Sequential QCQP for Bilevel Optimization with Line Search
This addresses the complex interdependencies in bilevel optimization for researchers and practitioners, offering a scalable and hyperparameter-free method, though it appears incremental as it builds on existing QCQP and line search techniques.
The paper tackles bilevel optimization by proposing a single-loop, tuning-free algorithm that guarantees anytime feasibility and descent of the upper-level objective, achieving an O(1/k) ergodic convergence rate and demonstrating effectiveness on representative tasks.
Bilevel optimization involves a hierarchical structure where one problem is nested within another, leading to complex interdependencies between levels. We propose a single-loop, tuning-free algorithm that guarantees anytime feasibility, i.e., approximate satisfaction of the lower-level optimality condition, while ensuring descent of the upper-level objective. At each iteration, a convex quadratically-constrained quadratic program (QCQP) with a closed-form solution yields the search direction, followed by a backtracking line search inspired by control barrier functions to ensure safe, uniformly positive step sizes. The resulting method is scalable, requires no hyperparameter tuning, and converges under mild local regularity assumptions. We establish an O(1/k) ergodic convergence rate in terms of a first-order stationary metric and demonstrate the algorithm's effectiveness on representative bilevel tasks.