ABSTRACT
We have shown (1) that vector fields were generalizations of gradients (definition 5.7.2), and (2) that optimization of a function f on a convex set K was related to a condition on the gradient of f and the normal cones NK(X) in points X of K (theorem 5.5.1): a solution X∗ is such that:
−∇f(X∗) ∈ NK(X∗) It therefore follows that the definition of a variational inequality is:
Definition 6.1.1 Let K ⊂ Rn a convex polyhedron and F : K → Rn a vector field. The variational inequality problem VI(F,K) consists in finding X∗ ∈ K such that −F (X∗) ∈ NK(X∗).
