Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods
Abstract
In this work we study a special minimax problem where there are linear constraints that couple both the minimization and maximization decision variables. The problem is a generalization of the traditional saddle point problem (which does not have the coupling constraint), and it finds applications in wireless communication, game theory, transportation, just to name a few. We show that the considered problem is challenging, in the sense that it violates the classical max-min inequality, and that it is NP-hard even under very strong assumptions (e.g., when the objective is strongly convex-strongly concave). We then develop a duality theory for it, and analyze conditions under which the duality gap becomes zero. Finally, we study a class of stationary solutions defined based on the dual problem, and evaluate their practical performance in an application on adversarial attacks on network flow problems.
Cite
@article{arxiv.2110.11210,
title = {Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods},
author = {Ioannis Tsaknakis and Mingyi Hong and Shuzhong Zhang},
journal= {arXiv preprint arXiv:2110.11210},
year = {2022}
}