Variational inequality ($F(x^{\ast })\cdot (x-x^{\ast })\ge 0(x\in K)$)

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.