On solving large-scale multistage stochastic problems with a new specialized interior-point approach

A novel approach based on a specialized interior-point method (IPM) is presented for solving large-scale stochastic multistage continuous optimization problems, which represent the uncertainty in strategic multistage and operational two-stage scenario trees, the latter being rooted at the strategic nodes. This new solution approach considers a split-variable formulation of the strategic and operational structures, for which copies are made of the strategic nodes and the structures are rooted in the form of nested strategic-operational two-stage trees. The specialized IPM solves the normal equations of the problem’s Newton system by combining Cholesky factorizations with preconditioned conjugate gradients, doing so for, respectively, the constraints of the stochastic formulation and those that equate the split-variables. We show that, for multistage stochastic problems, the preconditioner (i) is a block-diagonal matrix composed of as many shifted tridiagonal matrices as the number of nested strategicoperational two-stage trees, thus allowing the efficient solution of systems of equations; (ii) its complexity in a multistage stochastic problem is equivalent to that of a very large-scale two-stage problem. A broad computational experience is reported for large multistage stochastic supply network design (SND) and revenue management (RM) problems; the mathematical structures vary greatly for those two application types. Some of the most difficult instances of SND had 5 stages, 839 million variables, 13 million quadratic variables, 21 million constraints, and 3750 scenario tree nodes; while those of RM had 8 stages, 278 million variables, 100 million constraints, and 100,000 scenario tree nodes. For those problems, the proposed approach obtained the solution in 2.3 days using 167 gigabytes of memory for SND, and in 1.7 days using 83 gigabytes for RM; while the state-of-the-art solver CPLEX v20.1 required more than 24 days and 526 gigabytes for SND, and more than 19 days and 410 gigabytes for RMPeer ReviewedPreprin

On solving large-scale multistage stochastic optimization problems with a new specialized interior-point approach

© 2023 The Authors. Published by Elsevier B.VA novel approach based on a specialized interior-point method (IPM) is presented for solving largescale stochastic multistage continuous optimization problems, which represent the uncertainty in strategic multistage and operational two-stage scenario trees. This new solution approach considers a splitvariable formulation of the strategic and operational structures. The specialized IPM solves the normal equations by combining Cholesky factorizations with preconditioned conjugate gradients, doing so for, respectively, the constraints of the stochastic formulation and those that equate the split-variables. We show that, for multistage stochastic problems, the preconditioner (i) is a block-diagonal matrix composed of as many shifted tridiagonal matrices as the number of nested strategic-operational two-stage trees, thus allowing the efficient solution of systems of equations; (ii) its complexity in a multistage stochastic problem is equivalent to that of a very large-scale two-stage problem. A broad computational experience is reported for large multistage stochastic supply network design (SND) and revenue management (RM) problems. Some of the most difficult instances of SND had 5 stages, 839 million linear variables, 13 million quadratic variables, 21 million constraints, and 3750 scenario tree nodes; while those of RM had 8 stages, 278 million linear variables, 100 million constraints, and 100,000 scenario tree nodes. For those problems, the proposed approach obtained the solution in 1.1 days using 174 gigabytes of memory for SND, and in 1.7 days using 83 gigabytes for RM; while CPLEX v20.1 required more than 53 days and 531 gigabytes for SND, and more than 19 days and 410 gigabytes for RM.J. Castro was supported by the MCIN/AEI/FEDER grant RTI2018-097580-B-I00. L.E. Escudero was supported by the MCIN/AEI/10.13039/501100011033 grant PID2021-122640OB-I00. J.F. Monge was supported by the MCIN/AEI/10.13039/501100011033/ERDF grants PID2019-105952GB-I00 and PID2021-122344NB-I00, and by PROMETEO/2021/063 grant funded by the government of the Valencia Community, Spain.Peer ReviewedPostprint (published version

An exact approach for the reliable fixed-charge location problem with capacity constraints

Introducing capacities in the reliable fixed charge location problem is a complex task since successive failures might yield in high facility overloads. Ideally, the goal consists in minimizing the total cost while keeping the expected facility overloads under a given threshold. Several heuristic approaches have been proposed in the literature for dealing with this goal. In this paper, we present the first exact approach for this problem, which is based on a cutting planes algorithm. Computational results illustrate its good performancePostprint (published version

Introducing capacitaties in the location of unreliable facilities

The goal of this paper is to introduce facility capacities into the Reliability Fixed-Charge Location Problem in a sensible way. To this end, we develop and compare different models, which represent a tradeoff between the extreme models currently available in the literature, where a priori assignments are either fixed, or can be fully modified after failures occur. In a series of computational experiments we analyze the obtained solutions and study the price of introducing capacity constraints according to the alternative models both, in terms of computational burden and of solution cost.Peer ReviewedPostprint (author's final draft

Rank aggregation in cyclic sequences

In this paper we propose the problem of finding the cyclic sequence which best represents a set of cyclic sequences. Given a set of elements and a precedence cost matrix we look for the cyclic sequence of the elements which is at minimum distance from all the ranks when the permutation metric distance is the Kendall Tau distance. In other words, the problem consists of finding a robust cyclic rank with respect to a set of elements. This problem originates from the Rank Aggregation Problem for combining different linear ranks of elements. Later we define a probability measure based on dissimilarity between cyclic sequences based on the Kendall Tau distance. Next, we also introduce the problem of finding the cyclic sequence with minimum expected cost with respect to that probability measure. Finally, we establish certain relationships among some classical problems and the new problems that we have proposed.Ministerio de Economía y CompetitividadJunta de AndalucíaFondo Europeo de Desarrollo Regiona

Introducing capacitaties in the location of unreliable facilities

The goal of this paper is to introduce facility capacities into the Reliability Fixed-Charge Location Problem in a sensible way. To this end, we develop and compare different models, which represent a tradeoff between the extreme models currently available in the literature, where a priori assignments are either fixed, or can be fully modified after failures occur. In a series of computational experiments we analyze the obtained solutions and study the price of introducing capacity constraints according to the alternative models both, in terms of computational burden and of solution cost.Peer Reviewe

Proyecto E-MATH "Uso de las TIC en asignaturas cuantitativas aplicadas"

El projecte e-Math pretén fomentar i difondre la utilització i integració de les eines tecnològiques actuals (Internet i programari especialitzat) en els curricula de diverses assignatures quantitatives aplicades pertanyents a diferents titulacions universitàries. Els resultats del projecte (material docent, articles, conclusions, etc.) són de difusió general, ja que es pretén que puguin ser d'utilitat per a qualsevol universitat (presencial o virtual) interessada a fer ús dels recursos tecnològics en honor d'obtenir una millora substancial en la qualitat docent de les seves assignatures quantitatives aplicades.El proyecto e-Math pretende fomentar y difundir la utilización e integración de las herramientas tecnológicas actuales (Internet y software especializado) en los curricula de varias asignaturas cuantitativas aplicadas pertenecientes a diferentes titulaciones universitarias. Los resultados del proyecto (material docente, artículos, conclusiones, etc.) son de difusión general, ya que se pretende que puedan ser de utilidad para cualquier universidad (presencial o virtual) interesada en hacer uso de los recursos tecnológicos en aras de obtener una mejora sustancial en la calidad docente de sus asignaturas cuantitativas aplicadas.The project e-Math pretends to boost and spread the utilisation and integration of the current technological tools (Internet and skilled software) in the curricula of several quantitative subjects applied pertaining to different university degrees. The results of the project (educational material, articles, conclusions, etc.) are of general diffusion, since it pretends that they can be of utility for any university (face-to-face or virtual) interested in doing use of the technological resources in plough to obtain a substantial improvement in the educational quality of his quantitative subjects applied