# A Primal-Dual-Assisted Penalty Approach to Bilevel Optimization with Coupled Constraints

arXiv: https://arxiv.org/abs/2406.10148 · HTML: https://mlsanity.com/paper/2406.10148/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 77 references that are in the map.
- First posted on arXiv 14 Jun 2024.

## Builds on (most cited first, 30 of 39)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Truncated Back-propagation for Bilevel Optimization](https://mlsanity.com/paper/1810.10667/) (2018)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [Mini-batch Stochastic Approximation Methods for Nonconvex Stochastic Composite Optimization](https://mlsanity.com/paper/1308.6594/) (2013)
- [A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications](https://mlsanity.com/paper/1705.06270/) (2017)
- [A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum](https://mlsanity.com/paper/2102.07367/) (2021)
- [BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach](https://mlsanity.com/paper/2209.08709/) (2022)
- [A Fully First-Order Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2301.10945/) (2023)
- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
- [A Single-Timescale Method for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2102.04671/) (2021)
- [Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond](https://mlsanity.com/paper/2110.00455/) (2021)
- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [A Value-Function-based Interior-point Method for Non-convex Bi-level Optimization](https://mlsanity.com/paper/2106.07991/) (2021)
- [A Fully Single Loop Algorithm for Bilevel Optimization without Hessian Inverse](https://mlsanity.com/paper/2112.04660/) (2021)
- [On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation](https://mlsanity.com/paper/2309.01753/) (2023)
- [Principled Penalty-based Methods for Bilevel Reinforcement Learning and RLHF](https://mlsanity.com/paper/2402.06886/) (2024)
- [Will Bilevel Optimizers Benefit from Loops](https://mlsanity.com/paper/2205.14224/) (2022)
- [On the Convergence Theory for Hessian-Free Bilevel Algorithms](https://mlsanity.com/paper/2110.07004/) (2021)
- [Penalty Method for Inversion-Free Deep Bilevel Optimization](https://mlsanity.com/paper/1911.03432/) (2019)
- [Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity](https://mlsanity.com/paper/1802.01504/) (2018)
- [Automatic differentiation of nonsmooth iterative algorithms](https://mlsanity.com/paper/2206.00457/) (2022)

## Built on by (most cited first, 7 of 7)

- [A First-order Generative Bilevel Optimization Framework for Diffusion Models](https://mlsanity.com/paper/2502.08808/) (2025)
- [In-memory Training on Analog Devices with Limited Conductance States via Multi-tile Residual Learning](https://mlsanity.com/paper/2510.02516/) (2025)
- [A Fully First-Order Layer for Differentiable Optimization](https://mlsanity.com/paper/2512.02494/) (2025)
- [Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems](https://mlsanity.com/paper/2605.08006/) (2026)
- [Sequential QCQP for Bilevel Optimization with Line Search](https://mlsanity.com/paper/2505.14647/) (2025)
- [A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization](https://mlsanity.com/paper/2510.24710/) (2025)
- [Sample Complexity Analysis for Constrained Bilevel Reinforcement Learning](https://mlsanity.com/paper/2602.00282/) (2026)

## Abstract (arXiv)

Interest in bilevel optimization has grown in recent years, partially due to its applications to tackle challenging machine-learning problems. Several exciting recent works have been centered around developing efficient gradient-based algorithms that can solve bilevel optimization problems with provable guarantees. However, the existing literature mainly focuses on bilevel problems either without constraints, or featuring only simple constraints that do not couple variables across the upper and lower levels, excluding a range of complex applications. Our paper studies this challenging but less explored scenario and develops a (fully) first-order algorithm, which we term BLOCC, to tackle BiLevel Optimization problems with Coupled Constraints. We establish rigorous convergence theory for the proposed algorithm and demonstrate its effectiveness on two well-known real-world applications - hyperparameter selection in support vector machine (SVM) and infrastructure planning in transportation networks using the real data from the city of Seville.