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

A parallelizable algorithmic framework for solving large scale multi-stage stochastic mixed 0-1 problems under uncertainty

Preprint submitted to Computers & Operations Researchmulti-stage stochastic mixed 0-1 optimization, nonsymmetric scenario trees, implicit and explicit nonanticipativity constraints, splitting variable and compact representations, scenario cluster partitioning

Lagrangean decomposition for large-scale two-stage stochastic mixed 0-1 problems

In this paper we study solution methods for solving the dual problem corresponding to the Lagrangean Decomposition of two stage stochastic mixed 0-1 models. We represent the two stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangean Decomposition is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangean Decomposition schemes, as the Subgradient method, the Volume algorithm, the Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane scheme. We test the conditions and properties of convergence for medium and large-scale dimension stochastic problems. Computational results are reported.Progressive Hedging algorithm, volume algorithm, Lagrangean decomposition, subgradient method

A note on the implementation of the BFC-MSMIP algorithm in C++ by using COIN-OR as an optimization engine

The aim of this technical report is to present some detailed explanations in order to help to understand and use the algorithm Branch and Fix Coordination for solving MultiStage Mixed Integer Problems (BFC- MSMIP). We have developed an algorithmic approach implemented in a C++ experimental code that uses the optimization engine COmputational INfrastructure for Operations Research (COIN-OR) for solving the auxiliary linear and mixed 0-1 submodels. Now, we give the computational and implementational descrip- tion in order to use this open optimization software not only in the implementation of our procedure but also in similar schemes to be developed by the users.nonanticipativity constraints, cluster partitioning, COIN-OR library, branch-and-fix coordination, multi-stage stochastic mixed 0-1 programming

Lagrangean decomposition for large-scale two-stage stochastic mixed 0-1 problems

In this paper we study solution methods for solving the dual problem corresponding to the Lagrangean Decomposition of two stage stochastic mixed 0-1 models. We represent the two stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangean Decomposition is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangean Decomposition schemes, as the Subgradient method, the Volume algorithm, the Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane scheme. We test the conditions and properties of convergence for medium and large-scale dimension stochastic problems. Computational results are reported.This research has been partially supported by the projects ECO2008-00777 ECON from the Ministry of Education and Science, Grupo de Investigación IT-347-10 from the Basque Government, grant FPU ECO-2006 from the Ministry of Education and Science, grants RM URJC-CM-2008-CET-3703 and RIESGOS CM from Comunidad de Madrid, and PLANIN MTM2009-14087-C04-01 from Ministry of Science and Innovation, Spain

A parallelizable algorithmic framework for solving large scale multi-stage stochastic mixed 0-1 problems under uncertainty

Preprint submitted to Computers & Operations ResearchIn this paper we present a parallelizable scheme of the Branch-and-Fix Coordination algorithm for solving medium and large scale multi-stage mixed 0-1 optimization problems under uncertainty. The uncertainty is represented via a nonsymmetric scenario tree. An information structuring for scenario cluster partitioning of nonsymmetric scenario trees is also presented, given the general model formulation of a multi-stage stochastic mixed 0-1 problem. The basic idea consists of explicitly rewriting the nonanticipativity constraints (NAC) of the 0-1 and continuous variables in the stages with common information. As a result an assignment of the constraint matrix blocks into independent scenario cluster submodels is performed by a so-called cluster splitting-compact representation. This partitioning allows to generate a new information structure to express the NAC which link the related clusters, such that the explicit NAC linking the submodels together is performed by a splitting variable representation. The new algorithm has been implemented in a C++ experimental code that uses the open source optimization engine COIN-OR, for solving the auxiliary linear and mixed 0-1 submodels. Some computational experience is reported to validate the new proposed approach. We give computational evidence of the model tightening effect that have preprocessing techniques in stochastic integer optimization as well, by using the probing and Gomory and clique cuts identification and appending schemes of the optimization engine.This research has been partially supported by the projects ECO2008-00777 ECON from the Ministry of Education and Science, Grupo de Investigación IT-347-10 from the Basque Government, URJC-CM-2008-CET-3703 and RIESGOS CM from Comunidad de Madrid, and PLANIN MTM2009-14087-C04-01 from Ministry of Science and Innovation, Spain

Scenario Cluster Lagrangian Decomposition in two stage stochastic mixed 0-1 optimization

In this paper we introduce four scenario Cluster based Lagrangian Decomposition (CLD) procedures for obtaining strong lower bounds to the (optimal) solution value of two-stage stochastic mixed 0-1 problems. At each iteration of the Lagrangian based procedures, the traditional aim consists of obtaining the solution value of the corresponding Lagrangian dual via solving scenario submodels once the nonanticipativity constraints have been dualized. Instead of considering a splitting variable representation over the set of scenarios, we propose to decompose the model into a set of scenario clusters. We compare the computational performance of the four Lagrange multiplier updating procedures, namely the Subgradient Method, the Volume Algorithm, the Progressive Hedging Algorithm and the Dynamic Constrained Cutting Plane scheme for different numbers of scenario clusters and different dimensions of the original problem. Our computational experience shows that the CLD bound and its computational effort depend on the number of scenario clusters to consider. In any case, our results show that the CLD procedures outperform the traditional LD scheme for single scenarios both in the quality of the bounds and computational effort. All the procedures have been implemented in a C++ experimental code. A broad computational experience is reported on a test of randomly generated instances by using the MIP solvers COIN-OR and CPLEX for the auxiliary mixed 0-1 cluster submodels, this last solver within the open source engine COIN-OR. We also give computational evidence of the model tightening effect that the preprocessing techniques, cut generation and appending and parallel computing tools have in stochastic integer optimization. Finally, we have observed that the plain use of both solvers does not provide the optimal solution of the instances included in the testbed with which we have experimented but for two toy instances in affordable elapsed time. On the other hand the proposed procedures provide strong lower bounds (or the same solution value) in a considerably shorter elapsed time for the quasi-optimal solution obtained by other means for the original stochastic problem

A note on the implementation of the BFC-MSMIP algorithm in C++ by using COIN-OR as an optimization engine

The aim of this technical report is to present some detailed explanations in order to help to understand and use the algorithm Branch and Fix Coordination for solving MultiStage Mixed Integer Problems (BFC- MSMIP). We have developed an algorithmic approach implemented in a C++ experimental code that uses the optimization engine COmputational INfrastructure for Operations Research (COIN-OR) for solving the auxiliary linear and mixed 0-1 submodels. Now, we give the computational and implementational descrip- tion in order to use this open optimization software not only in the implementation of our procedure but also in similar schemes to be developed by the users.This research has been partially supported by the project ECO2008-00777 ECON from the Spanish Ministry of Education and Science, and Grupo de Investigación IT-347-10 from the Basque Government

On solving two stage stochastic linear problems by using a new approach, Cluster Benders Decomposition

The optimization of stochastic linear problems, via scenario analysis, based on Benders decomposition requires to appending feasibility and/or optimality cuts to the master problem until the iterative procedure reaches the optimal solution. The cuts are identified by solving the auxiliary submodels attached to the scenarios. In this work, we propose a so-called scenario cluster decomposition approach for dealing with the feasibility cut identification in the Benders method for solving large-scale two stage stochastic linear problems. The scenario tree is decomposed into a set of scenario clusters and tighter feasibility cuts are obtained by solving the auxiliary submodel for each cluster instead of each individual scenario. Then, this scenario cluster based scheme allows us to define tighter feasibility cuts that yield feasible second stage decisions in reasonable time consuming. Some computational experience by using the free software COIN-OR is reported to show the favorable performance of the new approach over traditional Benders decomposition.This research has been partially supported by the projects ECO2008-00777/ECON from the Ministry of Education and Science, PLANIN, MTM14087-C04-01 from the Ministry of Science and Innovation, Grupo de Investigación IT-347-10 from the Basque Government, and the project RIESGOS CM from the Comunidad de Madrid, Spain