This paper considers a bilevel program. To solve this bilevel program, it is generally necessary to transform it into some single-level optimization problem. One approach is to replace the lower-level program by its KKT conditions to transform the bilevel program as a mathematical program with complementarity constraints (MPCC). Another approach is to apply the lower-level Wolfe/Mond-Weir/extended Mond-Weir duality to transform the bilevel program into some duality-based single-level reformulations, called WDP, MDP, and eMDP respectively in the literature. In this paper, inspired by a conjecture from a recent publication that the tighter feasible region of a reformulation, the better its numerical performance, we present three new duality-based single-level reformulations, called TWDP/TMDP/eTMDP, with tighter feasible regions. Our main goal is to compare all above-mentioned reformulations by designing some direct and relaxation algorithms with projection and implementing these algorithms on 450 test examples generated randomly. Our numerical experiments show that, whether overall comparison or pairwise comparison, at least in our tests, the WDP/MDP/TWDP/TMDP reformulations were always better than the MPCC reformulation, while the eMDP/eTMDP reformulations were always the worst ones among six duality-based reformulations, which indicates that the above conjecture is incorrect. In particular, for the relaxation algorithms, the WDP/MDP/TWDP/TMDP reformulations performed 3-5 times better than the MPCC reformulation, while the eMDP/eTMDP reformulations performed 2 times better than the MPCC reformulation.
Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation
Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳应用数学国家研究中心）；City University of Hong Kong（香港城市大学）
A geometric reformulation of the bilevel parameter optimization problem to a single level non-linear programming problem with applications to phase equilibria
University of Bremen（不来梅大学）；Leibniz Institute for Materials Engineering - IWT（莱布尼兹材料工程研究所）
Directional first order approach for a class of bilevel programs
School of Mathematics, Hunan University（湖南大学数学学院）；School of Artificial Intelligence, Southern University of Science and Technology（南方科技大学人工智能学院）；Department of Mathematics and Statistics, University of Victoria（维多利亚大学数学与统计系）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
Moreau Envelope Based Difference-of-weakly-Convex Reformulation and Algorithm for Bilevel Programs
University of British Columbia（不列颠哥伦比亚大学）；University of Victoria（维多利亚大学）；Southern University of Science and Technology（南方科技大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）；Peng Cheng Laboratory（鹏城实验室）
Exact-Penalty Prox-Linear Methods for Bilevel Optimization with $\ell_1$ Lower-Level Gradient Penalty
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology（北京理工大学计算机辅助设计与图形学北京市重点实验室/信息安全数学理论计算北京市重点实验室）
Microsoft Research Cambridge（微软研究院剑桥）；Italian Institute of Technology（意大利理工学院）；AI Centre, Department of Computer Science, UCL（伦敦大学学院计算机科学系人工智能中心）；DIAG, Sapienza University of Rome（罗马智慧大学 DIAG 系）；School of Mathematical Sciences, University of Southampton（南安普顿大学数学科学学院）
Reliability of Single-Level Equality-Constrained Inverse Optimal Control
School of Electrical Engineering, University of Belgrade（贝尔格莱德大学电气工程学院）；Laboratory for Analysis and Architecture of Systems（系统分析与架构实验室）；Image and Pervasive Access Laboratory (IPAL)（图像与普及访问实验室）
Bilevel Optimization for Feature Selection in the Data-Driven Newsvendor Problem
Technical University of Munich（慕尼黑工业大学）；Munich Data Science Institute（慕尼黑数据科学研究所）；Polytechnique Montréal（蒙特利尔理工大学）；Pontifical Catholic University of Rio de Janeiro（里约热内卢天主教大学）
Joint Design of Time-of-Use Schedules and Price Levels: a Bilevel Model with Client Participation and Load Flexibility
École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）；EDF Lab Paris-Saclay（法国电力集团萨克雷实验室）；Institut National de Recherche en Informatique et en Automatique (Inria)（法国国家信息与自动化研究所）
