Decision-focused predictions via pessimistic bilevel optimization: complexity and algorithms

Dealing with uncertainty in optimization parameters is an important and longstanding challenge. Typically, uncertain parameters are predicted accurately, and then a deterministic optimization problem is solved. However, the decisions produced by this so-called predict-then-optimize procedure can be highly sensitive to uncertain parameters. In this work, we contribute to recent efforts in producing decision-focused predictions, i.e., to build predictive models that are constructed with the goal of minimizing a regret measure on the decisions taken with them. We begin by formulating the exact expected regret minimization as a pessimistic bilevel optimization model. Then, we show computational complexity results of this problem, including its membership in NP. In combination with a known NP-hardness result, this establishes NP-completeness and discards its hardness in higher complexity classes. Using duality arguments, we reformulate it as a non-convex quadratic optimization problem. Finally, leveraging the quadratic reformulation, we show various computational techniques to achieve empirical tractability. We report extensive computational results on shortest-path and bipartite matching instances with uncertain cost vectors. Our results indicate that our approach can improve training performance over the approach of Elmachtoub and Grigas (2022), a state-of-the-art method for decision-focused learning.