In the earlier paper “On solving large-scale multistage stochastic
optimization problems with a new specialized interior-point approach, European
Journal of Operational Research, 310 (2023), 268–285”
the authors presented a novel approach based on a
specialized interior-point method (IPM) for solving (risk neutral) large-scale multistage
stochastic optimization problems. The method computed the
Newton direction by combining Cholesky factorizations with preconditioned
conjugate gradients (PCG).
This work extends that approach to the risk averse setting by incorporating
coherent risk measures, namely expected conditional value-at-risk and second-order
expected conditional stochastic dominance. Introducing risk aversion makes the
optimization problems substantially more challenging and, for very large
instances, can even exhaust the available memory resources.
The contributions of this work are twofold. First, we propose a reformulation
of risk averse stochastic optimization models based on variable splitting,
which proves highly effective for general-purpose IPM solvers. Second, we
show that the reformulated problem remains fully compatible with the
specialized IPM. In particular, the new risk averse constraints extend the PCG
preconditioner with an additional diagonal matrix, thereby preserving the
efficiency of the solution process.
Extensive computational experiments are reported for large-scale risk averse
multistage stochastic revenue management problems with up to 278 million
variables and 99 million constraints. The results show that the proposed
splitting formulation significantly outperforms the classical compact
formulation when using state-of-the-art solvers, and that the specialized IPM
achieves superior performance on most of the largest problem instances.