Machine Learning-Augmented Optimization of Large Bilevel and Two-stage Stochastic Programs: Application to Cycling Network Design

TL;DR

This paper introduces a machine learning-augmented optimization framework for large bilevel and two-stage stochastic programs, demonstrated through a real-world cycling network design case in Toronto, achieving significant improvements over current practices.

Contribution

It presents a novel approach combining sampling, machine learning, and feature learning to efficiently solve large bilevel problems with many followers, with proven bounds on solution quality.

Findings

Improves cycling accessibility in Toronto by 19.2%.

Reduces potential costs by $18 million.

Outperforms existing methods significantly.

Abstract

A wide range of decision problems can be formulated as bilevel programs with independent followers, which as a special case include two-stage stochastic programs. These problems are notoriously difficult to solve especially when a large number of followers present. Motivated by a real-world cycling infrastructure planning application, we present a general approach to solving such problems. We propose an optimization model that explicitly considers a sampled subset of followers and exploits a machine learning model to estimate the objective values of unsampled followers. We prove bounds on the optimality gap of the generated leader decision as measured by the original objective function that considers the full follower set. We then develop follower sampling algorithms to tighten the bounds and a representation learning approach to learn follower features, which are used as inputs to the embedded machine learning model. Through numerical studies, we show that our approach generates leader decisions of higher quality compared to baselines. Finally, in collaboration with the City of Toronto, we perform a real-world case study in Toronto where we solve a cycling network design problem with over one million followers. Compared to the current practice, our approach improves Toronto's cycling accessibility by 19.2%, equivalent to $18M in potential cost savings. Our approach is being used to inform the cycling infrastructure planning in Toronto and outperforms the current practice by a large margin. It can be generalized to any decision problems that are formulated as bilevel programs with independent followers.