A Barrier Function Approach for Bilevel Optimization with Coupled Lower-Level Constraints: Formulation, Approximation and Algorithms

TL;DR

This paper introduces a barrier function approach for bilevel optimization problems with coupled lower-level constraints, providing new algorithms with convergence guarantees, including for linear lower-level problems, and demonstrating their effectiveness through experiments.

Contribution

It develops a novel barrier reformulation for bilevel problems with coupled constraints and proposes algorithms with convergence guarantees, including for linear lower-level problems.

Findings

Convergence of hyperfunction value and hypergradient to the original problem.

An adaptive algorithm with non-asymptotic convergence guarantees.

First algorithms with convergence guarantees for bilevel problems with linear lower-level constraints.

Abstract

In this paper, we consider bilevel optimization problem where the lower-level has coupled constraints, i.e. the constraints depend both on the upper- and lower-level variables. In particular, we consider two settings for the lower-level problem. The first is when the objective is strongly convex and the constraints are convex with respect to the lower-level variable; The second is when the lower-level is a linear program. We propose to utilize a barrier function reformulation to translate the problem into an unconstrained problem. By developing a series of new techniques, we proved that both the hyperfunction value and hypergradient of the barrier reformulated problem (uniformly) converge to those of the original problem under minimal assumptions. Further, to overcome the non-Lipschitz smoothness of hyperfunction and lower-level problem for barrier reformulated problems, we design an adaptive algorithm that ensures a non-asymptotic convergence guarantee. We also design an algorithm that converges to the stationary point of the original problem asymptotically under certain assumptions. The proposed algorithms require minimal assumptions, and to our knowledge, they are the first with convergence guarantees when the lower-level problem is a linear program. Numerical experiments are conducted to show the effectiveness of the proposed method.