Optimization Problems with Nonlinear Variational Inequalities Constraints

Abstract

Optimization problems with nonlinear variational inequalities (NVI) constraints represent a challenging class of mathematical problems with wide-ranging applications in engineering, economics, and the physical sciences. In these problems, the objective is to optimize a function subject to constraints defined by NVIs, which encompass both nonlinearity and inequality. This article provides an overview of these optimization problems, discussing their mathematical formulation, solution techniques, and practical applications. We explore numerical methods, such as the augmented Lagrangian method and the alternating direction method of multipliers, which are crucial for tackling NVI-constrained optimization problems. Additionally, we highlight applications in various domains, showcasing the significance of this field in addressing complex real-world challenges.