HGCN2SP: Hierarchical Graph Convolutional Network for Two-Stage Stochastic Programming

TL;DR

This work tackles the challenge of solving large-scale two-stage stochastic programming (2SP) by learning to select representative scenarios. It introduces HGCN2SP, a hierarchical graph model that encodes each scenario as a bipartite subgraph and captures cross-scenario relationships with a higher-level graph, processed by a hierarchical GCN and an attention-based decoder. The policy is trained with PPO using solver feedback as a reward, balancing solution quality and solving time. Empirical results on CFLP and NDP show that HGCN2SP achieves high-quality decisions in substantially less time than baselines and generalizes well to larger-scale problems and more scenarios, with ablations confirming the importance of the hierarchical architecture. The approach offers a practical, scalable path to efficient scenario reduction in 2SP, with potential impact on how uncertain, large-scale decision problems are solved in practice.

Abstract

Two-stage Stochastic Programming (2SP) is a standard framework for modeling decision-making problems under uncertainty. While numerous methods exist, solving such problems with many scenarios remains challenging. Selecting representative scenarios is a practical method for accelerating solutions. However, current approaches typically rely on clustering or Monte Carlo sampling, failing to integrate scenario information deeply and overlooking the significant impact of the scenario order on solving time. To address these issues, we develop HGCN2SP, a novel model with a hierarchical graph designed for 2SP problems, encoding each scenario and modeling their relationships hierarchically. The model is trained in a reinforcement learning paradigm to utilize the feedback of the solver. The policy network is equipped with a hierarchical graph convolutional network for feature encoding and an attention-based decoder for scenario selection in proper order. Evaluation of two classic 2SP problems demonstrates that HGCN2SP provides high-quality decisions in a short computational time. Furthermore, HGCN2SP exhibits remarkable generalization capabilities in handling large-scale instances, even with a substantial number of variables or scenarios that were unseen during the training phase.

Figures (4)

• Figure 1: An overview of HGCN2SP. For each 2sp instance, we initially combine individual scenarios with the first stage to form scenario subgraphs. These subgraphs are then input into a hierarchical graph convolutional network (GCN) to derive representations. In this network, each scenario subgraph first obtains embeddings from the low-level GCN, building an instance graph. The high-level GCN then utilizes this graph to generate the final and global embeddings. An attention-based decoder leverages these embeddings to select $k$ scenarios sequentially, which are transformed into an equivalent MIP problem. Ultimately, a solver is employed to derive the solution.
• Figure 2: The scenario-specific MIP is represented as a bipartite graph with two sets of nodes. On the left side are the variables $\{x_1, \cdots, x_{n_1}, y_1,\cdots, y_{n_2}\}$, and on the right are the constraints $\{c_1,\cdots,c_{m_1},d_1,\cdots,d_{m_2}\}$.
• Figure 3: The model's output is randomized using 100 seeds, with solving times plotted on a cumulative distribution function (CDF). The model's actual solving time, marked on the CDF, outperforms most randomized sequences.
• Figure 4: The decoder splices the encoder's output into a context embedding and calculates the selection probability by determining the attention between candidate points and the context embedding. Then, it selects them sequentially. (Credit to Kool2018AttentionLT)