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In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we…

Optimization and Control · Mathematics 2024-02-13 Jeongyeol Kwon , Dohyun Kwon , Stephen Wright , Robert Nowak

Bilevel optimization (BLO) problem, where two optimization problems (referred to as upper- and lower-level problems) are coupled hierarchically, has wide applications in areas such as machine learning and operations research. Recently, many…

Optimization and Control · Mathematics 2025-05-19 Xiaotian Jiang , Ioannis Tsaknakis , Prashant Khanduri , Mingyi Hong

Second-order optimality conditions are essential for nonsmooth optimization, where both the objective and constraint functions are Lipschitz continuous and second-order directionally differentiable. This paper provides no-gap second-order…

Optimization and Control · Mathematics 2025-11-05 Xiang Liu , Mengwei Xu , Liwei Zhang

Bilevel optimization is a hierarchical framework where an upper-level optimization problem is constrained by a lower-level problem, commonly used in machine learning applications such as hyperparameter optimization. Existing bilevel…

Optimization and Control · Mathematics 2026-03-03 Yuman Wu , Xiaochuan Gong , Jie Hao , Mingrui Liu

Bi-Level Optimization (BLO) is originated from the area of economic game theory and then introduced into the optimization community. BLO is able to handle problems with a hierarchical structure, involving two levels of optimization tasks,…

Machine Learning · Computer Science 2021-09-29 Risheng Liu , Jiaxin Gao , Jin Zhang , Deyu Meng , Zhouchen Lin

We propose efficient methods for solving stochastic simple bilevel optimization problems with convex inner levels, where the goal is to minimize an outer stochastic objective function subject to the solution set of an inner stochastic…

Optimization and Control · Mathematics 2025-11-25 Khanh-Hung Giang-Tran , Soroosh Shafiee , Nam Ho-Nguyen

Learning to Optimize (L2O) is a subfield of machine learning (ML) in which ML models are trained to solve parametric optimization problems. The general goal is to learn a fast approximator of solutions to constrained optimization problems,…

Optimization and Control · Mathematics 2025-12-04 James Kotary , Himanshu Sharma , Ethan King , Draguna Vrabie , Ferdinando Fioretto , Jan Drgona

In this paper, we study a class of bilevel optimization problems, also known as simple bilevel optimization, where we minimize a smooth objective function over the optimal solution set of another convex constrained optimization problem.…

Optimization and Control · Mathematics 2023-04-25 Ruichen Jiang , Nazanin Abolfazli , Aryan Mokhtari , Erfan Yazdandoost Hamedani

We study a class of two-stage stochastic programs, namely, those with fixed recourse matrix and fixed costs, and linear second stage. We show that, under mild assumptions, the problem can be solved with just one scenario, which we call an…

Optimization and Control · Mathematics 2025-10-29 Tito Homem-de-Mello , Juan Valencia , Felipe Lagos , Guido Lagos

We design and analyze a novel accelerated gradient-based algorithm for a class of bilevel optimization problems. These problems have various applications arising from machine learning and image processing, where optimal solutions of the two…

Optimization and Control · Mathematics 2023-11-20 Sepideh Samadi , Daniel Burbano , Farzad Yousefian

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic…

Optimization and Control · Mathematics 2023-08-16 Jincheng Cao , Ruichen Jiang , Nazanin Abolfazli , Erfan Yazdandoost Hamedani , Aryan Mokhtari

We introduce contextual stochastic bilevel optimization (CSBO) -- a stochastic bilevel optimization framework with the lower-level problem minimizing an expectation conditioned on some contextual information and the upper-level decision…

Optimization and Control · Mathematics 2023-10-31 Yifan Hu , Jie Wang , Yao Xie , Andreas Krause , Daniel Kuhn

Simple bilevel problems are optimization problems in which we want to find an optimal solution to an inner problem that minimizes an outer objective function. Such problems appear in many machine learning and signal processing applications…

Optimization and Control · Mathematics 2022-12-21 Lior Doron , Shimrit Shtern

Bilevel optimization is a central tool in machine learning for high-dimensional hyperparameter tuning. Its applications are vast; for instance, in imaging it can be used for learning data-adaptive regularizers and optimizing forward…

Optimization and Control · Mathematics 2025-11-11 Mohammad Sadegh Salehi , Subhadip Mukherjee , Lindon Roberts , Matthias J. Ehrhardt

Bilevel optimization (BO) is useful for solving a variety of important machine learning problems including but not limited to hyperparameter optimization, meta-learning, continual learning, and reinforcement learning. Conventional BO…

Machine Learning · Computer Science 2022-09-20 Mao Ye , Bo Liu , Stephen Wright , Peter Stone , Qiang Liu

A bilevel optimization problem consists of two optimization problems nested as an upper- and a lower-level problem, in which the optimality of the lower-level problem defines a constraint for the upper-level problem. This paper considers…

Machine Learning · Computer Science 2026-02-27 Takuya Kanayama , Yuki Ito , Tomoyuki Tamura , Masayuki Karasuyama

Many problems in machine learning involve bilevel optimization (BLO), including hyperparameter optimization, meta-learning, and dataset distillation. Bilevel problems consist of two nested sub-problems, called the outer and inner problems,…

Machine Learning · Computer Science 2022-12-29 Paul Vicol , Jonathan Lorraine , Fabian Pedregosa , David Duvenaud , Roger Grosse

Gradient-based minimax optimal algorithms have greatly promoted the development of continuous optimization and machine learning. One seminal work due to Yurii Nesterov [Nes83a] established $\tilde{\mathcal{O}}(\sqrt{L/\mu})$ gradient…

Machine Learning · Computer Science 2023-12-07 Yuanshi Liu , Hanzhen Zhao , Yang Xu , Pengyun Yue , Cong Fang

Bilevel learning refers to machine learning problems that can be formulated as bilevel optimization models, where decisions are organized in a hierarchical structure. This paradigm has recently gained considerable attention in machine…

Optimization and Control · Mathematics 2026-05-05 Riccardo Grazzi , Massimiliano Pontil , Saverio Salzo , Alain Zemkoho

Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel…

Machine Learning · Computer Science 2021-08-30 Kaiyi Ji , Junjie Yang , Yingbin Liang
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