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Our aim is to explain mathematical programs with equilibrium constraints (MPECs), motivate them through applications, present the main equivalent formulations of equilibrium constraints, and summarize the basic existence theory for optimal…

Optimization and Control · Mathematics 2026-05-04 Louis Shuo Wang

Bilevel optimization is characterized by a two-level optimization structure, where the upper-level problem is constrained by optimal lower-level solutions, and such structures are prevalent in real-world problems. The constraint by optimal…

Machine Learning · Computer Science 2025-05-29 Ruth Wan Theng Chew , Quoc Phong Nguyen , Bryan Kian Hsiang Low

This chapter presents a self-contained approach of variational analysis and generalized differentiation to deriving necessary optimality in problems of bilevel optimization with Lipschitzian data. We mainly concentrate on optimistic models,…

Optimization and Control · Mathematics 2019-07-16 Boris S. Mordukhovich

Due to the hierarchical structure of many machine learning problems, bilevel programming is becoming more and more important recently, however, the complicated correlation between the inner and outer problem makes it extremely challenging…

Machine Learning · Computer Science 2020-09-03 Junyi Li , Bin Gu , Heng Huang

In this paper, we study a class of bilevel programming problem where the inner objective function is strongly convex. More specifically, under some mile assumptions on the partial derivatives of both inner and outer objective functions, we…

Optimization and Control · Mathematics 2018-02-08 Saeed Ghadimi , Mengdi Wang

Bilevel Optimization has experienced significant advancements recently with the introduction of new efficient algorithms. Mirroring the success in single-level optimization, stochastic gradient-based algorithms are widely used in bilevel…

Optimization and Control · Mathematics 2024-11-12 Junyi Li , Heng Huang

Bilevel optimization has been successfully applied to many important machine learning problems. Algorithms for solving bilevel optimization have been studied under various settings. In this paper, we study the nonconvex-strongly-convex…

Optimization and Control · Mathematics 2022-06-14 Xuxing Chen , Minhui Huang , Shiqian Ma

Bilevel learning has gained prominence in machine learning, inverse problems, and imaging applications, including hyperparameter optimization, learning data-adaptive regularizers, and optimizing forward operators. The large-scale nature of…

Optimization and Control · Mathematics 2025-05-20 Mohammad Sadegh Salehi , Subhadip Mukherjee , Lindon Roberts , Matthias J. Ehrhardt

Gradient-based optimization has been critical to the success of machine learning, updating a single set of parameters to minimize a single loss. A growing number of applications rely on a generalization of this, where we have a bilevel or…

Machine Learning · Computer Science 2024-07-02 Jonathan Lorraine

This paper firstly proposes a convex bilevel optimization paradigm to formulate and optimize popular learning and vision problems in real-world scenarios. Different from conventional approaches, which directly design their iteration schemes…

Computer Vision and Pattern Recognition · Computer Science 2021-12-30 Risheng Liu , Long Ma , Xiaoming Yuan , Shangzhi Zeng , Jin Zhang

Bilevel programming has recently received attention in the literature due to its wide range of applications, including reinforcement learning and hyper-parameter optimization. However, it is widely assumed that the underlying bilevel…

Machine Learning · Computer Science 2024-10-11 Parvin Nazari , Ahmad Mousavi , Davoud Ataee Tarzanagh , George Michailidis

Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with…

Optimization and Control · Mathematics 2017-12-07 Ganzhao Yuan , Bernard Ghanem

In this letter, we consider a bilevel optimization problem in which the outer-level objective function is strongly convex, whereas the inner-level problem consists of a finite sum of convex functions. Bilevel optimization problems arise in…

Optimization and Control · Mathematics 2026-01-22 Sudkobfa Boontawee , Mootta Prangprakhon , Nimit Nimana

Bilevel optimization recently has attracted increased interest in machine learning due to its many applications such as hyper-parameter optimization and meta learning. Although many bilevel methods recently have been proposed, these methods…

Optimization and Control · Mathematics 2023-02-21 Feihu Huang , Junyi Li , Shangqian Gao

Evolutionary and bioinspired computation are crucial for efficiently addressing complex optimization problems across diverse application domains. By mimicking processes observed in nature, like evolution itself, these algorithms offer…

Neural and Evolutionary Computing · Computer Science 2025-01-14 Daniel Molina , Javier Del Ser , Javier Poyatos , Francisco Herrera

In many scenarios, one uses a large training set to train a model with the goal of performing well on a smaller testing set with a different distribution. Learning a weight for each data point of the training set is an appealing solution,…

Machine Learning · Statistics 2023-10-27 Anastasia Ivanova , Pierre Ablin

Bilevel optimization has been developed for many machine learning tasks with large-scale and high-dimensional data. This paper considers a constrained bilevel optimization problem, where the lower-level optimization problem is convex with…

Machine Learning · Computer Science 2023-08-22 Siyuan Xu , Minghui Zhu

Bilevel optimization is a key framework in hierarchical decision-making, where one problem is embedded within the constraints of another. In this work, we propose a control-theoretic approach to solving bilevel optimization problems. Our…

Optimization and Control · Mathematics 2025-03-25 Sina Sharifi , Nazanin Abolfazli , Erfan Yazdandoost Hamedani , Mahyar Fazlyab

Bilevel optimization and bilevel minimax optimization have recently emerged as unifying frameworks for a range of machine-learning tasks, including hyperparameter optimization and reinforcement learning. The existing literature focuses on…

Machine Learning · Computer Science 2026-04-23 Xuelin Zhang , Peipei Yuan

In this work we consider multitasking in the context of solving multiple optimization problems simultaneously by conducting a single search process. The principal goal when dealing with this scenario is to dynamically exploit the existing…

Neural and Evolutionary Computing · Computer Science 2021-08-20 Eneko Osaba , Aritz D. Martinez , Javier Del Ser