A Primal-Dual-Assisted Penalty Approach to Bilevel Optimization with Coupled Constraints

TL;DR

This paper introduces BLOCC, a first-order algorithm for bilevel optimization problems with coupled constraints, addressing complex applications in machine learning and infrastructure planning.

Contribution

It develops the first fully first-order method with convergence guarantees for bilevel problems with coupled constraints, expanding the scope of solvable problems.

Findings

Proposed BLOCC algorithm with proven convergence.

Effective in hyperparameter tuning for SVMs.

Successful application to transportation infrastructure planning.

Abstract

Interest in bilevel optimization has grown in recent years, partially due to its applications to tackle challenging machine-learning problems. Several exciting recent works have been centered around developing efficient gradient-based algorithms that can solve bilevel optimization problems with provable guarantees. However, the existing literature mainly focuses on bilevel problems either without constraints, or featuring only simple constraints that do not couple variables across the upper and lower levels, excluding a range of complex applications. Our paper studies this challenging but less explored scenario and develops a (fully) first-order algorithm, which we term BLOCC, to tackle BiLevel Optimization problems with Coupled Constraints. We establish rigorous convergence theory for the proposed algorithm and demonstrate its effectiveness on two well-known real-world applications - hyperparameter selection in support vector machine (SVM) and infrastructure planning in transportation networks using the real data from the city of Seville.