A Hybrid Subgradient Method for Nonsmooth Nonconvex Bilevel Optimization

TL;DR

This paper introduces a hybrid subgradient method for nonsmooth, nonconvex bilevel optimization, combining local convergence guarantees with a feasibility restoration scheme for global convergence.

Contribution

It develops a novel hybrid algorithm that alternates between a momentum-accelerated subgradient method and a feasibility restoration scheme, with adaptive hyperparameter estimation.

Findings

The proposed method converges locally when initialized near the feasible region.

The hybrid algorithm achieves global convergence under mild conditions.

Preliminary experiments show high efficiency and promising results.

Abstract

In this paper, we focus on the nonconvex-nonconvex bilevel optimization problem (BLO), where both upper-level and lower-level objectives are nonconvex, with the upper-level problem potentially being nonsmooth. We develop a two-timescale momentum-accelerated subgradient method (TMG) that employs two-timescale stepsizes, and establish its local convergence when initialized within a sufficiently small neighborhood of the feasible region. To develop a globally convergent algorithm for (BLO), we introduce a feasibility restoration scheme (FRG) that drives iterates toward the feasible region. Both (TMG) and (FRG) only require the first-order derivatives of the upper-level and lower-level objective functions, ensuring efficient computations in practice. We then develop a novel hybrid method that alternates between (TMG) and (FRG) and adaptively estimates its hyperparameters. Under mild conditions, we establish the global convergence properties of our proposed algorithm. Preliminary numerical experiments demonstrate the high efficiency and promising potential of our proposed algorithm.