María Araceli Garín

An algorithmic framework for solving large-scale multistage stochastic mixed 0-1 problems with nonsymmetric scenario trees

In this paper we present a parallelizable Branch-and-Fix Coordination algorithm for solving medium and large-scale multistage 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 multistage 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. Some computational experience is reported on a test of randomly generated instances as well as a large-scale real-life problem by using CPLEX as a solver of the auxiliary submodels within the open source engine COIN-OR.

A general algorithm for solving two-stage stochastic mixed 0-1 first-stage problems

We present an algorithmic approach for solving large-scale two-stage stochastic problems having mixed 0-1 first stage variables. The constraints in the first stage of the deterministic equivalent model have 0-1 variables and continuous variables, while the constraints in the second stage have only continuous. The approach uses the twin node family concept within the algorithmic framework, the so-called branch-and-fix coordination, in order to satisfy the nonanticipativity constraints. At the same time we consider a scenario cluster Benders decomposition scheme for solving large-scale LP submodels given at each TNF integer set. Some computational results are presented to demonstrate the efficiency of the proposed approach.

On BFC-MSMIP strategies for scenario cluster partitioning, and twin node family branching selection and bounding for multistage stochastic mixed integer programming

In the branch-and-fix coordination (BFC-MSMIP) algorithm for solving large-scale multistage stochastic mixed integer programming problems, we find it crucial to decide the stages where the nonanticipativity constraints are explicitly considered in the model. This information is materialized when the full model is broken down into a scenario cluster partition with smaller subproblems. In this paper we present a scheme for obtaining strong bounds and branching strategies for the Twin Node Families to increase the efficiency of the procedure BFC-MSMIP, based on the information provided by the nonanticipativity constraints that are explicitly considered in the problem. Some computational experience is reported to support the efficiency of the new scheme.

An exact algorithm for solving large-scale two-stage stochastic mixed-integer problems: Some theoretical and experimental aspects

We present an algorithmic framework, so-called BFC-TSMIP, for solving two-stage stochastic mixed 0-1 problems. The constraints in the Deterministic Equivalent Model have 0-1 variables and continuous variables at any stage. The approach uses the Twin Node Family (TNF) concept within an adaptation of the algorithmic framework so-called Branch-and-Fix Coordination for satisfying the nonanticipativity constraints for the first stage 0-1 variables. Jointly we solve the mixed 0-1 submodels defined at each TNF integer set for satisfying the nonanticipativity constraints for the first stage continuous variables. In these submodels the only integer variables are the second stage 0-1 variables. A numerical example and some theoretical and computational results are presented to show the performance of the proposed approach.

On time stochastic dominance induced by mixed integer-linear recourse in multistage stochastic programs

We propose in this work a new multistage risk averse strategy based on Time Stochastic Dominance (TSD) along a given horizon. It can be considered as a mixture of the two risk averse measures based on first- and second-order stochastic dominance constraints induced by mixed integer-linear recourse, respectively. Given the dimensions of medium-sized problems augmented by the new variables and constraints required by this new risk measure, it is unrealistic to solve the problem up to optimality by plain use of MIP solvers in a reasonable computing time, at least. Instead of it, decomposition algorithms of some type should be used. We present an extension of our Branch-and-Fix Coordination algorithm, so named BFC-TSD, where a special treatment is given to cross scenario group constraints that link variables from different scenario groups. A broad computational experience is presented by comparing the risk neutral approach and the tested risk averse strategies. The performance of the new version of the BFC algorithm versus the plain use of a state-of-the-art MIP solver is also reported.

Cluster Lagrangean decomposition in multistage stochastic optimization

We present a Lagrangean Decomposition approach for obtaining strong lower bounds on minimizing medium to large scale multistage stochastic mixed 0-1 problems. The problem is represented by a mixture of the splitting representation up to a given stage, so-named break stage, and the compact representation for the other stages along the time horizon. The dualization of the nonanticipativity constraints for the variables up to the break stage results in a model that can be decomposed into a set of independent scenario cluster submodels. The nonanticipativity constraints for the 0-1 and continuous variables in the cluster submodels are implicitly satisfied. Four scenario cluster schemes are compared for Lagrangean multipliers updating such as the Subgradient Method, the Volume Algorithm, the Lagrangean Progressive Hedging Algorithm and the Dynamic Constrained Cutting Plane scheme. We have observed in randomly generated instances that the smaller the number of clusters, the stronger the lower bound provided for the original problem (even, frequently, it is the solution value). HighlightsA Multistage scenario Cluster Lagrangian Decomposition (MCLD) approach for obtaining strong lower bounds on the solution value of large sized instances of the multistage stochastic mixed 0-1 problem is presented.MCLD frequently gives the solution value of the original stochastic mixed 0-1 problem by using merged or pure scenario cluster submodels either without Lagrange multipliers usage or considering any of the four Lagrange multipliers schemes that we have selected to work with.It is easy to prove that the performance of the MCLD procedure outperforms the traditional Lagrangian Decomposition scheme based on single scenarios in both the bounds quality and elapsed time.The efficiency of the four updating schemes, as contrasted in the testbed we have experimented with, is very similar in quality of the MCLD bound. However, a careful implementation of the Subgradient Method makes it more robust than the other three updating schemes. Notice that the SM based MCLD approach obtains the solution value in the original problem in thirteen out of the fourteen instances included in the testbed.The proposed approach requires an elapsed time that is one order of magnitude smaller than the time required by CPLEX, in the worst case.

On efficient matheuristic algorithms for multi-period stochastic facility location-assignment problems

In this work we present two matheuristic procedures to build good feasible solutions (frequently, the optimal one) by considering the solutions of relaxed problems of large-sized instances of the multi-period stochastic pure 0–1 location-assignment problem. The first procedure is an iterative one for Lagrange multipliers updating based on a scenario cluster Lagrangean decomposition for obtaining strong (lower, in case of minimization) bounds of the solution value. The second procedure is a sequential one that works with the relaxation of the integrality of subsets of variables for different levels of the problem, so that a chain of (lower, in case of minimization) bounds is generated from the LP relaxation up to the integer solution value. Additionally, and for both procedures, a lazy heuristic scheme, based on scenario clustering and on the solutions of the relaxed problems, is considered for obtaining a (hopefully good) feasible solution as an upper bound of the solution value of the full problem. Then, the same framework provides for the two procedures lower and upper bounds on the solution value. The performance is compared over a set of instances of the stochastic facility location-assignment problem. It is well known that the general static deterministic location problem is NP-hard and, so, it is the multi-period stochastic version. A broad computational experience is reported for 14 instances, up to 15 facilities, 75 customers, 6 periods, over 260 scenarios and over 420 nodes in the scenario tree, to assess the validity of proposals made in this work versus the full use of a state-of the-art IP optimizer.