Leveraging Decision Diagrams to Solve Two-stage Stochastic Programs with Binary Recourse and Logical Linking Constraints

Two-stage stochastic programs with binary recourse are challenging to solve and efficient solution methods for such problems have been limited. In this work, we generalize an existing binary decision diagram-based (BDD-based) approach of Lozano and Smith (Math. Program., 2018) to solve a special class of two-stage stochastic programs with binary recourse. In this setting, the first-stage decisions impact the second-stage constraints. Our modified problem extends the second-stage problem to a more general setting where logical expressions of the first-stage solutions enforce constraints in the second stage. We also propose a complementary problem and solution method which can be used for many of the same applications. In the complementary problem we have second-stage costs impacted by expressions of the first-stage decisions. In both settings, we convexify the second-stage problems using BDDs and parametrize either the arc costs or capacities of these BDDs with first-stage solutions depending on the problem. We further extend this work by incorporating conditional value-at-risk and we propose, to our knowledge, the first decomposition method for two-stage stochastic programs with binary recourse and a risk measure. We apply these methods to a novel stochastic dominating set problem and present numerical results to demonstrate the effectiveness of the proposed methods

Approximate and exact convexification approaches for solving two-stage mixed-integer recourse models

Many practical decision-making problems are subject to uncertainty. A powerful class of mathematical models designed for these problems is the class of mixed-integer recourse models. Such models have a wide range of applications in, e.g., healthcare, energy, and finance. They permit integer decision variables to accurately model, e.g., on/off restrictions or natural indivisibilities. The additional modelling flexibility of integer decision variables, however, comes at the expense of models that are significantly harder to solve. The reason is that including integer decision variables introduces non-convexity in the model, which poses a significant challenge for state-of-the-art solvers.In this thesis, we contribute to better decision making under uncertainty by designing efficient solution methods for mixed-integer recourse models. Our approach is to address the non-convexity caused by integer decision variables by using convexification. That is, we construct convex approximating models that closely approximate the original model. In addition, we derive performance guarantees for the solution obtained by solving the approximating model. Finally, we extensively test the solution methods that we propose and we find that they consistently outperform traditional solution methods on a wide range of benchmark instances