Bilinear modeling solution approach for fixed charge network flow problems

Rebennack, Steffen; Nahapetyan, Artyom G.; Pardalos, Pãnos M.

doi:10.1007/s11590-009-0114-0

Cited by 25 publications

(10 citation statements)

References 27 publications

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“…The problem is transformed into a continuous one with a bilinear cost function, through the use of a nonlinear relaxation technique and it is then solved with a dynamic slope scaling method, based on the one proposed by (Kim and Pardalos, 1999). Rebennack et al (2009) propose a continuous bilinear formulation for the fixed-charge network flow problem from which an exact algorithm is derived, based on these two previous works. Ortega and Wolsey (2003) also provide an exact algorithm using a branch-and-cut method by extending the cutting planes previously used to solve uncapacitated lot sizing problems.…”

Section: Literature Review On Mcnfpsmentioning

confidence: 99%

Concave minimum cost network flow problems solved with a colony of ants

Monteiro

Fontes

2012

J Heuristics

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In this work we address the Single-Source Uncapacitated Minimum Cost Network Flow Problem with concave cost functions. This problem is NP-hard, therefore we propose a hybrid heuristic to solve it. Our goal is not only to apply an Ant Colony Optimization (ACO) algorithm to such a problem, but also to provide an insight on the behaviour of the parameters in the performance of the algorithm. The performance of the ACO algorithm is improved with the hybridization of a local search procedure. The core ACO procedure is used to mainly deal with the exploration of the search space, while the Local Search is incorporated to further cope with the exploitation of the best solutions found. The method we have developed has proven to be very efficient while solving both small and large size problem instances. The problems we have used to test the algorithm were previously solved by other authors using other population based heuristics. Our algorithm was able to improve upon some of their results in terms of solution quality, proving that the HACO algorithm is a very good alternative approach to solve these problems. In addition, our algorithm is substantially faster at achieving these improved solutions. Furthermore, the magnitude of the reduction of the computational requirements grows with problem size.

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Section: Literature Review On Mcnfpsmentioning

confidence: 99%

Concave minimum cost network flow problems solved with a colony of ants

Monteiro

Fontes

2012

J Heuristics

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“…MIPs without continuous variables are integer programs (IPs). Sometimes, the problems can be equivalently formulated in several classes, e.g., Rebennack et al [42] derive an exact algorithm from a continuous, bilinear formulation of the fixed charge network flow problem. The authors reformulate this classical MILP problem with a continuous QP programming formulation.…”

Section: Lp Linear Programmingmentioning

confidence: 99%

Polylithic modeling and solution approaches using algebraic modeling systems

Kallrath

2011

Optim Lett

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Based on the Greek term monolithos (stone consisting of one single block) Kallrath (Comput Chem Eng 33:1983-1993, 2009) introduced the term polylithic for modeling and solution approaches in which mixed integer or non-convex nonlinear optimization problems are solved by tailor-made methods involving several models and/or algorithmic components, in which the solution of one model is input to another one. This can be exploited to initialize certain variables, or to provide bounds on them (problem-specific preprocessing). Mathematical examples of polylithic approaches are decomposition techniques, or hybrid methods in which constructive heuristics and local search improvement methods are coupled with exact MIP algorithms. Tailormade polylithic solution approaches with thousands or millions of solve statements are challenges on algebraic modeling languages. Local objects and procedural structures are almost necessary. Warm-start and hot-start techniques can be essential. The effort of developing complex tailor-made polylithic solutions is awarded by enabling us to solve real-world problems far beyond the limits of monolithic approaches and general purpose solvers.

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“…Many important applications can be formulated as a disjoint bilinear program including fixed charge network flows (Rebennack et al [27]), concave cost facility location (Soland [29]), bilinear assignment problems ( Ćustić et al [11]), non-convex cutting-stock problems (Harjunkoski et al [23]), multicommodity flow network interdiction problems ( [25]), game theory (Mangasarian and Stone [26], Firouzbakht et al [17]) pooling problems (Gupte et al [22]).…”

Section: Introductionmentioning

confidence: 99%

LP-based Approximations for Disjoint Bilinear and Two-Stage Adjustable Robust Optimization

Housni¹,

Foussoul²,

Goyal³

2021

Preprint

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We consider the class of disjoint bilinear programs max {x T y | x ∈ X , y ∈ Y} where X and Y are packing polytopes. We present an O( log log m1 log m1 log log m2 log m2 )-approximation algorithm for this problem where m 1 and m 2 are the number of packing constraints in X and Y respectively. In particular, we show that there exists a near-optimal solution (x, y) such that x and y are "nearintegral". We give an LP relaxation of this problem from which we obtain the near-optimal near-integral solution via randomized rounding. As an application of our techniques, we present a tight approximation for the two-stage adjustable robust problem with covering constraints and right-hand side uncertainty where the separation problem is a bilinear optimization problem. In particular, based on the ideas above, we give an LP restriction of the two-stage problem that is an O( log n log log n log L log log L )-approximation where L is the number of constraints in the uncertainty set. This significantly improves over state-of-the-art approximation bounds known for this problem.

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Bilinear modeling solution approach for fixed charge network flow problems

Cited by 25 publications

References 27 publications

Concave minimum cost network flow problems solved with a colony of ants

Concave minimum cost network flow problems solved with a colony of ants

Polylithic modeling and solution approaches using algebraic modeling systems

LP-based Approximations for Disjoint Bilinear and Two-Stage Adjustable Robust Optimization

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