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

TL;DR

This paper formulates and analyzes a decision-focused prediction approach using pessimistic bilevel optimization, addressing complexity challenges and proposing algorithms that improve decision quality under uncertainty.

Contribution

It introduces a novel bilevel optimization model for decision-focused predictions, analyzes its computational complexity, and develops algorithms demonstrating empirical effectiveness.

Findings

The problem is NP-complete, indicating high computational complexity.

The quadratic reformulation enables practical algorithms for real instances.

Empirical results show improved decision quality over existing methods.

Abstract

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.