Sequential QCQP for Bilevel Optimization with Line Search

TL;DR

This paper introduces a scalable, tuning-free bilevel optimization algorithm that guarantees feasibility and descent, using a single-loop QCQP approach with line search, and proves its convergence under mild conditions.

Contribution

It presents a novel single-loop, tuning-free bilevel optimization method with guaranteed feasibility, convergence, and practical effectiveness demonstrated on benchmark tasks.

Findings

Achieves an O(1/k) convergence rate in ergodic sense.

Requires no hyperparameter tuning and is scalable.

Demonstrates effectiveness on representative bilevel problems.

Abstract

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.