[Paper Review] Scenario Aggregation using Binary Decision Diagrams for Stochastic Programs with Endogenous Uncertainty

Utz‐Uwe Haus, Carla Michini+1 more|arXiv (Cornell University)|Jan 15, 2017

Reliability and Maintenance Optimization30 references3 citations

TL;DR

This paper proposes a novel method to model endogenous uncertainty in stochastic programs using binary decision diagrams (BDDs) for compact scenario aggregation. By encoding scenario sets as BDDs, the approach transforms non-linear, non-anticipative models into small mixed-integer linear programs (MILPs) through linear factorization of decision-dependent probabilities, significantly reducing problem size and solution time for problems like stochastic network interdiction and reliability optimization.

ABSTRACT

Modeling decision-dependent scenario probabilities in stochastic programs is difficult and typically leads to large and highly non-linear MINLPs that are very difficult to solve. In this paper, we develop a new approach to obtain a compact representation of the recourse function using a set of binary decision diagrams (BDDs) that encode a nested cover of the scenario set. The resulting BDDs can then be used to efficiently characterize the decision-dependent scenario probabilities by a set of linear inequalities, which essentially factorizes the probability distribution and thus allows to reformulate the entire problem as a small mixed-integer linear program. The approach is applicable to a large class of stochastic programs with multivariate binary scenario sets, such as stochastic network design, network reliability, or stochastic network interdiction problems. Computational results show that the BDD-based scenario representation reduces the problem size, and hence the computation time, significant compared to previous approaches.

Motivation & Objective

• To address the challenge of modeling decision-dependent scenario probabilities in stochastic programs, which typically results in large, non-linear MINLPs.
• To develop a compact, exact representation of the recourse function for multivariate binary scenario sets using BDDs.
• To enable efficient characterization of decision-dependent scenario probabilities via linear inequalities, avoiding non-linear terms.
• To reformulate complex stochastic programs with endogenous uncertainty as small mixed-integer linear programs (MILPs).
• To demonstrate the method’s effectiveness on real-world problems such as stochastic network interdiction and network reliability.

Proposed method

• Construct a set of binary decision diagrams (BDDs) that encode a nested cover of the scenario set, capturing the structure of decision-dependent scenario probabilities.
• Use the BDDs to represent the scenario set in a compressed, exact form that preserves all relevant outcomes and probabilities.
• Derive a system of linear inequalities from the BDD structure to model the probability distribution as a function of decisions, enabling linear reformulation.
• Leverage the monotonicity of the recourse function (e.g., shortest path length or maximum flow) to ensure BDDs capture all relevant scenario behavior without approximation.
• Apply the BDD-based reformulation to convert the original non-linear stochastic program into a small MILP, solvable by standard MIP solvers.
• Use a joint generation procedure with a max-flow oracle (via lemon-1.3.1 library) to compute minimal edge sets that reduce flow values, enabling scenario enumeration for BDD construction.

Experimental results

Research questions

• RQ1 Can BDDs be used to compactly represent exponentially large scenario sets in stochastic programs with endogenous uncertainty?
• RQ2 Can decision-dependent scenario probabilities be modeled via linear inequalities derived from BDD structure, avoiding non-linear terms?
• RQ3 Does the BDD-based reformulation significantly reduce problem size and computation time compared to traditional MINLP or sampling-based approaches?
• RQ4 To what extent can this method be applied to real-world problems such as network interdiction and reliability optimization?
• RQ5 Can the reformulated MILP be solved efficiently across varying interdiction budgets and success probabilities without re-generation?

Key findings

• The BDD-based scenario aggregation reduces the problem size and computation time substantially compared to previous approaches, enabling exact solution of previously intractable instances.
• The method allows exact reformulation of stochastic programs with endogenous uncertainty into small mixed-integer linear programs, avoiding non-linearities from product terms of decision variables.
• For the SNIP(IB) benchmark instances, the approach enables solving the exact reformulation across all interdiction budgets and success probabilities without re-generating the MIP.
• The formulation separates network structure from parameters (e.g., budget, probabilities), allowing efficient parameter studies and reusability of the MILP model.
• Despite the compact formulation, current MIP solvers (CPLEX, Gurobi, FICO Xpress, SCIP) struggle with the problem structure, suggesting potential for decomposition methods in future work.
• The approach is applicable to a broad class of problems with multivariate binary scenario sets, including stochastic network design, reliability, and interdiction.