A New Lagrangian Relaxation Approach for Multistage Stochastic Programs under Endogenous Uncertainties

Optimization problems with endogenous (decision-dependent) uncertainties are commonly observed in process industry. Most optimization problems with endogenous uncertainties, by nature, can be modelled as multi-period multi-stage stochastic programs (MSSPs), where possible future states of the system are modelled as scenarios by enumerating all possible outcomes of uncertain parameters. However, MSSPs rapidly grow and quickly become computationally intractable for real-world problems. This paper presents a new Lagrangian relaxation for obtaining valid dual bounds for MSSPs under endogenous uncertainties. By exploiting the structure of the MSSP, a tight dual problem is formulated, which reduces the total number of Lagrangian multipliers. The paper also introduces a modified multiplier-updating scheme. We applied the new Lagrangian relaxation to bound instances of artificial lift infrastructure planning (ALIP) problem under uncertain production rates and the clinical trial planning (CTP) problem under uncertain clinical trial outcomes. The computational results reveal that the proposed Lagrangian relaxation generates tight dual bounds compared to the original Lagrangian relaxation formulation and that the proposed multiplier-updating scheme reduces the zigzagging behavior of the Lagrangian dual solutions as iterations progress.