# An Improved Unconstrained Approach for Bilevel Optimization

arXiv: https://arxiv.org/abs/2208.00732 · HTML: https://mlsanity.com/paper/2208.00732/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Aug 2022.

## Built on by (most cited first, 11 of 11)

- [First-order penalty methods for bilevel optimization](https://mlsanity.com/paper/2301.01716/) (2023)
- [Bilevel reinforcement learning via the development of hyper-gradient without lower-level convexity](https://mlsanity.com/paper/2405.19697/) (2024)
- [Non-Convex Bilevel Optimization with Time-Varying Objective Functions](https://mlsanity.com/paper/2308.03811/) (2023)
- [A Single-Loop Algorithm for Decentralized Bilevel Optimization](https://mlsanity.com/paper/2311.08945/) (2023)
- [Adam-family Methods with Decoupled Weight Decay in Deep Learning](https://mlsanity.com/paper/2310.08858/) (2023)
- [LancBiO: dynamic Lanczos-aided bilevel optimization via Krylov subspace](https://mlsanity.com/paper/2404.03331/) (2024)
- [Stochastic Subgradient Methods with Guaranteed Global Stability in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2307.10053/) (2023)
- [Solving bilevel optimization via sequential minimax optimization](https://mlsanity.com/paper/2511.07398/) (2025)
- [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024)
- [On Stability and Generalization of Bilevel Optimization Problem](https://mlsanity.com/paper/2210.01063/) (2022)
- [Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems](https://mlsanity.com/paper/2605.08006/) (2026)

## Abstract (arXiv)

In this paper, we focus on the nonconvex-strongly-convex bilevel optimization problem (BLO). In this BLO, the objective function of the upper-level problem is nonconvex and possibly nonsmooth, and the lower-level problem is smooth and strongly convex with respect to the underlying variable y. We show that the feasible region of BLO is a Riemannian manifold. Then we transform BLO to its corresponding unconstrained constraint dissolving problem (CDB), whose objective function is explicitly formulated from the objective functions in BLO. We prove that BLO is equivalent to the unconstrained optimization problem CDB. Therefore, various efficient unconstrained approaches, together with their theoretical results, can be directly applied to BLO through CDB. We propose a unified framework for developing subgradient-based methods for CDB. Remarkably, we show that several existing efficient algorithms can fit the unified framework and be interpreted as descent algorithms for CDB. These examples further demonstrate the great potential of our proposed approach.