A variational inequality is a mathematical formulation that seeks to find a vector within a convex set such that a given function evaluated at that vector satisfies a specific inequality involving a linear functional. This concept connects to various mathematical problems, including complementarity problems, which often arise in optimization and equilibrium models, as well as the study of monotone operators, which play a crucial role in understanding the properties of these inequalities. Variational inequalities also find applications in areas like machine learning and data science, where they are used to model optimization challenges and constraints.
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