Variational inequality is a concept in mathematical analysis and optimization that involves finding a function or point that satisfies certain conditions related to inequality constraints. It is particularly relevant in the study of equilibrium problems, optimization problems, and differential inclusions.
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For a feasible set and a vector field , a variational inequality asks for satisfying the displayed inequality for every feasible . If is convex and is the gradient of a differentiable convex function, it is the necessary and sufficient first-order condition for minimizing that function. It also describes Wardrop equilibria when no scalar objective has the route-cost vector as its gradient.