Using the primal-dual interior point algorithm within the branch-price-and-cut method

Munari, Pedro; Gondzio, Jacek

doi:10.1016/j.cor.2013.02.028

Cited by 33 publications

(32 citation statements)

References 51 publications

(93 reference statements)

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“…Among the further tasks to be done we find: improving the efficiency of the PCG by adaptive selection of φ, the number of terms in the preconditioner, using the estimation of ρ by the Ritz values; adaptive selection of either Newton or second-order directions, according to the quality of the preconditioner at each interior-point iteration; testing other (linear and nonlinear) classes of block-angular problems (e.g., routing problems in telecommunication networks, formulated as nonlinear multicommodity flows); using this approach within a more general framework for the solution of large mixed integer problems with block-angular structure (e.g., [28]); and implementing within BlockIP other type of preconditioners, such as, e.g., the hybrid approach described in [9]. Some of these tasks are already under development.…”

Section: Discussionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

“…The main difference between (28) and (29) is that µQ R tends to zero with µ and therefore (29) approximates (27) better than (28). Therefore, in general, B Q should be preferred to B P .…”

Section: Improving the Spectral Radiusmentioning

confidence: 99%

Interior-point solver for convex separable block-angular problems

Castro

2015

Optimization Methods and Software

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“…, q n y t n ] ∈ R n is the right-hand side vector for the linking constraints and contains the supply capacities of the designed locations. Note that the block constraints e x t j = d t j , j ∈ J , correspond to (19), whereas the linking constraints ∑ j∈J Lx t j + x t 0 = q t are (20). Formulation (34)-(36) exhibits a primal block-angular structure, and thus it can solved by the interior-point method of [5,8].…”

Section: Solving the Subproblem By A Specialized Interior-point Methodsmentioning

confidence: 99%

“…It is worth noting that interiorpoints methods have already been used in the past for the solution of integer optimization problem using cutting-plane approaches, such as in [19] for linear ordering problems. More recently, primal-dual interior-point methods have shown to be very efficient in the stabilization of column-generation procedures for the solution of vehicle routing with time windows, cutting stock, and capacitated lot sizing problems with setup times [11,20].…”

Section: Introductionmentioning

confidence: 99%

A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

2016

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We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The effect of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. This approach allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour, outperforming other stateof-the-art methods, which exhausted the (144 Gigabytes of) available memory in the largest instances.

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“…Publications [26] and [20] propose branch-and-bound algorithms for solving problem (1), and [13,7] explore valid inequalities in these techniques. In [22] a branch-price-and-cut strategy combined with decomposition techniques is proposed to provide valid inequalities and strong bounds to…”

Section: Introductionmentioning

confidence: 99%

Regularized optimization methods for convex MINLP problems

Oliveira

2016

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We propose regularized cutting-plane methods for solving mixed-integer nonlinear programming problems with nonsmooth convex objective and constraint functions. In order to avoid tailing-off effect that makes calculations unstable as the iteration process progresses, the given algorithms require solving a regularized mixed-integer linear programming subproblem at each iteration. The goal is to perform as few function evaluations as possible in the quest of solving convex mixed-integer nonlinear optimization problems. This is an important feature for some industrial applications resulting in hard-to-evaluate objective and/or constraint functions. Numerical experiments comparing the proposed algorithms with classical methods in this area show the effectiveness of our approach.

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Using the primal-dual interior point algorithm within the branch-price-and-cut method

Cited by 33 publications

References 51 publications

Interior-point solver for convex separable block-angular problems

Interior-point solver for convex separable block-angular problems

A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

Regularized optimization methods for convex MINLP problems

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