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One of the most efficient interior-point methods for some classes of primal block-angular problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. Its efficiency depends on the spectral radius-in [0, 1)-of a certain matrix in the definition of the preconditioner. Spectral radius close to 1 degrade the performance of the approach. The purpose of this work is twofold. First, to show that a separable quadratic regularization term in the objective reduces the spectral radius, significantly improving the overall performance in some classes of instances. Second, to consider a regularization term which decreases with the barrier function, thus with no need for an extra parameter. Computational experience with some primal block-angular problems confirms the efficiency of the regularized approach. In particular, for some difficult problems, the solution time is reduced by a factor of two to ten by the regularization term, outperforming state-of-the-art commercial solvers.Keywords interior-point methods · primal block-angular problems · multicommodity network flows · preconditioned conjugate gradient · regularizations · large-scale computational optimization Mathematics Subject Classification (2000) 90C06 · 90C08 · 90C51

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Section: Computational Results For Primal Block-angular Problemsmentioning

confidence: 99%

Section: Outline Of the Interior-point Algorithm For Primal Block-angmentioning

confidence: 99%

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Quadratic regularizations in an interior-point method for primal block-angular problems

Castro

Cuesta

2010

Math. Program.

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“…The minimum congestion problem (also known as the maximum concurrent flow [7]) is defined on an infeasible nonoriented multicommodity flow problems. Its purpose is to minimize y ∞ , where y is the vector of relative increments in arc capacities needed to make the multicommodity flow problem feasible.…”

Section: Minimum Congestion Problemsmentioning

confidence: 99%

Interior-point solver for convex separable block-angular problems

Castro

2015

Optimization Methods and Software

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Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work we present an efficient solver based on this algorithm. Some of its features are: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor-corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the preconditioner). The solver has been hooked to SML, a structure-conveying modelling language based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using ℓ 1 and ℓ 2 distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to ℓ 1 which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300000 constraints, this approach is from two to three orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.

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“…The solution of the corresponding subproblem, in general a vertex of the polyhedron of the constraints, is used to perform a line search on the segment linking that optimal vertex and the current point. When applied to non linear flow problems, the LP subproblem yields a shortest path computation and the line search corresponds to the deviation of a uniform quantity of flow from the old active paths ti the new shortest path, simplifying dramatically the method in the form of the popular "Flow Deviation" method (see [8,10]). …”

Section: Proposition 2 the Function φ(X µ) Is Convex On S For Allmentioning

confidence: 99%

k‐Splittable delay constrained routing problem: A branch‐and‐price approach

2009

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Routing problems, which include a QoS-based path control, play a key role in broadband communication networks. We analyze here an algorithmic procedure based on branch-and-price algorithm and on the flow deviation method to solve a nonlinear k -splittable flow problem. The model can support end-to-end delay bounds on each path and we compare the behavior of the algorithm with and without these constraints. The trade-off between QoS guarantees and CPU time is clearly established and we show that minimizing the average delay on all arcs will yield solutions close to the optimal one at a significant computational saving.

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Asymptotic analysis of the flow deviation method for the maximum concurrent flow problem

Cited by 40 publications

References 21 publications

Quadratic regularizations in an interior-point method for primal block-angular problems

Quadratic regularizations in an interior-point method for primal block-angular problems

Interior-point solver for convex separable block-angular problems

k‐Splittable delay constrained routing problem: A branch‐and‐price approach

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