Extended Semismooth Newton Method for Functions with Values in a Cone

This paper deals with variational inclusions of the form 0 ∈ K-f(x) where f : ℝ^n→ℝ ^m is a semismooth function and K is a nonempty closed convex cone in ℝ^m . We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of f . The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341–347, 1972 ). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence.