 Variational inequality

For a feasible set $K$ and a vector field $F$, a variational inequality asks for $x^{\ast }\in K$ satisfying the displayed inequality for every feasible $x$. If $K$ is convex and $F$ 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.

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.