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 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; 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 RM.

## Citation

J. Castro, L.F. Escudero, J.F. Monge, On solving large-scale multistage stochastic problems with a new specialized interior-point approach, Research Report UPC-DEIO-JC-2021-02, Dept. of Statistics and Operations Research, Universitat Politècnica de Catalunya, Barcelona, Catalonia, 2021.