Solving Large-Scale Linear Multicommodity Flow Problems with an Active Set Strategy and Proximal-ACCPM

Babonneau, Frédéric Louis François; Merle, O. du; Vial, Jean–Philippe

doi:10.1287/opre.1050.0262

Cited by 35 publications

(48 citation statements)

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“…Regularizations based on proximal terms have also been applied in the proximal analytic center cutting plane method (proximal-ACCPM) for multicommodity flows [3]. However, unlike our approach, the proximal term did not improve the performance of proximal-ACCPM, but just simplified the tuning of parameters with respect to ACCPM.…”

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confidence: 98%

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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confidence: 98%

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

Castro

Cuesta

2010

Math. Program.

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“…By duality, any feasible solution of (13) provides a lower bound for the original problem (1). Taking the values returned by the two oracles at the successive query points, we obtain the lower bound θ = max …”

Section: Upper and Lower Boundsmentioning

confidence: 99%

“…This results in a reduction of the dimension of the Lagrangian dual space and easier calculation of analytic center. This strategy has been implemented with success in [1] and can be described as an active set strategy aiming to find and eliminate unsaturated arcs. In the nonlinear case, the strategy is applied to arcs on which the optimal flow lies in a region where the cost function is well approximated by a linear function.…”

Section: Introductionmentioning

confidence: 99%

“…In the nonlinear case, the strategy is applied to arcs on which the optimal flow lies in a region where the cost function is well approximated by a linear function. The idea of active set has already been used in the context of linear multicommodity flow problems [1,9,22], but not within an approximation scheme for the nonlinear component in NLMCF.…”

Section: Introductionmentioning

confidence: 99%

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ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems

Babonneau

Vial

2007

Math. Program.

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This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self-concordant augmented barrier. Our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component. The paper also proposes an approximation scheme for the original objective. An active set strategy can be applied to the transfomed problem: it reduces the dimension of the dual space and accelerates computations. The new approach solves huge instances with high accuracy. The method is compared to alternative approaches proposed in the literature.

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“…Solution strategies to deal with this class of problems can be broadly classified into simplex-based methods [14,23], decomposition methods [17,1,2,27], approximation methods [5], and interiorpoint methods [10,21]. One of the most efficient interior-point methods (IPMs) for some classes of block-angular problems solves normal equations by a combination of Cholesky factorizations for the block constraints and preconditioned conjugate gradient (PCG) iterations for the linking constraints [10,11].…”

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confidence: 99%

On Geometrical Properties of Preconditioners in IPMs for Classes of Block-Angular Problems

Castro¹,

Nasini²

2017

SIAM J. Optim.

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Abstract. One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. In this work we show that the choice of a good preconditioner depends on geometrical properties of the constraints structure. In particular, it is seen that the principal angles between the subspaces generated by the diagonal blocks and the linking constraints can be used to estimate ex-ante the efficiency of the preconditioner. Numerical validation is provided with some generated optimization problems. An application to the solution of multicommodity network flow problems with nodal capacities and equal flows of up to 64 million variables and up to 7.9 million constraints is also presented.Key words. interior-point methods, structured problems, preconditioned conjugate gradient, principal angles, large-scale optimization AMS subject classifications. 90C06, 90C08, 90C511. Introduction. Many real-world optimization problems exhibit a primal block-angular structure, where decision variables and constraints can be grouped in different blocks which are dependent due to some linking constraints Applications of this class of problems can be found, for instance, in multicommodity telecommunications networks, statistical data protection, multistage control and planning, and complex networks.Solution strategies to deal with this class of problems can be broadly classified into simplex-based methods [14,23], decomposition methods [17, 1, 2, 27], approximation methods [5], and interiorpoint methods [10,21]. One of the most efficient interior-point methods (IPMs) for some classes of block-angular problems solves normal equations by a combination of Cholesky factorizations for the block constraints and preconditioned conjugate gradient (PCG) iterations for the linking constraints [10,11]. The spectral radius of a certain matrix in the preconditioner, which is always in [0, 1) plays an important role in the efficiency of this approach. It was observed that for separable convex problems with nonzero Hessians this spectral radius is reduced, and the PCG becomes more efficient [13]. When the spectral radius approaches 1, switching to a suitable preconditioner may be an efficient alternative [7]. It is worth noting that computing approximate Newton directions by PCG does not destroy the good convergence properties of IPMs, as shown in [20]. There is an extensive literature on the use of PCG within IPMs for other types of problems (e.g., [3,4,9,18,25,28] to mention a few).However, it is not yet clear why for some classes of block-angular problems the above approach may be very efficient (see, for instance, the results of [13,12]), while it may need a large number of PCG iterations in others. The spectral radius may be used to monitor the good or bad behaviour, but an ex-ante explanation has to be found in the structural information of the block-angular constr...

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Solving Large-Scale Linear Multicommodity Flow Problems with an Active Set Strategy and Proximal-ACCPM

Cited by 35 publications

References 16 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

ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems

On Geometrical Properties of Preconditioners in IPMs for Classes of Block-Angular Problems

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