Stable optimisation-based scenario generation via game theoretic approach

TL;DR

The paper tackles instability in optimisation-based scenario generation for stochastic programming, caused by user-defined weights in distribution and moment matching (DMP) models. It introduces a Nash bargaining extension of the DMP MILP, using a log-transform and separable programming to linearise the Nash product and treat objective terms as players. Through a capacity-planning case study, the authors show that the Nash-extended DMP (NASH) improves in-sample and out-of-sample stability and reduces bias compared to traditional NLP/MILP formulations, especially under varying error-weight configurations. This approach yields more robust stochastic solutions and mitigates under-specification issues, with potential for extending to multi-stage SG and explainable stochastic solutions in process systems engineering contexts.

Abstract

Systematic scenario generation (SG) methods have emerged as an invaluable tool to handle uncertainty towards the efficient solution of stochastic programming (SP) problems. The quality of SG methods depends on their consistency to generate scenario sets which guarantee stability on solving SPs and lead to stochastic solutions of good quality. In this context, we delve into the optimisation-based Distribution and Moment Matching Problem (DMP) for scenario generation and propose a game-theoretic approach which is formulated as a Mixed-Integer Linear Programming (MILP) model. Nash bargaining approach is employed and the terms of the objective function regarding the statistical matching of the DMP are considered as players. Results from a capacity planning case study highlight the quality of the stochastic solutions obtained using MILP DMP models for scenario generation. Furthermore, the proposed game-theoretic extension of DMP enhances in-sample and out-of-sample stability with respect to the challenging problem of user-defined parameters variability.

Figures (9)

• Figure 1: Two-stage stochastic problem and scenario set representation, where $x$ are the first-stage decisions, $y_k$ are the second-stage decisions for each scenario $k \in K$, with values $\xi_k$ and probabilities $p_k$.
• Figure 2: Types of under-specification issues that may arise from the solution of original NLP MMP problems and worsen their performance.
• Figure 3: Proposed extended DMP MILP optimisation model can be incorporated in the data-driven scenario generation framework presented by Bounitsis2022.
• Figure 4: Process network for the investigated case study.
• Figure 5: Initial steps (1 to 4) of the proposed scenario generation framework of Algorithm \ref{['alg:framework']}.