Variational Inequalities in Nonconvex Optimization: Challenges and Solutions

Abstract

Variational inequalities are powerful mathematical tools extensively employed in optimization, equilibrium modeling, and economics. While they have proven highly effective in convex settings, dealing with nonconvex optimization problems presents unique challenges. This article explores the application of variational inequalities in the context of nonconvex optimization, discussing the mathematical foundations, challenges, and emerging solutions. We delve into techniques such as global optimization algorithms, nonconvex relaxation methods, and the use of machine learning approaches to tackle the complexities of nonconvex variational inequalities. By addressing these challenges, researchers aim to extend the reach of variational inequalities into broader optimization landscapes.